Characterization of Laser Voltage Probing

Cheng Yi Chiao

SCHOOL OF ELECTRICAL AND ELECTRONIC ENGINEERING

A DISSERTATION SUBMITTED IN PARTIAL FULFILMENT OF

THE REQUIREMENTS FOR THE DEGREE OF

MASTER OF SCIENCE IN MICROELECTRONICS

2012

Abstract

The effects that laser power on 32nm SOI for a repeater circuitry are explored. Other parameters such as voltage and frequency are also tested to determine the variations that may arise from adjusting such variables. The noise is also characterized for the same process using the same parameters. The noise is analysed using the Wigner Ville Distribution to determine frequency anomalies in time that would normally be missing using the conventional time or frequency domain. Laser power and voltage have a positive correlation with the final signal strength while frequency does not seem to play a major role. Whilst for noise analysis, low frequency is very pronounced in the noise signal. At the same time there are signs of peaks and troughs in the PSD analysis of the noise. These are believed to be contributed by best case and worst case scenario of impedance matching of the circuitry.

Acknowledgements

Many people contributed to this final end result. Of course, most importantly, my supervisors Professor Tan Cher Ming and Mr. Venkat Krishnan Ravikumar. I am also grateful to my previous supervisors for their support in this program and the help they have extended to me, Professor Siek Liter and Professor Chang Chip Hong. At the same time, Ms. Vivienne Ho from GIST has been of great help and support during this grueling endeavor. Lastly of course, I would like to thank my friends at AMD and during my coursework as well as my girlfriend for her understanding.

Table of Contents Abstract ..................................................................................................................................................... 2

Acknowledgements ................................................................................................................................... 3

Table of figures ......................................................................................................................................... 5

I Introduction ............................................................................................................................................ 8

1. Motivation ..................................................................................................................................... 8

2. Objective ....................................................................................................................................... 8

II Literature Review ................................................................................................................................... 9

1. Introduction .................................................................................................................................. 9

2. Sample preparation/ Chemical Mechanical Polishing (CMP)[1-4] ................................................ 9

3. Flip chip[8-13] ............................................................................................................................. 12

4. Franz Keldysh effect [14] ............................................................................................................. 14

5. Laser voltage probe (LVP)[16-18] ............................................................................................... 15

6. Device under test (DUT) .............................................................................................................. 16

C. Repeater circuitry........................................................................................................................ 23

8. Noise ........................................................................................................................................... 27

9. Time Frequency Analysis ............................................................................................................. 29

8. Data collection ............................................................................................................................ 53

III Noise Characterization ........................................................................................................................ 78

1. Overview ..................................................................................................................................... 78

2. Frequency.................................................................................................................................... 80

3. Laser Power ................................................................................................................................. 82

4. Voltage ........................................................................................................................................ 85

5. Focus ........................................................................................................................................... 88

6. Limitations and Problems encountered ...................................................................................... 90

7. Conclusion ................................................................................................................................... 90

IV Conclusion and future work................................................................................................................ 91

Conclusion ........................................................................................................................................... 91

Future works ....................................................................................................................................... 91

V References ........................................................................................................................................... 93

Table of figures

Figure 1 Multiprep for CMP ........................................................................................................................ 10 Figure 2 CMP ............................................................................................................................................... 10 Figure 3 Llano FS1 chip ................................................................................................................................ 11 Figure 4 Flip Chip pad .................................................................................................................................. 12 Figure 5 Flip Chip bumps ............................................................................................................................ 12 Figure 6 Flip Chip flipped ............................................................................................................................ 12 Figure 7 Mounting of flip chip ..................................................................................................................... 12 Figure 8 Alignment and mounting of flip chip ............................................................................................ 12 Figure 9 Melting the solder through hot air reflow .................................................................................... 12 Figure 10 Mount underfill ........................................................................................................................... 13 Figure 11 Flip chip final ............................................................................................................................... 13 Figure 12 U.S. Patent 5888297 ................................................................................................................... 17 Figure 13 SIMOX .......................................................................................................................................... 17 Figure 14 Wafer bonding ............................................................................................................................ 17 Figure 15 US Patent 5882987 ..................................................................................................................... 18 Figure 16 Smart cut process flow ................................................................................................................ 18 Figure 17 Conventional Silicon .................................................................................................................... 19 Figure 18 Partially Depleted SOI ................................................................................................................. 19 Figure 19 Fully Depleted SOI ....................................................................................................................... 19 Figure 20 Kink effect ................................................................................................................................... 21 Figure 21 Floating body effect .................................................................................................................... 21 Figure 22 Conventional Process versus SOI ................................................................................................ 22 Figure 23 Repeater insertion ...................................................................................................................... 23 Figure 24 Llano FS1 Die Area ......................................................................... Error! Bookmark not defined. Figure 25 DCG Emiscope ............................................................................................................................. 26 Figure 26 Flicker Noise ................................................................................................................................ 27

Figure 27 Time-frequency distributions of four different signals with similar frequency components ..... 30 Figure 28 Short-time Fourier Transform ..................................................................................................... 37 Figure 29 Wavelet Transform ..................................................................................................................... 37 Figure 30 Basic Wavelet function Mexican hat function ............................................................................ 38 Figure 31 Morlet function ........................................................................................................................... 38 Figure 32 Compared time-frequency resolution of spectrograms and scalograms. (a) Spectrogram with window neither short nor long; (b) Spectrogram with long window; (c) Spectrogram with short window; (d) Scalograms. [29] .................................................................................................................................... 39 Figure 33 Crossterm interference of WVD; (a) WVD of the sum of two Gaussian functions [60]. (b) the crossterm in the middle of the two sinusoidal function [29] ..................................................................... 44 Figure 34 The worst-case signal component for the fastest oscillating frequency [61] ............................. 45 Figure 35 Real signal (a) and analytical signal (b) spectrum. <ω> is the average frequency ...................... 47 Figure 36 The reduction of Wigner-Ville distribution’s anti-lasing by applying the analytical signal[62]. Signal on the left is real whilst signal on the right is analytical .................................................................. 47 Figure 37 (a) t-f plane of 2 signal components, (b) Corelative-domain analysis of their anti-lasing [63] .. 49 Figure 38 Effective support of ambiguity kernel: (a) WVD (no smoothing), (b) PWD, (c) SPWD ............... 51 Figure 39 10X magnification ....................................................................................................................... 53 Figure 40 SIL Magnification ......................................................................................................................... 53 Figure 41 Digital 7X after SIL ....................................................................................................................... 53 Figure 42 Identification using LVI ................................................................................................................ 53 Figure 43 Generic example of moving average .......................................................................................... 54 Figure 44 Generic waveform ....................................................................................................................... 56 Figure 45 Demonstration of SNR ................................................................................................................ 56 Figure 46 Definitions for rise/ fall time ....................................................................................................... 57 Figure 47 NMOS2 1V Laser Power 269 10Mhz ........................................................................................... 58 Figure 48 NMOS4 1V Laser Power 269 10Mhz ........................................................................................... 58 Figure 49 PMOS1 1V Laser Power 269 10Mhz ............................................................................................ 59 Figure 50 PMOS2 1V Laser Power 269 10Mhz ............................................................................................ 59 Figure 51 PMOS4 1V Laser Power 269 10Mhz ............................................................................................ 60 Figure 52 Laser power at 240 ...................................................................................................................... 61 Figure 53 Laser power at 260 ...................................................................................................................... 61 Figure 54 Laser power at 269 ...................................................................................................................... 61 Figure 55 Laser power at 279 ...................................................................................................................... 61 Figure 56 Chip 1 NMOS4 1V Laser Power 240 10Mhz ................................................................................ 62 Figure 57 Chip 1 NMOS4 1V Laser Power 269 10Mhz ................................................................................ 63 Figure 58 Chip 1 NMOS4 1V Laser Power 279 10Mhz ................................................................................ 63 Figure 59 Chip 2 NMOS4 1V Laser Power 240 10Mhz ................................................................................ 64 Figure 60 Chip 2 NMOS4 1V Laser Power 269 10Mhz ................................................................................ 64 Figure 61 Chip 2 NMOS4 1V Laser Power 279 10Mhz ................................................................................ 65 Figure 62 Chip 1 PMOS4 1V Laser Power 240 10Mhz ................................................................................. 66 Figure 63 Chip 1 PMOS4 1V Laser Power 269 10Mhz ................................................................................. 66 Figure 64 Chip 1 PMOS4 1V Laser Power 279 10Mhz ................................................................................. 67

Figure 65 Chip 2 PMOS4 1V Laser Power 240 10Mhz ................................................................................. 67 Figure 66 Chip 2 PMOS4 1V Laser Power 269 10Mhz ................................................................................. 68 Figure 67 Chip 2 PMOS4 1V Laser Power 279 10Mhz ................................................................................. 68 Figure 68 Voltage at 0.9V ............................................................................................................................ 69 Figure 69 Voltage at 1.0V ............................................................................................................................ 69 Figure 70 Voltage at 1.1V ............................................................................................................................ 69 Figure 71 Voltage at 1.2V ............................................................................................................................ 69 Figure 72 Chip 1 NMOS4 0.9V Laser Power 269 10Mhz ............................................................................. 70 Figure 73 Chip 1 NMOS4 1V Laser Power 269 10Mhz ................................................................................ 71 Figure 74 Chip 1 NMOS4 1.1V Laser Power 269 10Mhz ............................................................................. 71 Figure 75 Chip 1 NMOS4 1.2V Laser Power 269 10Mhz ............................................................................. 72 Figure 76 Chip 2 NMOS4 0.9V Laser Power 269 10Mhz ............................................................................. 72 Figure 77 Chip 2 NMOS4 1V Laser Power 269 10Mhz ................................................................................ 73 Figure 78 Chip 2 NMOS4 1.1V Laser Power 269 10Mhz ............................................................................. 73 Figure 79 Chip 2 NMOS4 1.2V Laser Power 269 10Mhz ............................................................................. 74 Figure 80 Chip 2 NMOS4 1.2V Laser Power 279 10Mhz ............................................................................. 74 Figure 81 Chip 2 NMOS4 1.2V Laser Power 269 5Mhz ............................................................................... 76 Figure 82 Figure 54 Chip 2 NMOS4 1.2V Laser Power 269 10Mhz ............................................................. 76 Figure 83 Figure 54 Chip 2 NMOS4 1.2V Laser Power 269 25Mhz ............................................................. 77 Figure 84 Repeater circuitry at LVI and location of probe .......................................................................... 78 Figure 85 Time domain and frequency domain of signals varying in frequency ........................................ 81 Figure 86 PSD at 1Mhz ................................................................................................................................ 81 Figure 87 PSD at 10Mhz .............................................................................................................................. 81 Figure 88 PSD at 50Mhz .............................................................................................................................. 81 Figure 89 PSD at 200Mhz ............................................................................................................................ 81 Figure 90 Time domain and frequency domain of signals varying in laser power ..................................... 83 Figure 91 PSD at laser power 240 ............................................................................................................... 83 Figure 92 PSD at laser power 269 ............................................................................................................... 83 Figure 93 PSD at laser power 279 ............................................................................................................... 83 Figure 94 Time domain and frequency domain of signals varying in voltage ............................................ 85 Figure 95 PSD at voltage 0p5 ...................................................................................................................... 86 Figure 96 PSD at voltage 1p0 ...................................................................................................................... 86 Figure 97 PSD at voltage 1p2 ...................................................................................................................... 86 Figure 98 Time domain and frequency domain of signals varying in focus ................................................ 88 Figure 99 PSD In Focus ................................................................................................................................ 89 Figure 100 PSD Out of Focus ....................................................................................................................... 89

I Introduction

1. Motivation Following Moore’s law, the semiconductor industry has managed to shrink feature size of devices by half every 18 months. The scaling down of the devices creates an ever more challenging task to conduct tests on the devices. The motivation for the project was to fill the gaps concerning the laser voltage probe test equipment. Limited research has been conducted with concerns to the effects of laser power targeted on a piece of silicon. The literature on the laser voltage probe does not detail the effects of laser on the device under test. At the same time, there is little documentation on the disparity in noise due to different conditions, namely a change in laser power, voltage, frequency and focus. This project aims to fill in those voids. A better understanding of laser power and its effects on silicon would allow users to select an optimum value when performing probes. At the same time, the understanding of the noise effects would further fine tune the optimum operating point.

2. Objective The objective of the project is twofold:

To characterize the laser power and the effects it has on voltage and timing of the waveform

To characterize the noise from changes in the setup due to frequency, voltage, laser power and focusing

II Literature Review

1. Introduction The literature review will cover the processes, test equipment as well as the underlying physics that is at work to yield results from the test equipment. The historical reason for using the laser voltage probe is discussed, namely the change of packaging technology to flipchip. At the same time, the noise effects were analysed using time frequency analysis analysis as opposed to the conventional time or frequency stand alone analysis. The literature review covers the benefits and disadvantages of time frequency analysis.

2. Sample preparation/ Chemical Mechanical Polishing (CMP)[1-4]

CMP is a process of removing bumps from the die surface using chemical and mechanical methods. The die is planarized by undergoing mechanical grinding. This process however induces a high surface damage. Standalone chemical methods cannot achieve planarization. The operating physics involved behind CMP is to induce higher pressure where the die is thicker this results in the removal of the thicker area first. The polishing head not only oscillates but rotates about an axis to bring about smoother polishing. The axis of rotation of the polishing head is rotated in tandem with the plate to bring about maximum effect. The silicon is removed micron by micron and the irregular topography is slowly evened out. Lapping paper is added to the polishing plate. The lapping paper is akin to sandpaper, however with very small grains ranging from 3µm to 30µm in size. The lapping paper is changed from small grain size to large grain size and back to a small grain size again to ensure a fine and even grinding. The slurry is normally water at the beginning stages and then finally a 0.2µm silica colloid for the final mirror polish.

Figure 1 Multiprep for CMP[5]

Figure 2 CMP[6]

The chips are being probed from the backside due to the flipchip packaging. The samples need to be thinned down to the order of 100 microns in order for the solid immersion lens (SIL) to have optimum contact and allow optimum laser to pass through. The silicon normally starts at 800 microns thickness after the chip has been de lidded and the metallic cover has been removed. The chip is processed and attached to a jig which is then attached to the Multiprep machine and thinned down through chemical mechanical polishing.

Figure 3 Llano FS1 chip[7]

There are certain difficulties when performing the CMP

• Over polishing • Under polishing • Uneven polishing • Cracking during polishing

For over polished samples and cracked samples, the chips ets are immediately rendered useless. As for under polished samples, the image quality deteriorates. A thicker sample tends to exhibit more fringe effects when undergoing imaging and causes features to be hidden within the fringes. This is especially pronounced at lower magnifications. At the same time, the thicker sample requires much more laser power and a higher pressure from the SIL. Despite the compensation, the image quality will still be vastly inferior to be a properly polished sample. Uneven samples on the other hand causes a need to refocus whilst doing imaging and may produce sporadic bad images with heavy fringing at critical areas of interest. Unevenness in the sample cannot be avoided and can range between 10 to 20 microns. This effect is known as warpage and the highest point is normally the center of the chip if attached to the jig properly.

3. Flipchip Technology [8-13]

Figure 4 Flip Chip pad[10]

Figure 5 Flip Chip bumps[10]

Figure 6 Flip Chip flipped[10]

Figure 7 Mounting of flip chip[10]

Figure 8 Alignment and mounting of flip chip[10]

Figure 9 Melting the solder through hot air reflow[10]

Figure 10 Mount underfill[10]

Figure 11 Flip chip final[10]

The flip chip is also known as C4 (controlled collapse chip connection). The silicon die is connected to external PCB through solder bumps. During wafer fabrication the solder bumps are deposited on the topside of the chip pads. The chip is then flipped over and aligned to the correct connections on the PCB external circuitry. The solder is then melted to join the silicon with the external circuitry.

The process steps in creating the flip chip are:[10] 1. The integrated circuits is created on the wafer 2. Pads are metalized on the surface of the chips 3. Solder dots are deposited on each pad individually 4. The chips are cut into the correct die size 5. The chips are flipped and aligned to ensure the solder balls are connected to the correct

external connectors on the external circuitry 6. The solder is melted through hot air reflow 7. The mounted chip is filled at the bottom with an electrically insulating adhesive known as

underfill

The advent of the flipchip packaging technology helped to improve the speed and performance of integrated circuits (IC) however; it posed a problem when it came to the analysis of the chip. This is discussed later in the section on Laser Voltage Probe.

4. Franz Keldysh effect [14] An electroabsorption modulator is a semiconductor device which can be used for modulating the intensity of a laser beam via an electric voltage.[15] The physical principle behind it is the Franz–Keldysh effect i.e. This effect is a change in the absorption spectrum caused by an applied electric field, which changes the bandgap energy (thus the photon energy of an absorption edge) but usually does not involve the excitation of carriers by the electric field.

The Franz Keldysh effect occurs when there is an uniform electric field and a crystal is placed in the field thereby changing the absorption coefficient for the crystal. The uniform electric field causes a chande in the structure and electron states in the crystal. The absorption threshold shifted to a lower energy state. The absorption curve on the low energy side of the zero field thresholds will demonstrate an exponential tail. The Bloch functions help to model the conditions for the electrons

𝜑𝑗 ( 𝑝0, 𝑟, 𝑡) = exp { - 1𝜆 ∫ 𝜀𝑗 ( 𝑝0 − 𝑒 𝐸 𝑥)𝑑𝑥} 𝜑𝑗 ( 𝑝0 − 𝑒 𝐸 𝑡, 𝑟) 10

The function satisfies the time dependent Schrodinger equation for the model.

When the electric field is absent, the integrand varies harmonically with time. This demonstrates that the energy remains constant.

Electroabsorption modulators are constructed using an electric field that is orthogonal to the modulated light beam. The field is applied to form the wave. In order to attain a high extinction ratio, the quantum-confined Stark effect in a quantum well structure is employed.

Advantages of the electroabsorption modulators versus electro-optic modulators are that electroabsorption modulators can operate at much lower. They can be operated at very high speeds. The modulation bandwidth can reach tens of gigahertz, which makes these devices useful for optical fiber communications. A convenient feature is that an electroabsorption modulator can be constructed into photonic integrated circuit. The modulator is integrated with a distributed feedback laser diode on a single die to form a data transmitter. This is crucial in the experiment in order to collect data points. Compared with direct modulation of the laser diode, a higher bandwidth and reduced chirp can be obtained.

5. Laser voltage probe (LVP)[16-18] The laser voltage probe (LVP) is mainly used for waveform analysis. The tool was designed for debugging of flipchip packaged ICs. As discussed in the previous section, with the advent of flichip technology, only the backside of the chips became accessible for probing. LVP is a a viable technique to probe diffusions in flipchip ICs. Traditional techniques such as mechanical probing are more expensive and become more inefficient with decreasing feature size. The main disadvantages include:

• limited access to sites due to area of interest versus probe size • Probe impedance • Probe bandwidth • Probe size • Probe speed with concerns to multiple locations

Electron beam probing is also heavily relied on in the testing industry. This tool solves some of the problems with mechanical probing; however it presents its own limitations. There is limited bandwidth, the inability to reach metal layers and destructive milling is needed to reach the silicon. This destroys the functionality of the chip and renders it useless.

The waveform analysis includes the following:

• Recording the voltage amplitude at specific locations of the device • The switching characteristics at that location • Compare the parameters between different locations

The main physical principal behind the LVP is the Franz Keldysh effect. Based on the effect, the effective bandgap of the semiconductor will change when an electric field is applied. This is the result of the mixing of the conduction and valence bands. The refractive index of silicon changes accordingly. The absorption of light will alter due to this change in the refractive index. The intensity and phase of the reflected beam are modulated and these measurements could be taken to track the correlated changes in the electric field. The free carrier absorption also plays a role in the use of the technique. The 1.34um laser is the main wavelength of the laser. Phase modulation tends to dominate over the absorptive component. The free carrier component mostly contributes to the amplitude modulation. This is mostly from the laser power of the LVP system.

The amplitude of the laser beam is attenuated when passing through the silicon, therefore for best results the silicon thickness is normally between 50um to 100um. The polishing process was described earlier in the CMP section. The minimum basic system consists of a diffraction limited confocal laser scanning microscope (LSM) and a diode pumped Nd laser. There are normally two lasers provided the 1340nm laser as well as a 1064nm laser. The 1340nm laser is normally utilized because it is less destructive to the silicon. During signal acquisition, the supplied and received intensities of the two lasers are measured. Vibrations and its effects are reduced by using information from the reflected low power laser intensity. A separate circuitry helps to synchronize the entire sequence of the waveform

measurement. A phase-interference-detector can be added to the LSM to further minimize vibrations. The addition of the solid immersion lens (SIL) improves magnification and resolution of the image. The SIL allowed a better physical magnification and a greater resolution to the area of interest.

6. Device under test (DUT) A. Silicon on insulator process

Overview

Silicon on Insulator (SOI)[19, 20] is essentially the sandwiching of an insulating material between two pieces of silicon as compared to conventional bulk silicon. Parasitic device capacitance is greatly reduced and the overall performance of the chip is greatly improved. Normally silicon dioxide or sapphire is chosen as the insulation. SOI technology simplifies manufacturing process by eliminating the wells and field implantation steps. Smaller features can be fabricated and cross talk reduced.

B. Methods of manufacture

Separation by Implantation of Oxygen (SIMOX)[19, 21, 22]

Simox is a two stage process. It begins with an implantation with an oxygen beam. The silicon dioxide layer is then created through high temperature annealing.

Figure 12 Simox process [21]

Figure 13 SIMOX[20]

Wafer bonding [23]

The basics of wafer bonding are illustrated in the figure below.

Figure 14 Wafer bonding[20]

There are a few different technologies available for the wafer bonding process. One of them is the Smart Cut method from Soitec. [24]

Figure 15 Process steps for Smart Cut technology[24]

Figure 16 Smart cut process flow[24]

Fully Depleted and Partially Depleted SOI

Figure 17 Conventional Silicon[20]

Figure 18 Partially Depleted SOI[20]

Figure 19 Fully Depleted SOI[20]

Fully Depleted SOI (FD SOI)[20]

Fully depleted mode occurs when the channel depletion region extends through the entire thickness of the silicon layer.

Partially Depleted SOI (PD SOI)[20]

Partially depleted transistors are built with thicker silicon layers as compared to the FD SOI. The fully powered MOS channel will have a depletion depth less than the thickness of the silicon.

Table 1 illustrates the different properties and attributes between FD SOI and PD SOI.

Table 1 Differences between FD SOI and PD SOI[20] FD SOI[20] PD SOI[20] FD SOI devices do not suffer from the kink effect. This arises because the majority carriers can penetrate the source more easily. This prevents the excess carriers from accumulating

PD SOI tend to suffer from the kink effect which is described later

FD SOI MOSFETs will have a reduced body effect and a nearly ideal 𝑔𝑚 𝐼𝑑� ratio when biased in the weak region. There is still a weak current voltage kink still exists in the strong inversion region

Once again, the PD SOI suffers from the kink effect

They are by the interface coupling effect. The interface coupling is unavoidable for FD SOI. All parameters: threshold voltage, transconductance, interface-trap response etc. of one channel are affected by the opposite gate voltage

PD SOI is not affected by the interface coupling effect

FD SOI has a better subthreshold swing, S. FD SOI has 1/S = 65 to 70 mV/decade. The ideal characteristic of a MOS transistor at room temperature is 1/S = 60 mV/decade

For the bulk and PD devices, 1/S = 85 to 90 mV/decade

Fully-depleted SOI devices perform better in general. The circuit characteristics show higher gains in circuit speed, reduced power requirements and high levels of soft-error immunity. The sharper subthreshold gradient allows the FD devices operate faster. The reduced threshold voltage allows for faster switching of the MOS transistors. These transistors also have increased drive currents at relatively low voltages

PD SOI have poorer performance in general

Threshold voltage is directly related to SOI thickness. Because the FD SOI is relatively thin, a slight variation would be huge in percentage change of the thickness thereby affecting the threshold voltage. This is one of the most serious problems in FD SOI MOSFETs

PD SOI devices have a thick silicon layer and is less susceptible to manufacturing drift

Porting of FD SOI is not as straight forward and can incur higher costs

Porting of PD SOI designs from bulk Silicon is relatively easy

Electrical anomalies

Overview

Both types of SOI suffer from certain electrical anomalies such as the kink effect, floating body effect and the self heating effect. However some of the effects are more pronounced in PD SOI as compared to FD SOI.

Kink Effect[20]

There is a sudden discontinuity in the drain current. This is most pronounced when device is biased in the saturation region as seen in figure 20.

Figure 20 Kink effect[20]

Floating body effect[20]

The Floating body effect is a common occurrence in the partially depleted devices. A parasitic bipolar structure exists in parallel with the MOS structure. The base of the BJT is not connected and floating, hence the name floating body effect. This is demonstrated in figure 21.

Partially depleted SOI MOSFETs for the floating body devices exhibit a larger drain current as compared to a tied body device. For VDS that is very much lower in magnitude than the kink voltage, the effect arises because of the storage of holes thermally generated at the drain-body junction.

Figure 21 Floating body effect[20]

Self heating effect[20]

Thermal insulation is provided by the oxide surface. The material is not a good conductor and heat dissipation is not efficient. The toggling of logic states is when the effect is most pronounced.

Advantage and disadvantages of SOI over conventional bulk silicon[20]

Implementing SOI allows for the feature sizes of transistors to continue to shrink and hence allow Moore’s Law to survive. The main advantages of SOI are:

• Lower parasitic capacitance due to the isolation form bulk silicon • Better of power consumption because of smaller feature size and lower parasitic • Higher device density • Easier to isolate the device • Reduced and eliminate latchup by the complete isolation of p and n wells

Figure 22 Conventional Process versus SOI[20]

Manufacturing of SOI does not require a change in the manufacturing tools. The machines used are the same and only requires a change in process steps and recipe. This makes it more adaptable for foundries when considering implementation as it does not require additional sunk in costs.

There are however several drawbacks:

• Despite no extra sunk in costs, the manufacturing costs is 15% higher because of the added complexity and additional steps needed to realize the final wafer

• There will be floating body effect • The electrical properties differ from bulk silicon and may require additional design steps to port

the technology fully

C. Repeater circuitry Repeater circuitry’s main function is to reduce propagation delay of long wires. They serve as intermediate buffers between interconnects. An m times reduction in interconnect line length will reduce propagation delay by a quadratic function. This is offsets the extra delay caused by the repeaters when the wire is extremely long. [25-29]

Figure 23 Repeater insertion[25]

Figure 24 Repeater circtuitry layout

Figure 25 Repeater Circuitry in LSM

The optimum number of repeaters can be derived:

𝑚𝑜𝑝𝑡 = 𝐿�0.38𝑟𝑐𝑡𝑝𝑏𝑢𝑓

= �𝑡𝑝𝑤𝑖𝑟𝑒(𝑢𝑛𝑏𝑢𝑓𝑓𝑒𝑟𝑒𝑑)

𝑡𝑝𝑏𝑢𝑓

Where, 𝑚𝑜𝑝𝑡 is the optimum number of repeaters

𝑡𝑝𝑏𝑢𝑓 is the fixed delay

The minimum delay for the wire is

𝑡𝑝,𝑜𝑝𝑡 = 2�𝑡𝑝𝑤𝑖𝑟𝑒(𝑏𝑢𝑓𝑓𝑒𝑟𝑒𝑑)𝑡𝑝𝑏𝑢𝑓

The optimum is obtained when the delay of the individual wire segments are made equal to that of a repeater.

Short circuit power consumption[25]

Short-circuit current flows when both transistors of the inverting repeater are simultaneously on. The thickness of the line thickness determines overall dynamic power dissipated. It also determines the short circuit power dissipation. The relation of the line thickness with short circuit power is direct proportionality. In the case of dynamic power it is inversely proportional. Thin resistive lines allow for a large number of repeaters. Short-circuit power also depends on both the input signal transition time and the load characteristics. The short-circuit power dissipation of a repeater driving an RC load is

𝑃𝑠𝑐−𝑠𝑒𝑐𝑡𝑖𝑜𝑛 = 12𝐼𝑝𝑒𝑎𝑘𝑡𝑏𝑎𝑠𝑒𝑣𝑑𝑑𝑓

Where, 𝐼𝑝𝑒𝑎𝑘 is peak current that flows from Vdd to ground

𝑡𝑏𝑎𝑠𝑒 is the time period where both transitions are on

Vdd is supply voltage

f is the switching frequency

Total short circuit power is

𝑃𝑠𝑐−𝑡𝑜𝑡𝑎𝑙 = 𝑚𝑜𝑝𝑡𝑃𝑠𝑐−𝑡𝑜𝑡𝑎𝑙

Dynamic Power dissipation[25]

Toggling of the device as well as the power consumption of the interconnect capacitances contribute to the dynamic power. The total dynamic power is the sum of all the 𝐶𝑉2𝑓 power of the line capacitance and repeaters.

𝑃𝑑𝑦𝑛−𝑡𝑜𝑡𝑎𝑙 = 𝑃𝑑𝑦𝑛−𝑙𝑖𝑛𝑒 + 𝑃𝑑𝑦𝑛−𝑟𝑒𝑝𝑒𝑎𝑡𝑒𝑟𝑠

𝑃𝑑𝑦𝑛−𝑟𝑒𝑝𝑒𝑎𝑡𝑒𝑟𝑠 is dependent on the size and the number of repeaters. When the repeater number decreases the size increases. 𝑃𝑑𝑦𝑛−𝑙𝑖𝑛𝑒 is proportional to line capacitance.

7. Test equipment

Figure 26 DCG Emiscope[30]

DCG Emiscope, Laser Voltage Imaging and Probing (LVx)[30]

The machine was the DCG Emiscope. It provided a non invasive method to test the circuitry. The main function of the equipment is for timing analysis and time resolved emission.

Laser Voltage Imaging (LVI), shows the physical locations of transistors that are active at a specific frequency. LVI can be tuned to target frequencies, and may also be used to show exactly where to get the best signal strength for specific waveform measurements.

The Laser Scanning Microscope (LSM) visually maps locations of transistors. By concentrating on a specific area of the DUT, one can scan for the dominant frequencies. LVI locates the transistors and thus maps circuits operating at those frequencies. LVI also enables signal tracing through circuitry, and even non-periodic signals can be monitored.

8. Noise Flicker Noise[31]

Flicker noise is present in all active devices as well as some passive devices. Flicker noise is a low frequency noise which occurs when a direct current is flowing.

Figure 27 Flicker Noise[31]

The power spectral density of flicker noise is

S(f) = 𝑘𝑓𝐸𝐹

K, EF are constants

1/ f noise is ubiquitous. As of now there is still no single mechanism to account for flicker noise. 1/f noise cannot be predicted from dc or other device characteristics and can only be computed through noise measurements. The main focus of the project is on flicker noise.

Shot Noise[31]

Shot noise arises due to the random flow of carriers across a potential barrier. Shot noise was documented by Schottky as random fluctuations in the plate current as a result of the flow of discrete charges. The fluctuation of the current I is

𝚤2�=(𝐼 − 𝐼𝐷𝐶)2��������������

Shot noise is given by the Schottky formula as

𝚤2� = 2q𝐼𝐷𝐶Δf

Q is electronic charge

Δf is the bandwidth in Hz over which the noise is measured

Thermal Noise[31]

Even in the absence of current, thermal noise is present in a resistor. The resistor should be in thermal equilibrium. The physical mechanism that causes this source of noise is from the collision of the free moving electrons against the thermally agitated atoms. The effect is best pictured as similar to Brownian motion.

Brownian motion states that for a system in equilibrium, the average energy associated with each

degree of freedom of the system is 𝑘𝑇2

, k is the Boltzmann constant and T is the temperature in Kelvin. A

suspended particle has 3 degrees of freedom (x, y, z axis). Each stores a mean energy of 𝑘𝑇2

. The electron

gas in semiconductors is modeled to observe the rules of Brownian motion. Therefore the mean square noise current in an inductor L is given by

12L𝚤2� = 1

2kT

For a capacitor C, the mean square noise voltage is

12𝐶𝑣2��� = 1

2kT

For a resistor R, the mean square thermal noise current would be

𝚤2�= 4kT1𝑅

Δf

Δf is the bandwidth in Hz

Mean square thermal noise voltage would be

𝑣2���= 4kTR Δf

This equation is strictly valid only in equilibrium. This can only be met when no current is passing through the resistor.

9. Time Frequency Analysis A signal is normally analysed through the time or frequency domain. The time domain would give information such as amplitude while frequency analysis would show the changes that take place. The combination of running in both time and frequency is akin to reading musical notes on a score sheet. The time frequency analysis allows users to determine a certain effect at a particular time. Fourier transform is the fundamental bridge between the two domains. Fourier Transform will provide the frequencies within the signal duration, however it does not provide the time it happens. In real world situations, many signals vary with time such as acoustic signal, biomedical signal and seismic signal. Hence it is necessary to go beyond from individual analysis on the time and frequency domain to the one on time-frequency joint domain.

The Time-Frequency analysis helps the user to understand where the frequency is changing with time.

This chapter presents the basic time-frequency distribution characteristic. This is followed by brief description of some existing time-frequency distributions.

Introduction

The main aim of using Time-Frequency analysis is to devise a function that will describe the amplitude/energy density of a signal simultaneously in both the time and frequency domain.

Basic idea

The figure below demonstrates an example of time-frequency analysis. Fourier transform would not be able to show the difference between the four different data sets. However, from the time-frequency plane, we can have a clear idea of the signal component distribution in both time and frequency domain.

Figure 28 Time-frequency distributions of four different signals with similar frequency components[32]

There are multiple methods to do the mapping of the two domains. First, it is possible to construct ‘time-frequency atoms’ to decompose the signal. This is the easiest. Short time Fourier transform is formed based on this principle. The second way is to obtaining the energy/power distribution of the original signal in the time-frequency domain. The Wigner-Ville distribution is an example of such an application and the main focus of the project.

Uncertainty principle

It is not possible to achieve high resolution in both the time and frequency domain. This is mainly because of the uncertainty principle.

If TF(t,ω) is a time-frequency distribution, given that,

∫∫ ><−= ωωσ dtdtTFttt ),()( 22

∫∫ ><−= ωωωωσω dtdtTF ),()( 22

where σt2 : standard derivation in time domain

σf 2 : standard derivation in frequency domain

ησσ ω ≥t

where η is a universal constant. The smallest possible η is ½.

Uncertainty principle demonstrates the limitations of achieving high resolution for both time and frequency components. By this principle, only time or frequency can be chosen to have high resolution [32]

Application of Time-Frequency Distribution

There are many fields that currently employ the use of time frequency distribution for analysis of data sets. These include industries such as medicine, music and telecommunications.

Multi component Signal

When analyzing signals with multiple frequencies, the time frequency analysis would be able to separate the data and give a clearer picture. It shows the different frequencies that are present in the signal and when they are changing. A simple example of the signal will be the speech acoustic signal. Time-frequency distribution has been widely used in blind-separation in multi component signal with the help of array sensor [33-35].

Frequency Dependent Physical Events

Certain physical events are different to analyze using just time or frequency domain. To be able to observe the duration of the change of the signal of a specific frequency, there is more information about the signal to make a more informed judgment. Biomedical applications are common; especially heart ECGs [36-38]. Time-frequency distribution would allow doctors to observe abnormal activities or abnormal durations and make a better diagnosis.

Instantaneous Frequency and Instantaneous Power Spectrum

One of the important concepts closely related to time-frequency distribution is Instantaneous Frequency (IF) defined as:

2)(

),()(21)(

ts

dfftIPSf

dttdtfi

∫∞

∞−⋅

==φ

π

where fi(t) is the instantaneous frequency at time t, φ(t) is the phase of the signal under consideration, and IPS refer to Instantaneous Power Spectrum. The bilinear time-frequency distribution combines the signal’s IPS at any particular time within the signal duration. These are some parameters not obvious in a conventional Fourier Transform. Applications of IF had been found in several areas in communications systems or maritime applications like estimation of Doppler velocity [39] or military applications such as landmine detection[40].

Desirable Properties of Time-Frequency Distribution

Time-frequency distribution comes in various forms and is not a unique distribution. It can be constructed with various methods. It is always desirable that there is minimal loss of information when performing an operation to obtain the distribution. The properties are documented in the subsequent section [32, 41].

Correct total energy property

There is conservation of energy and the total energy of the distribution should be the same as that as the signal

∫ ∫∫ ∫ === ωωωω dSdttsdtdtTFE 22 )()(),(

The total energy criteria will be fulfilled if the distribution satisfies the limits, however the reverse situation is not true.

Correct marginal properties

At a particular instance in time, the instantaneous energy will be given through the summation of the energy distribution for all the frequencies. Likewise, for any frequency, the summation over time will yield the energy density spectrum. Hence, a time-frequency analysis would ideally be

∫ = 2)(),( tsdtTF ωω

∫ = 2)(),( ωω SdttTF

where P(t,ω) : time-frequency representative.

s(t) : signal

S(ω) : Fourier transform of signal

This is usually called time and frequency marginal conditions.

Reality property

For all signal, s(t), we have

RtRtTF ∈∈ ωω ,,),(

for TF(t,ω) is the time-frequency distribution of s(t).

Invertibility

The time frequency distribution must be unique in order for the original to be recovered when it undergoes transformation. This can only be achieved while no signal information if lost during the time frequency transforming operation.

Symmetry property

If

),()(

),()(

** ω

ω

tTFts

tTFts

sTFD

sTFD

→

→

then we have,

),(),( ** ωω −= tTFtTF ss

Time and frequency shift invariance property

),(),()()( 00 ωω ttTFtTFthenttstsif −→−→),(),()()( 00 ωωωωωω −→−→ tTFtTFthenSSif

),(),()()( 0000 ωωωω −−→−→ ttTFtTFthenttsetsif tj

Linear scaling property

)()(1)( atsatsifa

Sa

S scsc =

=ωω

the scaled distribution,

=

aatTFtTFsc

ωω ,),(

and it satisfies,

22 )()(),(∫ == atsatsdtTF scsc ωω

22 1)(),(∫

==

aS

aSdtTF scsc

ωωωω

Moyal’s condition

),()(

),()(

2

1

2

1

ω

ω

tTFts

tTFts

sTFD

sTFD

→

→

2*21

* )()(),(),(21 ∫∫∫ = dttstsdtdtTFtTF ss ωωω

Convolution property

),()()(

),()(

),()(

21

2

1

21

2

1

ω

ω

ω

tTFtsts

tTFts

tTFts

ssTFD

sTFD

sTFD

∗ →∗

→

→

),(),(),(2121

ωωω tTFtTFtTF stsss ∗=∗

*t stands for convolution with respect to the time variable.

Positivity property

For all s(t), we have

TF(t,ω) ≥ 0

Frequency/time support property

The signal and the corresponding TFD should have the same frequency band and time duration; this property is called frequency and time support.

TF(t,ω) = 0 for t outside (t1, t2) if s(t) is zero outside (t1, t2)

TF(t,ω) = 0 for ω outside (ω1, ω2) if S(ω) is zero outside (ω1, ω2)

Types of Time-Frequency Distribution

All time-frequency representation (except Wavelet Transform) can be obtained from

∫∫∫ +−−+−= θττθφττπ

ω θτωθ ddudeusustTF ujjtj),()21()

21(*

41),( 2

where φ(θ,τ) is the kernel.

Table 2 Different distributions and their kernels[32]

Name Kernel : φ(θ,τ) Distribution : C(t,ω)

General class φ(θ,τ) ∫∫∫ +−−+− θττθφττπ

θτωθ ddudeusus ujjtj),()21()

21(*

41

2

Short Time Fourier Transform

∫ −+− dueuhuh ujθττ )21()

21(*

2

)()(21 τττπ

ωτ dthse j −∫ −

Page τθje 2

' ')'(21 dtets

tt tj∫ ∞−

−

∂∂ ω

π

Choi-Williams στθ /22−e τττστπ

τωτσ dudususe jtu )21()

21(*

/

14

1 22 /)(22/3 +−−−−∫∫

Wigner 1 τττπ

τω dtstse j )21()

21(*

21∫ +−−

In this chapter only the more popular representatives, namely Short-Time Fourier Transform (STFT), Wavelet Transform (WT) and Wigner Ville distribution (WVD), are discussed.

Short-Time Fourier Transform

The short-time Fourier Transform (STFT) is one of the most widely used time-frequency distribution due to its simplicity. A conventional Fourier transform is performed within a windowed. The window function is shifted slowly to construct the full picture. The mathematic evaluation is as follows,

)()()( thsst −= τττ

where st(τ) : windowed signal segment

s(t) : signal

h(t) : window function

τ : running time

the windowed signal have the following characteristic

st(τ) ~ s(τ) for τ near t

~ 0 for τ far away from t

The window function gives a new set of signal which put more emphasis on the signal within the time of interest, t. By taking the Fourier transform on this windowed signal, we thus derive the distribution of frequency within the time of interest,

∫

∫

−=

=

−

−

τττπ

ττπ

ω

ωτ

ωτ

dthse

dsetSTFT

j

tj

)()(21

)(21),(

The power distribution of the STFT is called Spectrogram, (SP)

22 )()(

21),(),( ∫ −== − τττπ

ωω ωτ dthsetSTFTtSP j

Figure 29 Short-time Fourier Transform[32]

Wavelet Transform

The wavelet theory was introduced in 1964 by Calderon [42] for the studying integral operators. It is an alternative to linear time-frequency distribution. The fundamental idea of a wavelet transform is to replace the frequency shifting operation in STFT by a time-scaling operation,

τττ da

thsa

atWT ∫∞

∞−

−

= *)(1),(

where h(t), basic wavelet, is localized in time, and a is dilation (scale) parameter.

Figure 30 Wavelet Transform[42]

There are various wavelet functions including Morlet function, and Mexican hat function. The original signal can be recovered from WT if the basic wavelet satisfies the ‘admissibility condition’,

1)( 2 =∫∞

∞− fdffH ( 3.28 )

where H(f) is the Fourier transform of basic wavelet, h(t).

Figure 31 Basic Wavelet function Mexican hat function[32]

Figure 32 Morlet function[32]

The power distribution of the wavelet transform is called Scalogram (SC),

22 )(1),(),( ∫

−

== τττω da

thsa

atWTtSC

According to the Uncertainty Principle, there is a tradeoff between time and frequency resolution. For STFT, once the window is chosen, the resolution is constant over the entire time-frequency plane

whereas for wavelets this is not the case. The resolution changes with the analyzed frequency even when a fix window is chosen.

Figure 33 Compared time-frequency resolution of spectrograms and scalograms. (a) Spectrogram with window neither short nor long; (b) Spectrogram with long window; (c) Spectrogram with short window; (d) Scalograms. [32]

The dark areas in the above figure represent the resolution scale at the time-frequency for Spectrogram (a), (b (c) and Scalogram (Figure (d)). In the spectrogram, the resolution scale is fixed for each point only after the window size is fixed. However, for Scalogram, given a window size, the resolution scale is time and frequency dependent. Nevertheless, both distributions follow the “Uncertainty Principle”.

Wigner-Ville Distribution

The Wigner-Ville distribution is another tool in the time frequency analysis toolbox. It provides a straight forward derivation of the power density. Unlike linear time-frequency distribution, it does not take extra steps of squaring the linear component. The distribution is represented as

∫ −+−= τττπ

ω τωdetststWVD j)21()

21(*

21),(

Wigner-Ville takes Fourier Transform of the signal product is present in the past and future time. Wigner-Ville distribution, unlike STFT, weight equally for far and near times and this causes it to become a highly non-localized distribution.

There are two major disadvantages to using this distribution, nonpositivity and cross interference term. The non-positivity makes the interpretation of the distribution unrealistic. However, because the Wigner Ville Distribution yields very high resolution in both time and frequency domain, this is considered an acceptable tradeoff. There has also been work done to overcome them and it will be discussed later.

Comparison of Different Time-Frequency Analysis

The desirable property

The table below documents the desirable properties, WVD is superior compared to the spectrogram and the scalogram. By satisfying those desirable properties, WVD retain the integrity in the signal interpretation.

Table 3 Desirable Properties[32]

Spectrogram Scalogram WVD

Correct Energy 2 2 2

Correct Marginal 2

Reality 2 2 2

Invertibility 2

Shift Invariance 2

Linear Scaling 2

Moyal’s condition 2

Convolution 2

Positivity 2 2

Time support 2

Frequency support 2

Anti-lasing smoothing

When converting short-time Fourier transform (STFT) and Wavelet transform (WT) into their power distribution form, anti-lasing are introduced [43].The Wigner Ville Distribution does not share this problem. The anti-lasing is heavily dependent on the nature of the signal’s components and the window function. With the implication of smoothing window, the anti-lasing in WVD can be effectively reduced. This is not possible for Spectrogram and Scalogram. The reduction of the anti-lasing can only be done by choosing a suitable window function and depending on the nature of signal component. This presents a problem. The knowledge of the signal is required beforehand and because of this limitation, it can only exist in the realm of academia and not have much practical value.

Time/frequency resolution

Spectrogram resolution tends to be lower than that of Scalogram and WVD[44, 45]. The time and frequency resolution of spectrogram is restricted by uncertainty principle. A better resolution in time can be achieved in the expense of that of frequency, and vice-versa. However, for WVD, the separated smoothing effects can allow for an improvement in the resolution of time and frequency component individually, hence a better overall resolution can be obtained.

Discrete Smoothed pseudo Wigner-Ville Distribution (SPWD)

Wigner-Ville is also known as a bilinear or 2nd order time-frequency distribution. The distribution gives a distribution of the power spectrum directly. Wigner-Ville distribution has found popularity in multiple applications including various forms of engineering and Biomedical fields.

Wigner-Ville distribution was first introduced by Wigner [46]but it was Ville [47] who created a practical application and incorporated it into signal analysis. Wigner laid the groundwork of the mathematics behind the distribution, while Ville found and applied the application of the mathematical relation to signal analysis. This is actually the first distribution introduced for time-frequency analysis and have been studied for a long time.

Wigner-Ville Distribution

Wigner-Ville distribution has found its wide applications in the nonstationary signal processing [48-51].Once again, the distribution is used because of the high resolution in both the time and frequency domains. It also satisfies the conditions for accurate marginal property giving an accurate total energy property

In Biomedical field, Wigner-Ville distribution applications are widespread. It has been intensively developed for the analysis of the heart sound and murmur [52-54], blood pressure[55], electrocardiogram (ECG) study [56, 57].As discussed previously, communications engineering also make

use of this powerful distribution for analysis in Radar systems [58, 59]. Likewise the time-frequency distribution of Wigner-Ville is also used in the acoustic signal analysis [60-62].

Theory of the distribution

For a given signal of interest s(t), the instantaneous correlations can be expressed as follows [63],

)2

(*)2

(),( τττ −+= tststR

For conventional method, the average of this instantaneous correlation, ∫∞

∞−= dttRR ),()( ττ , is taken

and thus the information of time t is lost. In Wigner-Ville distribution a Fourier transform is performed without the averaging of correlation function,

τττ

ττω

ωτ

ωτ

detsts

detRtWVD

j

j

∫

∫∞

∞−−

∞

∞−−

−+=

=

)2

(*)2

(

),(),(

Similarly, if the signal consists of, for instance, 2 components, the corresponding WVD is shown as follows,

∫∞

∞−−++= τττω ωτ dets

dtstxWVD j)

2(*)(),( 21

It is sometimes refers as cross-Wigner-Ville distribution, xWVD.

General properties

Symmetry

For a real signal, a symmetrical Wigner-Ville distribution can be obtained in both frequency and time plane. It means,

spectralrealfortWtWsignalrealfortWtW

),(),(),(),(

ωωωω

−=−=

Signal inversion

One of the very attractive features of the Wigner-Ville distribution is the uniqueness. As a result, it is possible to recover the original signal given its Wigner-Ville distribution. The inversion can be done by follows,

∫∞

∞−= dfeftW

sts tfj π2

21 ),(

)0(*1)(

Instantaneous frequency

Traditional definition of the instantaneous frequency is through the first derivation of the phase function[58],

dttdtfi)(

21)( φπ

=

However the multiple frequencies signal can provide more information versus the traditional definition only provides single frequency value.

Wigner-Ville distribution, instead, provide a weighted average instantaneous frequency with integrity at time t, which take into consideration the existence of multiple frequency component of the signal,

∫∫

∞

∞−

∞

∞−=ωω

ωωω

π dtWVD

dtWVDtfi

),(

),(

21)(

Negative distribution

One of the shortcomings of Wigner-Ville distribution is their non-positivity. The negative power value makes interpreting the WVD difficult. It is a worthwhile tradeoff to spend some time to eliminate the negativities by either excluding them or taking an absolute value of WVD.

Anti-lasing

Wigner-Ville distribution is a distribution relates to signal correlation function. When 2 frequency components are present, one term reflects the correlation of two signal components. This term exists in the middle of the 2 components in the time-frequency plane, as shown below.

The amplitude of cross term may exceed the amplitude of the original signal and it cause a distortion of the interpretation of the original signal.

A lot of research has been carried out to overcome this issue. One of the most direct ways is to convert the real signal into an analytical signal. Analytical signal is derived for the mathematical convenient in signal processing application by turning the amplitude of negative frequency into zero. Details of analytical signal will be given in next section.

Figure 34 Crossterm interference of WVD; (a) WVD of the sum of two Gaussian functions [63]. (b) the crossterm in the middle of the two sinusoidal function [32]

Sampling for Wigner-Ville distribution

Minimum sampling frequency

Given a real signal of interest, s(t) and allow the signal to be bandlimited to frequency |fmax| < Bandwidth/2. This condition must be met to satisfy the Nyquist sampling theorem. The minimum sampling frequency should be equal to the double of the faster frequency component or Bandwidth of the signal, i.e. Nyquist Rate RN = 2fmax.

For a single component signal, the sampling frequency is simply equal to the RN as usual. However if signal consists for few frequency components, a double of the Nyquist sampling Rate is required. This is illustrated [64]

Figure 35 The worst-case signal component for the fastest oscillating frequency [64]

In the above, it shows the Fourier Transform of a signal which consists of two components, S1(f) and S2(f). S1(f) and S2(f) are the Fourier Transform of the signal components s1(t) and s2(t) respectively.

tfjetsts 2

2201

max

)()(ε

π−

= , tfj

etsts 222

02

max

)()(ε

π−

−=

where ε is the bandwidth of signal component.

By placing these 2 signals at the edge of bandwidth, the “worst case” or highest oscillating frequency can be obtained. The WVD distribution of the interference between them is shown as follows,

])2(2cos[2),( max)(

021tfWVDftWVD s

Iss επ −=

The oscillating frequency is 2fmax-ε, hence the sampling frequency fsampling ≥ 2 ( 2fmax - ε ) ∼ 4×fmax which is two times of Nyquist Rate (2×fmax). In conclusion, for Wigner-Ville distribution, a double of Nyquist Rate is needed for sampling of a real signal.

Analytical signal

Theory

Given a real signal, s(t), and its spectrum, S(ω), the complex signal, z(t), can be defined as follows,

ωωπ

ω deStz tj∫∞

=0

)(212)(

It simply means taking an inverse Fourier Transform of the spectrum in positive frequency. The factor 2 is needed because z(t) complies of the symmetrical negative frequency plane.

∫

∫∫

∫ ∫

∫ ∫

−+=

−+−=

−+−=

=

=

∞ −

∞ −

∞ −

'')'()(

')'('

')'()'(1

')'(1

')'(212)(

0)'(

0)'(

0'

dttt

tsjts

ttjttdewheredt

ttjttts

ddtets

ddteetstz

ttj

ttj

tjtj

π

πδωπδπ

ωπ

ωπ

ω

ω

ωω

The second part of the last term is the Hilbert Transform,

'')'(1)(ˆ)]([ dttt

tststsH ∫ −==π

Hence the analytical signal is as follows,

)]([)()( tsjHtstz +=

There are many reasons why an analytical signal is used for WVD and other signal processing, the following are some of them;

It gives a correct indication of the physical situation. For the real signal’s spectrum will give an incorrect average frequency of zero as shown.

Figure 36 Real signal (a) and analytical signal (b) spectrum. <ω> is the average frequency[32]

The average frequency of any signal is found to be zero, due to the existence of a negative frequency component. However, by omitting the negative frequency component (analytical signal) the average frequency will be found at the middle of the signal as shown.

The fundamental definition of instantaneous frequency involved with phase derivation, with the complex analytical signal. The phase function of the signal can be easily calculated.

Likewise it helps to avoid doubling of the fundamental Nyquist sampling frequency as well as reduce the anti-lasing. This would reduce distortions on the Wigner-Ville distribution when multi-component signal is analyzed.

Figure 37 The reduction of Wigner-Ville distribution’s anti-lasing by applying the analytical signal[65]. Signal on the left is real whilst signal on the right is analytical[65]

Smoothed Wigner-Ville Distribution

The cross term interference in WD is mainly due to the multi component of the signal. This presents a major limitation for the usefulness of this distribution. The number of the anti-lasing, m, increases

quadratically, m =

2n

, with the number of signal components, n. As a result, they will often mask out

the original signal. Research has been conducted in the amplitude reduction of these cross term. Other than the conversion to analytical signal, using a smoothing function is another viable alternative. The smoothing can be done on time and/or frequency variables. The result distribution is called Smoothed Wigner-Ville Distribution (SWD).

Let us denote the smoothing function as ψ (t,ω) and its two-dimension Fourier Transform function asΨ(τ,υ) [66].

ωωψυτ τωυ dtdet tj )(),(),( −−∫∫=Ψ

which is called ambiguity kernel.

To ensure that the insertion of the smoothing function does not distort the original Wigner-Ville distribution, here are some requirements of ψ(t,ω),

Linear and independent of time and frequency,

∫∫ −−= '')','()','(),( ωωωωψω ddttWDtttSWD

Smoothing function can be treated as a shift-invariant function for the Wigner-Ville distribution.

Normalization, to guarantee the correct total energy property

∫∫ ==Ψ=Ψ 1),()0,0(),(max ωωψυτ dtdt

Real value, Rt ∈),( ωψ , to make sure SWD is real also.

Below, the two signal components, separated by τ12 and υ12, are plotted at t-f domain. Let us illustrate the anti-lasing reduction effect through smoothing by looking at the kernel ambiguity function,Ψ(τ,υ). If we plot Ψ(τ,υ) according to τ and υ (correlative domain), the crossterm interference will be located away from the origin, as shown in Figure 3.10(b). Whereas the signal, having a zero τ and υ, will be positioned at the central of τ-υ plane.

Figure 38 (a) t-f plane of 2 signal components, (b) Corelative-domain analysis of their anti-lasing [66]

The cross term interference will be attenuated by the factor 1,(),( 12121212 ≤−−Ψ=Ψ υτυτ .

However the attenuation through smoothing may not work well if the signal and the cross term interference are closely spaced.

A highly attenuated SWD can be achieved with Ψ(τ,υ) as narrow as possible around the origin of τ-υ plane. Due to the Fourier Transform relationship of ψ (t,ω) and Ψ(τ,υ), a narrow Ψ(τ,υ) will result in a broad ψ (t,ω). And this broad ψ(t,ω) will result in a spread of the signal component and hence a loss of time-frequency concentration of the signal. There is a fundamental tradeoff of interference attenuation versus time-frequency concentration. As a result, a suitable design of the smooth function is important and careless choice may result in low signal concentration yet without effective reduction of anti-lasing.

Hence, the separation of the time and frequency smoothing will be desirable as different degree and function of window can be applied for different signal structure. This separation can be obtained by the formulation of Smoothed pseudo Wigner-Ville Distribution.

Pseudo Wigner-Ville distribution

Pseudo Wigner-Ville (PWD) distribution is a short-time version of the Wigner-Ville distribution with a running window h(t)[67], known as frequency smoothing window.

ττττω ωτ dehtststPWD js

−∫ −+=2

* )2

()2

()2

(),(

To satisfy the correct total energy property, we normally desire a h(t) with real-valued, even, normalized (h(0) = 1 ) and finite-length, i.e.

20)( hTtifth >=

where Th is the window length. Wigner-Ville distribution can be viewed as a PWD with h(t) having a infinitive window length and constant height, h(t) ≡ 1.

PWD can reduce the anti-lasing occur in conventional Wigner-Ville distribution. To illustrate this, let us express PWD in following form [66],

∫ −='

')',()'(),(f hs dtWVDtPWD ωωωωξω

where

∫ −==τ

ωτ ττωξ dehWVD jhh )

2(),0( 2

A previous form of SWD was seen earlier. Therefore, pseudo Wigner-Ville distribution and the SWD are somewhat interchangeable. The smoothing only occurs with respect to the frequency variable, since t=0. As a result, we can preserve the time concentration and yet attenuate the interference due to the oscillation in frequency direction.

Smoothed pseudo Wigner-Ville distribution

The finite-length window applied for pseudo Wigner-Ville distribution has done the smoothing in the frequency. Insertion of a time-smoothing window function, g(t) will yield a Smoothed pseudo Wigner-Ville distribution (SPWD). It is defined as follows,

∫∫∫

−−=

−=

'')','()'()'(

'),'()'(),('

ωωωωξ

ωω

ddttWDttg

dttPWDttgtSPWD

h

t

g(t) is a smoothing function for time direction, which serve as a low-pass function here. The separation of the smoothing operation can be further illustrated below. (C) describes a complete smoothing effect for both time and frequency domain.

Figure 39 Effective support of ambiguity kernel: (a) WVD (no smoothing), (b) PWD, (c) SPWD

Discrete Smoothed pseudo Wigner-Ville Distribution

Data collection was done in a discrete format and a corresponding discrete version of the above formula for smoothed pseudo Wigner-Ville distribution is needed.

Given a discrete function s(n), the discrete-Wigner-Ville distribution (DWD) is as shown [68],

( )∑∞

−∞=

−−+=m

mjmnzmnznDWD ωω 2exp)(*)(),(

where z(n) is the analytical signal for s(n), n is the time-index. However this definition of DWD suggests a noncausal operation, which requires the knowledge of the signal in all time. This can be overcome by introducing a running window function just like what we do in pseudo Wigner-Ville distribution, the discrete version of PWD is as follows [55],

∑−

+−=

−−+⋅=

1

1

221 2exp)(*)()(),(

N

Nk NkmjknzknzkhNmnDPWD π

where h(k) : frequency smoothing window with the length of 2N-1.

m : frequency index.

N : time index

z(n) : analytical signal

After inserting the time smoothing function, we have SPWD in discrete form as follows,

−

−+++⋅= ∑ ∑

−

+−=

−

+−= NkmjkpnzkpnzpgkhNmnDSPWD

N

Nk

M

Mp

π2exp)(*)()()(),(1

1

1

1

221

g(p) : time smoothing window with the length of 2M-1.

This equation is the complete algorithm for the Smoothed pseudo Wigner-Ville distribution in discrete form.

8. Data collection As stated in the literature review, the circuitry of interest is the repeater circuit. The circuitry is identified after running a Laser Voltage Image (LVI) and scanning for the transistors that are toggling at the frequency of desire. Once the location is identified, we zoom into the circuitry first using the solid immersions lens and followed by a digital zoom.

Figure 40 10X magnification

Figure 41 SIL Magnification

Figure 42 Digital 7X after SIL

Figure 43 Identification using LVI

The test equipment Emiscope III collects voltage amplitude data over a specified period of time. The time window is normally spaced into 1000 equal segments, also known as bins. For this experiment, the window was set to 1ns giving 1000 bins of 1ps.

The data waveform is averaged over 15 minutes to remove as much noise as possible. During the averaging only overlap data is retained, therefore signal would become stronger and stronger over time versus noise which is random. 1000 data points are collected and this data is filtered through a moving

average function to obtain a trend line. The moving period is selected to provide the best fit for the waveform. Through trial and error, the moving average of 20 points is considered the best fit for this system.

Generally moving average calculations are used widely in financial calculations especially in stock market analysis to track trends. Simple moving averages are normally plotted together with exponential moving averages to determine distinct patterns. The analysis does not involve data of different time periods, only within a single window of 1ns; therefore a simple moving average is enough. The mathematical argument for moving average is illustrated as follows.

Given a sequence {𝑎𝑖}𝑖𝑁, the n-moving average of the sequence is {𝑠𝑖}𝑖𝑁−𝑛+1. The n-moving average is calculated as

𝑠𝑖 = 1𝑛

∑ 𝑎𝑗𝑖+𝑛−1𝑗=1

Once a trend line is established, the data is then put through a piecewise linear manipulation for ease of analysis of the data. It becomes obvious that the curve can be broken down into three sections. The first and last are relatively flat, while the middle section displays a gradient (positive for rise time and negative for fall time).

Figure 44 Generic example of moving average

The piecewise linear approximation of the trend line should follow a function of:

f(x) = �0 𝑤ℎ𝑒𝑛 𝑥 < 𝑎

𝑥 + 𝑐 𝑤ℎ𝑒𝑛 𝑎 < 𝑥 < 𝑏𝑌 𝑤ℎ𝑒𝑛 𝑥 > 𝑏

�

0

20

40

60

80

100

120

0 200 400 600 800 1000 1200

Voltage

Number of bins

voltage

where Y is the maximum amplitude,

a, b and c are constants

The breakpoints are determined by the following criteria:

For the first part where x < a and for the last part where x > b, the gradient must be 0 or close to 0.

For the center portion, the magnitude of the gradient must be adjusted until a maximum value is obtained.

The two criteria are satisfied through trial and error by setting a start point and going through an iteration of incrementing values and decrementing values until the optimum point is found. Due to time constraints, the final points are rounded off to the nearest 10 value.

The data is further normalized by shifting the waveform to the origin. The first part of the piecewise linear curve is averaged to determine the offset from the origin. The offset difference is added to the curve to normalize all the curves to start from the point of origin 0. This allows for the amplitude values to be compared.

Metrics of comparison

When the project was planned, the consideration the SNR, gradient of the rise/fall time, amplitude and rise/ fall time were considered as the standard for comparison.

Signal to Noise Ratio (SNR)

The SNR is a measure of the signal power to noise power

SNR = 𝑃𝑠𝑖𝑔𝑛𝑎𝑙𝑃𝑛𝑜𝑖𝑠𝑒

𝑆𝑁𝑅𝑑𝐵 = 10 𝑙𝑜𝑔10𝑃𝑠𝑖𝑔𝑛𝑎𝑙𝑃𝑛𝑜𝑖𝑠𝑒

= 10 𝑙𝑜𝑔10(𝐴𝑠𝑖𝑔𝑛𝑎𝑙𝐴𝑛𝑜𝑖𝑠𝑒

)2

= 20 𝑙𝑜𝑔10𝐴𝑠𝑖𝑔𝑛𝑎𝑙𝐴𝑛𝑜𝑖𝑠𝑒

The SNR is a good measure of the strength of the signal that is being transferred. The standard deviation of the flat parts of the trend line is the basis of measuring the SNR. SNR was not investigated further because of the difficulty in reproducing the results. The deviation of the values differed too much for different data points and the underlying reason was difficult to pinpoint. Therefore, the SNR was not further explored and is left for future work.

Gradient

The rise time is an absolute number. The gradient is better able to provide the rate of chance of gradient versus time. The curves are further normalized to show the same amplitude to determine a change in the gradient. The gradient would have provided a clearer picture concerning the rate of change; however, once again, it was difficult to reproduce the results. It was difficult to determine where the signal started to rise or fall based on the linear piecewise approximation and the margin of error was very drastic from data point to data point. Hopefully this can be further explored in future work.

Figure 45 Generic waveform

Figure 46 Demonstration of SNR

Amplitude

The amplitude is based on the difference of the highest point of the trend line minus the lowest point of the trendline. The curves are normalized to the origin. The absolute mean of the last linear part of the piecewise approximation would equate to the amplitude. However, because both sets of linear approximations displays a gradient, the equations of both lines are equated to calculate the interception point and that is assumed to be the amplitude.

Rise time/ Fall time

There are many different definitions for rise (fall) time. This could range from the time taken to rise from 10% to 90%. However for simplicity, the rise and fall time is taken to be the difference between the end of the first linear piecewise approximation line and the beginning of the last line.

0 50

100 150 200 250

0 100 200 300

Voltage

Number of bins

voltage

0 50

100 150 200 250

0 100 200 300

Voltage

Number of bins

voltage

Figure 47 Definitions for rise/ fall time[69]

Observations

General

This section would discuss the effects of the different inverters in the repeater circuitry. The figure below illustrates the terminology in the following sections. Two 32nm SOI chips were used. Chip 1 was polished to around 98 micron thickness, while chip 2 was polished to around 115 microns thickness.

Figure 48 NMOS and PMOS of Repeater

PMOS4 NMOS4

NMOS

PMOS PMOS

NMOS2

PMOS

Figure 49 NMOS2 1V Laser Power 269 10Mhz

Figure 50 NMOS4 1V Laser Power 269 10Mhz

y = 0.0787x + 156.38 y = 1.6041x - 510.46

-150

-100

-50

0

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150

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250

300

350

400

0 200 400 600 800 1000 1200

Number of bins

y = 0.0664x + 107.93 y = 1.517x - 665.48

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-50

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0 200 400 600 800 1000 1200

Number of bins

Figure 51 PMOS1 1V Laser Power 269 10Mhz

Figure 52 PMOS2 1V Laser Power 269 10Mhz

y = 0.0551x + 23.657

y = 0.4662x - 189.74

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-100

-50

0

50

100

150

200

0 200 400 600 800 1000 1200

Number of bins

y = 0.0712x + 57.113 y = 1.0267x - 463.08

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-100

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Number of bins

Figure 53 PMOS4 1V Laser Power 269 10Mhz

It can be observed that there is a slight gradient in the last section of the linear piecewise approximation when it should be flat. This could arise due to a variety of reasons such as leakage or short channel effects.

The sizing of the inverters does not seem to have any discernible pattern with regards to rise time. The amplitude however seems to show a negative correlation with the increment of NMOS sizing and a positive correlation with an increase in PMOS sizing. The only data that falls out of this observation is Chip 2 PMOS4. The contact between the SIL and the die loosened during the collection and resulted in a loss of focus.

Table 4 Comparison of Rise Time and Amplitude with the different inverter sizes

Chip1 Rise Time Amplitude Chip2 Rise Time Amplitude

NMOS2 145 190 NMOS2 120 103

NMOS4 130 143 NMOS4 130 88

PMOS1 150 46 PMOS1 200 40

PMOS2 115 89 PMOS2 170 95

PMOS4 180 91 PMOS4 150 64

For the comparison between laser power ratings, voltage and frequency, the focus will be on the NMOS4 inverter and the PMOS4 inverter.

y = 0.07x + 63.019 y = 0.5941x - 239.38

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-50

0

50

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250

0 200 400 600 800 1000 1200

Number of bins

Laser power

The laser power is provided by the manufacturer of the equipment DCG. The values of laser power range from 0.35mW to 19.8mW. The range of values used during the experiment was between 1.16mW (240) to 3.15mW (269).The final power was based on the attenuation value of the laser ranging from 196 to 330. The final value when decreasing power and increasing laser power are totally different. Extreme caution was used to ensure that laser powers were consistent by ensuring the laser power was only measured when there was an increment of laser power.

Figure 54 Laser power at 240

Figure 55 Laser power at 260

Figure 56 Laser power at 269

Figure 57 Laser power at 279

The rise time of the laser power seems to increment with higher power. Chip2 at 279 is an exception and once again due to a loss of focus the data collected seems to be inaccurate. The increase in rise time could be attributed to an increase in temperature of the chip due to the power supplied by the laser. This in turn caused the internal atoms to become more agitated and created more resistance. With regards to the amplitude, it is very consistent upward increment with increasing laser power.

Table 5 Comparison between Chip 1 and Chip 2 for Laser Power

NMOS

Chip1 Chip2

Laser Power

Rise Time Amplitude Laser Power

Rise Time Amplitude

240 140 38 240 120 51

269 150 137 269 150 90

279 190 186 279 100 168

Figure 58 Chip 1 NMOS4 1V Laser Power 240 10Mhz

y = 0.0402x + 20.833

y = 0.518x - 229.62

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-100

-50

0

50

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200

0 200 400 600 800 1000 1200

Number of bins

Figure 59 Chip 1 NMOS4 1V Laser Power 269 10Mhz

Figure 60 Chip 1 NMOS4 1V Laser Power 279 10Mhz

y = 0.0641x + 109.9

y = 1.3259x - 576.67

-150

-100

-50

0

50

100

150

200

250

300

0 200 400 600 800 1000 1200

Number of bins

y = 0.0599x + 161.49

y = 1.2933x - 537.8

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-100

-50

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300

350

0 200 400 600 800 1000 1200

Number of bins

Figure 61 Chip 2 NMOS4 1V Laser Power 240 10Mhz

Figure 62 Chip 2 NMOS4 1V Laser Power 269 10Mhz

y = 0.028x + 40.666 y = 0.4575x - 173.75

-250

-200

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-100

-50

0

50

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150

200

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300

0 200 400 600 800 1000 1200

Number of bins

y = 0.0775x + 59.54

y = 0.8197x - 319.71

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0

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400

0 200 400 600 800 1000 1200

Number of bins

Figure 63 Chip 2 NMOS4 1V Laser Power 279 10Mhz

Table 6 Comparison between Chip 1 and Chip 2 for Laser Power

PMOS

Chip1 Chip2

Rise Time Amplitude Rise Time Amplitude

240 230 44 240 100 12

269 200 98 269 180 65

279 140 135 279 140 69

y = 0.0382x + 153.55 y = 2.3596x - 975.35

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-200

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0

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0 200 400 600 800 1000 1200

Number of bins

Figure 64 Chip 1 PMOS4 1V Laser Power 240 10Mhz

Figure 65 Chip 1 PMOS4 1V Laser Power 269 10Mhz

y = 0.0149x + 38.307 y = 0.1823x - 73.092

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-50

0

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0 200 400 600 800 1000 1200

Number of bins

y = 0.0534x + 78.005 y = 0.4806x - 182.46

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-50

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0 200 400 600 800 1000 1200

Number of bins

Figure 66 Chip 1 PMOS4 1V Laser Power 279 10Mhz

Figure 67 Chip 2 PMOS4 1V Laser Power 240 10Mhz

y = 0.1272x + 95.535 y = 1.6179x - 787.28

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-50

0

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300

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0 200 400 600 800 1000 1200

Number of bins

y = 0.0242x + 10.397

y = 0.0513x - 4.5752

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Number of bins

Figure 68 Chip 2 PMOS4 1V Laser Power 269 10Mhz

Figure 69 Chip 2 PMOS4 1V Laser Power 279 10Mhz

y = 0.0159x + 59.066 y = 0.4745x - 201.79

-200

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-50

0

50

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300

0 200 400 600 800 1000 1200

Number of bins

y = 0.0526x + 43.494

y = 0.8152x - 407.43

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0

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0 200 400 600 800 1000 1200

Number of bins

Voltage

Figure 70 Voltage at 0.9V

Figure 71 Voltage at 1.0V

Figure 72 Voltage at 1.1V

Figure 73 Voltage at 1.2V

The general trend for the increase in voltage leads to an increase in amplitude. Even in the LSM mode, it can be seen that the intensity of the signal increases and it saturates and overflows into nearby circuitry. This makes sense because when the supply voltage is increased the voltage swing provided to the internal circuitry is also increased. There seems to be no discernible pattern to the rise time of the circuitry and no conclusions can be drawn.

Table 7 Comparison between Chip 1 and Chip 2 for Voltage

Rise Time Amplitude Rise Time Amplitude

0.9V 160 109 0.9V 80 25

1.0V 130 143 1.0V 120 88

1.1V 100 149 1.1V 100 112

1.2V 130 167 1.2V 90 79

Figure 74 Chip 1 NMOS4 0.9V Laser Power 269 10Mhz

y = 0.0502x + 85.374

y = 0.9833x - 463.9

-150

-100

-50

0

50

100

150

200

250

0 200 400 600 800 1000 1200

Number of bins

Figure 75 Chip 1 NMOS4 1V Laser Power 269 10Mhz

Figure 76 Chip 1 NMOS4 1.1V Laser Power 269 10Mhz

y = 0.0664x + 107.93 y = 1.517x - 665.48

-150

-100

-50

0

50

100

150

200

250

300

0 200 400 600 800 1000 1200

Number of bins

y = 0.0367x + 137.23

y = 1.3049x - 417.86

-150

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-50

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350

0 200 400 600 800 1000 1200

Number of bins

Figure 77 Chip 1 NMOS4 1.2V Laser Power 269 10Mhz

Figure 78 Chip 2 NMOS4 0.9V Laser Power 269 10Mhz

y = 0.0617x + 140.64

y = 1.6348x - 710.68

-150

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-50

0

50

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300

0 200 400 600 800 1000 1200

Number of bins

y = 0.0703x + 71.193

y = 1.1934x - 553.28

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400

0 200 400 600 800 1000 1200

Number of bins

Figure 79 Chip 2 NMOS4 1V Laser Power 269 10Mhz

Figure 80 Chip 2 NMOS4 1.1V Laser Power 269 10Mhz

y = 0.0775x + 58.54 y = 0.8106x - 316.1

-300

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-100

0

100

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400

0 200 400 600 800 1000 1200

Number of bins

y = 0.0479x + 31.255 y = 0.2135x - 40.643

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Number of bins

Figure 81 Chip 2 NMOS4 1.2V Laser Power 269 10Mhz

Figure 82 Chip 2 NMOS4 1.2V Laser Power 279 10Mhz

y = 0.084x + 57.353

y = 1.0369x - 472.67

-200

-100

0

100

200

300

400

0 200 400 600 800 1000 1200

Number of bins

y = 0.0177x + 56.519

y = 0.5863x - 261.69

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Number of bins

Frequency

In general the frequency does not contribute much to the change in rise time nor amplitude of the results. The slight fluctuation could be caused by the noise as detailed in the following section. This noise could be minimized through good circuit design and hence the effects are not pronounced and do no manifest during testing.

Table 8 Comparison between Chip 1 and Chip 2 for Frequency

Rise Time Amplitude

5Mhz 110 109

10Mhz 130 143

25Mhz 100 122

Figure 83 Chip 2 NMOS4 1.2V Laser Power 269 5Mhz

Figure 84 Figure 54 Chip 2 NMOS4 1.2V Laser Power 269 10Mhz

y = 1.3641x - 601.71

y = 0.0682x + 79.836

-150

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-50

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0 200 400 600 800 1000 1200

Number of bins

y = 0.0664x + 107.93 y = 1.517x - 665.48

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Number of bins

Figure 85 Figure 54 Chip 2 NMOS4 1.2V Laser Power 269 25Mhz

CONCLUSION

The frequency does not seem to have any effects on the final results. While increasing laser power and voltage seems to produce generally stronger signals.

Preliminary tests were conducted concerning high laser power and there are signs that high laser power may lead to permanent damage and hysteresis of the silicon die. However due to time constraints, this must be left for future work.

y = 1.5012x - 925.71

y = 0.109x + 55.179

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Number of bins

III Noise Characterization

1. Overview A spot with no signal was chosen for the probing of noise. For all the situations, the top left hand corner of the repeater circuitry was chosen to maintain consistency.

Figure 86 Repeater circuitry at LVI and location of probe

The voltage amplitude data was collected over a span of 1 hour in 5 minute blocks to ensure consistency. Once again the window was set to 1 ns over 1000 bins of 1ps. The data was then put through a Wigner

Ville Distribution Matlab file to generate the distribution. The results were then plotted using the PSD Burg plot method to observe the output. A criteria for comparison was to use the peak to valley difference to determine the effects of the variables with noise.

function [wv, ff, tt] = wvdc(x, res, win, sps); % wvdc creates a Wigner-Ville spectrogram % % (wv,ff,tt)=wvd(x,res,win,sps) % % x= real input time series % res= resolution, number of samples between windows % (for full resolution: 1) % win= window, becomes length of frequency axis % (a good default: length(x)/2) % sps= samples per second of signal % % wv= the W-V spectrum, each row represents a % frequency, each column a time instant % ff= frequency vector (optional) % tt= time vector (optional) % % Display using: % % imagesc(tt,ff,log10(abs(wv)));axis xy % or: % surf(tt,ff,log10(abs(wv)));shading interp % % ...of course modifying the abs or log10 as desired. % % -Case Bradford, February 2005 % % Adapted from Rene Laterveer, 1999, wvd.m package available from: % http://www.mathworks.com/matlabcentral/link_exchange/MATLAB/Signal_processing/ % z=hilbert(x); % make even number of points, at given resolution npts = floor(floor(length(z)/res)/2)*2; % make sure that we entered in an integer for the window win=floor(win); % round window length down to nearest odd integer oddwin = (floor((win-1)/2)*2)+1; % half point (for indexing reasons we need it later, we're % filling two columns per loop, so we only index through half) halfwin = (oddwin+1)/2-1;

% create tt and ff tt=[0:npts-1]*res/sps; ff=[0:(win-1)]*(sps/2)/(win-1); % pad with zeros z = [zeros(1,oddwin-1), z, zeros(1,oddwin-1)]; % initialize (important when creating huge arrays) wv = zeros(win,npts); R = zeros(1, win); idx = 1:halfwin; for n=0:npts/2-1 t = 2*n*res+oddwin; R(1) = z(t)*conj(z(t)) + i*z(t+res)*conj(z(t+res)); v1 = z(t+idx).*conj(z(t-idx)); v2 = z(t+res+idx).*conj(z(t+res-idx)); R(idx+1) = v1+i*v2; R(win-idx+1) = conj(v1)+i*conj(v2); RF = fft(R, win); wv(:,2*n+1) = real(RF); wv(:,2*n+2) = imag(RF); end return;

The result outputs were plotted in Power Spectral Density (PSD) (Burg Method) to determine the PSD of the final WVD.

2. Frequency The noise was characterized using the same parameters from the previous section. As the frequency was reduced, the amount of noise increased exponentially. As seen in the time domain, the amplitude of the low frequency is a lot higher than that of the higher frequencies. The SNR was determined by using the standard deviations and the result is plotted below and a curve is fitted to demonstrate the results.

Figure 87 Time domain and frequency domain of signals varying in frequency

Figure 88 PSD at 1Mhz

Figure 89 PSD at 10Mhz

Figure 90 PSD at 50Mhz

Figure 91 PSD at 200Mhz

Figure 92 Voltage Amplitude versus Frequency

3. Laser Power As discussed above in the literature review, the time domain and frequency domain cannot provide a distinction between the differing laser powers used on the die and the Wigner Ville distribution is employed.

0

50

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350

400

1Mhz 10Mhz 50Mhz 200Mhz

Votla

ge A

mpl

itude

Frequency

Figure 93 Time domain and frequency domain of signals varying in laser power

Figure 94 PSD at laser power 240

Figure 95 PSD at laser power 269

Figure 96 PSD at laser power 279

Figure 97 Peak Vs Valley values for laser power

After performing an operation to convert it to become a Wigner Ville distribution and plotting the data using power spectral density (PSD) it can be observed that the signals have a different effect on the PSD. From the results, the turning points appear to be at a lower normalized frequency. As the laser power increases, the power/ frequency drops slightly. The curve also becomes more linear.

The data collected makes sense, because as the laser power increases, the signal becomes stronger and hence the power of flicker noise decreases. The noise that is picked up would probably be thermal noise as the power is increased and the electrons become more agitated.

-5

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1

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3

1.15 2.5 3.15

Laser Power

4. Voltage

Figure 98 Time domain and frequency domain of signals varying in voltage

Figure 99 PSD at voltage 0p5

Figure 100 PSD at voltage 1p0

Figure 101 PSD at voltage 1p2

Figure 102 Peak Vs Valley values for voltage

From the plotted PSD graphs, it is observed that as the voltage increases, the turning points become more pronounced. At the same time the power/ frequency also seems to have a positive correlation with the increasing voltage. The turning points however seem to appear at lower normalized frequencies as the voltage increases.

The data points to the fact that as voltage becomes higher, the noise also becomes higher. As the voltage swing is increased it creates more space for noise margin as well. It is interesting to observe that the noise is moving towards lower frequencies as the voltage is incremented.

0

1

2

3

4

5

6

0.5 1 1.2 Voltage

5. Focus

Figure 103 Time domain and frequency domain of signals varying in focus

Figure 104 PSD In Focus

Figure 105 PSD Out of Focus

Figure 106 Peak Vs Valley values for focus

From the figure104 and figure 105 above, the turning points of the PSD are the same. However, the Power/ Frequency is substantially lower for the die which is out of focus. The difference is roughly 5dB/rad. When the die is out of focus, the values reflected would be substantially lower because the optical receiver is not absorbing at the optimal efficiency.

Figure 106 demonstrates the increase in peak to valley value. The loss in focus possibly created a greater increment to noise.

0

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In Focus Out of Focus

6. Limitations and Problems encountered The CMP process is unable to produce a tolerance better than 20 microns and causes the samples to vary in terms of thickness. This led to certain samples having stronger or weaker signals, depending on the sample thickness.

At the same time, there was a substantial amount of jitter and noise contributed by the wires, connectors and test equipment. There was no way to minimize the amount of jitter and noise being introduced into the test set up. The LVP would go through averaging and try to yield the best possible signal, however jitter and noise was still not removed completely.

With concerns to noise analysis. The machine averaged out the data. This led to problems when doing time frequency analysis as the injected events were masked by the averaging. Therefore the final results presented in this dissertation only showcased individual variables instead of changing variables.

7. Conclusion It is interesting to note the peaks and valleys that are observed throughout the plotting of the PSD graphs. From theory, noise is supposed to follow an exponential decay in PSD. However in the observed cases here, there are peaks and valleys at specific frequencies. The peaks could be attributed to the resonant frequencies when the impedance is closely matched. The better power matching causes the spike in the power spectrum. The valleys could be frequencies where the impedance matching is the worse case scenario and causes the drop in value.

The only instance where the valley is lower than the peak and the curve follows the traditional 1/f noise curve is when laser power is at 3.15 mW. The result shows that the higher laser power would yield more realistic results, especially when studying noise. The higher laser power also contributes to a much better signal with the noise decreasing.

For voltage, based on the incrementing peak to valley ratio, we can conclude that the noise is exponentially increasing with increasing voltage swing.

IV Conclusion and future work

Conclusion It is certain that induction of laser power on the silicon die will create a change to the die structure. From the results it is concluded that there is a deterioration of the rise time with increasing laser power. However there is also a better signal with increasing laser power. This presents a contradiction and it is up to the discretion of the tester to determine the best trade off and achieve the best signal needed to find a fault using the LVP.

With results to noise, it is interesting to note the effects the variables could have on the test results and could prove to be an invaluable tool for quick and efficient diagnostics to ensure the correct set of values are inputted. This data could be used by industry to create efficient operating procedures for efficient manufacturing and testing.

Future works Future work could be done on other circuitry and other processes to try to replicate and reproduce the results or determine if the characterizations only work with the current circuitry and current process.

Preliminary tests were also carried out concerning the damage by laser power on the die and there seems to be evidence that a higher laser power does cause some hysteresis on the silicon. This would be a good direction to move towards in order to maximize the LVP tool.

Due to time constraints, some of the criteria for comparison were not fully explored such as SNR and gradient of rise/ fall time. An algorithm to standardize the processing of the data is needed to remove variation and produce results that can be replicated.

At the same time, more work needs to be done to find a faster frequency filtering algorithm that can detect multiple components and make it cost efficient and more viable for industry to adopt time frequency analysis into main stream testing apparatus.

V References

[1] V. N. Khmelev, A. V. Shalunov, and E. S. Smerdina, "Increasing Efficiency of a Chemical-Mechanical Polishing of the Silicon Wafer," in Electron Devices and Materials, 2006. Proceedings. 7th Annual 2006 International Workshop and Tutorials on, 2006, pp. 263-269.

[2] G. Zhixue, L. Qingzhong, Z. Hui, and Z. Jing, "Research on material removal rate of each element of chemical mechanical polishing," in Mechanic Automation and Control Engineering (MACE), 2011 Second International Conference on, 2011, pp. 608-611.

[3] K. A. Reinhardt and R. F. Reidy, Handbook of cleaning for semiconductor manufacturing : fundamentals and applications. Salem, Mass. Scrivener ; Hoboken, N.J.: John Wiley & Sons, Inc., 2011.

[4] I. D. Marinescu, E. Uhlmann, and T. K. Doi, Handbook of lapping and polishing. Boca Raton, FL: CRC Press, 2007.

[5] "http://www.alliedhightech.com/grinding/polishsystem/." [6] "http://www.horiba.com/us/en/scientific/products/particle-

characterization/applications/cmp/." [7] "www.amd.com." [8] T. C. Chai, C. Lee, Y. Y. Ma, E. H. Wong, X. W. Zhang, D. Pinjala, S. S. Murthy, and P. S. Teo,

"Reliability assessment of flipchip on build up laminate based multi-chip package," in Electronic Components and Technology Conference, 2002. Proceedings. 52nd, 2002, pp. 747-752.

[9] L. Weiping, H. Fields, and R. Parker, "Flipchip on board (FCOB): solderability, reliability and the role of surface finish," in Advanced Packaging Materials: Processes, Properties andInterfaces, 2000. Proceedings. International Symposium on, 2000, pp. 172-174.

[10] R. Windemuth and T. Ishikawa, "New flipchip technology," in Microelectronics and Packaging Conference, 2009. EMPC 2009. European, 2009, pp. 1-6.

[11] P. Viswanadham, Essentials of Electronic Packaging: A Multidisciplinary Approach: ASME Press, 2011.

[12] T. M. Moore and R. G. McKenna, Characterization of integrated circuit packaging materials: Butterworth-Heinemann, 1993.

[13] Y. Jin, Z. Wāng, and J. Chen, Introduction to Microsystem Packaging Technology: Taylor & Francis Group, 2010.

[14] L. V. Keldysh, "Behaviour of Non-Metallic crystals in strong electric fields," Physics: J. Exptl. Theoret. Phy, vol. 33, pp. 994–1003, 1957.

[15] C. Mee Koy, "An analysis of the performance of Franz-Keldysh electroabsorption waveguide modulators," Photonics Technology Letters, IEEE, vol. 7, pp. 309-311, 1995.

[16] S. Kolachina, "Introduction to Laser Voltage Probing (LVP) of Integrated Circuits," Microelectronics Failure Analysis: Desk Reference Fifth Edition, pp. p426 - 430, 2004.

[17] T. L. Yin S Ng, Dmitry Skvortsov, "Laser Voltage Imaging: A New Perspective of Laser Voltage Probing," ISTFA 2010: Proceedings from the 36th International Symposium for Testing and Failure Analysis, 2010.

[18] W. L. Yin S Ng, Kenneth Wilsher, "Next Generation Laser Voltage Probing," ISTFA 2008: Proceedings from the 34th International Symposium for Testing and Failure Analysis, 2008.

[19] M. Akiya, K. Ohwada, and S. Nakashima, "High-voltage buried-channel MOS fabricated by oxygen implantation into silicon," Electronics Letters, vol. 17, pp. 640-641, 1981.

[20] J.-P. Colinge, Silicon-on-insulator technology : materials to VLSI, 3rd ed. Boston: Kluwer Academic Publishers, 2004.

[21] A. Ogura, "Method of fabricating SOI substrate," U.S. Patent 5888297, 1995. [22] "U.S. Patent 5,061,642." [23] Q. Y. Tong, U. Gösele, and Electrochemical Society., Semiconductor wafer bonding : science and

technology. New York: John Wiley, 1999. [24] K. V. Srikrishnan, "Smart-cut process for the production of thin semiconductor material films,"

U.S. Patent 5882987, 1997. [25] T. Chia-Chun, W. Jan-Ou, L. Trong-Yen, and H. Rong-Shue, "Propagation Delay Minimization on

RLC-Based Bus with Repeater Insertion," in Circuits and Systems, 2006. APCCAS 2006. IEEE Asia Pacific Conference on, 2006, pp. 1285-1288.

[26] Y. I. Ismail and E. G. Friedman, "Effects of inductance on the propagation delay and repeater insertion in VLSI circuits," in Design Automation Conference, 1999. Proceedings. 36th, 1999, pp. 721-724.

[27] Y. I. Ismail and E. G. Friedman, "Repeater insertion in RLC lines for minimum propagation delay," in Circuits and Systems, 1999. ISCAS '99. Proceedings of the 1999 IEEE International Symposium on, 1999, pp. 404-407 vol.6.

[28] Y. I. Ismail and E. G. Friedman, "Effects of inductance on the propagation delay and repeater insertion in VLSI circuits," Very Large Scale Integration (VLSI) Systems, IEEE Transactions on, vol. 8, pp. 195-206, 2000.

[29] Y. I. Ismail and E. G. Friedman, "Effects of inductance on the propagation delay and repeater insertion in VLSI circuits: A summary," Circuits and Systems Magazine, IEEE, vol. 3, pp. 24-28, 2003.

[30] "http://www.dcgsystems.com/product.emiscope.html." [31] A. V. Oppenheim, A. S. Willsky, and S. H. Nawab, Signals and systems: Prentice Hall, 1997. [32] L. Cohen, Time-frequency analysis. Englewood Cliffs, N.J: Prentice Hall PTR, 1995. [33] M. G. Amin and A. Belouchrani, "Blind source separation using the spatial ambiguity functions,"

in Time-Frequency and Time-Scale Analysis, 1998. Proceedings of the IEEE-SP International Symposium on, 1998, pp. 413-416.

[34] A. Belouchrani, K. Abed-Meraim, J. F. Cardoso, and E. Moulines, "A blind source separation technique using second-order statistics," Signal Processing, IEEE Transactions on, vol. 45, pp. 434-444, 1997.

[35] A. R. Leyman, Z. M. Kamran, and K. Abed-Meraim, "Higher-order time frequency-based blind source separation technique," Signal Processing Letters, IEEE, vol. 7, pp. 193-196, 2000.

[36] A. Ramanathan, S. G. Pan, M. Gauthier, H. J. Jacob, M. J. Brody, and G. A. Myers, "Mechanisms regulating blood pressure in the conscious rat: investigation by Wigner-Ville analysis," in Computers in Cardiology 1994, 1994, pp. 317-319.

[37] M. Unser and A. Aldroubi, "A review of wavelets in biomedical applications," Proceedings of the IEEE, vol. 84, pp. 626-638, 1996.

[38] P. R. White, W. B. Collis, and A. P. Salmon, "Analysing heart murmurs using time-frequency methods," in Time-Frequency and Time-Scale Analysis, 1996., Proceedings of the IEEE-SP International Symposium on, 1996, pp. 385-388.

[39] N. Allan, D. B. Trizna, and D. J. McLaughlin, "Numerical comparison of techniques for estimating Doppler velocity time series from coherent sea surface scattering measurements," Radar, Sonar and Navigation, IEE Proceedings -, vol. 145, pp. 367-373, 1998.

[40] B. Barkat, A. M. Zoubir, and C. L. Brown, "Application of time-frequency techniques for the detection of anti-personnel landmines," in Statistical Signal and Array Processing, 2000. Proceedings of the Tenth IEEE Workshop on, 2000, pp. 594-597.

[41] L. Cohen and P. Loughlin, Recent developments in time-frequency analysis. Boston: Kluwer Academic Publishers, 1998.

[42] A. P. Calderon, "Interpolation spaces by complex methods - the complex method," Studia Math, vol. 24, 1964.

[43] S. Kadambe and G. F. Boudreaux-Bartels, "A comparison of the existence of `cross terms' in the Wigner distribution and the squared magnitude of the wavelet transform and the short-time Fourier transform," Signal Processing, IEEE Transactions on, vol. 40, pp. 2498-2517, 1992.

[44] R. H. Layton and A. Murray, "Comparison of techniques for time-frequency analysis of the ECG during human ventricular fibrillation," Science, Measurement and Technology, IEE Proceedings -, vol. 145, pp. 301-306, 1998.

[45] P. S. Wright, "Short-time Fourier transforms and Wigner-Ville distributions applied to the calibration of power frequency harmonic analyzers," Instrumentation and Measurement, IEEE Transactions on, vol. 48, pp. 475-478, 1999.

[46] E. Wigner, "On the Quantum Correction For Thermodynamic Equilibrium," Physical Review, vol. 40, pp. 749-759, 1932.

[47] J. Ville., "Theorie et Applications de la Notion de Signal Analytique," Cables et Transmission, pp. 1:61–74, 1948.

[48] L. L. Presti and G. Olmo, "An interpretation of the measured spectra of non-stationary integrated processes," in Information Theory, 1994. Proceedings., 1994 IEEE International Symposium on, 1994, p. 298.

[49] A. M. Sayeed and D. L. Jones, "Optimal kernels for nonstationary spectral estimation," Signal Processing, IEEE Transactions on, vol. 43, pp. 478-491, 1995.

[50] D. J. Thomson, "Multiple-window spectrum estimates for non-stationary data," in Statistical Signal and Array Processing, 1998. Proceedings., Ninth IEEE SP Workshop on, 1998, pp. 344-347.

[51] E. F. Velez and R. G. Absher, "Smoothed Wigner-Ville parametric modeling for the analysis of nonstationary signals," in Circuits and Systems, 1989., IEEE International Symposium on, 1989, pp. 507-510 vol.1.

[52] C. Hsiao-Lung, L. Jiunn-Lee, D. Chao-Cheng, and W. Chien-Ping, "Time-frequency analysis of 24-hour heart rate fluctuation for the characterization of very low-frequency component," in Engineering in Medicine and Biology Society, 1994. Engineering Advances: New Opportunities for Biomedical Engineers. Proceedings of the 16th Annual International Conference of the IEEE, 1994, pp. 1250-1251 vol.2.

[53] L. Jeongwhan, L. Joonyoung, K. Jeonghoon, K. Joonsoo, and L. Myoungho, "Power spectral estimation of HRV in patients with vasovagal syncope using time-frequency analysis," in

Engineering in Medicine and Biology Society, 2000. Proceedings of the 22nd Annual International Conference of the IEEE, 2000, pp. 953-955 vol.2.

[54] A. Z. Sha'ameri, S. Hussain, and S. Salleh, "Window width estimation and the application of the windowed Wigner-Ville distribution in the analysis of heart sounds and murmurs," in TENCON 2000. Proceedings, 2000, pp. 114-119 vol.2.

[55] P. Novak and V. Novak, "Time/frequency mapping of the heart rate, blood pressure and respiratory signals," Med Biol Eng Comput, vol. 31, pp. 103-10, Mar 1993.

[56] M. E. Tagluk, M. J. English, and R. Vincent, "Diagnostic application of a universal kernel to time-frequency energy distribution of ECG signals," in Computers in Cardiology 1997, 1997, pp. 395-398.

[57] M. Stridh, L. Sornmo, C. J. Meurling, and S. B. Olsson, "Characterization of atrial fibrillation using the surface ECG: time-dependent spectral properties," Biomedical Engineering, IEEE Transactions on, vol. 48, pp. 19-27, 2001.

[58] P. K. Kumar and K. M. M. Prabhu, "Classification of radar returns using Wigner-Ville distribution," in Acoustics, Speech, and Signal Processing, 1996. ICASSP-96. Conference Proceedings., 1996 IEEE International Conference on, 1996, pp. 3105-3108 vol. 6.

[59] G. J. Frazer and S. J. Anderson, "Wigner-Ville analysis of HF radar measurement of an accelerating target," in Signal Processing and Its Applications, 1999. ISSPA '99. Proceedings of the Fifth International Symposium on, 1999, pp. 317-320 vol.1.

[60] B. Boashash and G. Frazer, "Time-varying higher-order spectra, generalised Wigner-Ville distribution and the analysis of underwater acoustic data," in Acoustics, Speech, and Signal Processing, 1992. ICASSP-92., 1992 IEEE International Conference on, 1992, pp. 193-196 vol.5.

[61] M. Alfaouri, "Synthesizer to speech analysis based on pseudo Wigner distribution," in Electronics, Circuits, and Systems, 1996. ICECS '96., Proceedings of the Third IEEE International Conference on, 1996, pp. 497-501 vol.1.

[62] S. Ghaemmaghami and M. Deriche, "An improved model for speech excitation using time-frequency characterisation," in Signal Processing and Its Applications, 1999. ISSPA '99. Proceedings of the Fifth International Symposium on, 1999, pp. 47-50 vol.1.

[63] S. Qian and D. Chen, Joint time-frequency analysis : methods and applications. Upper Saddle River, N.J.: PTR Prentice Hall, 1996.

[64] F. Hlawatsch and F. Auger, Time-frequency analysis : concepts and tools. London

Hoboken, NJ: ISTE ;

Wiley, 2008. [65] P. Flandrin, Time-frequency/time scale analysis. San Diego: Academic Press, 1999. [66] F. Hlawatsch, Time-frequency analysis and synthesis of linear signal spaces : time-frequency

filters, signal detection and estimation, and range-Doppler estimation. Boston: Kluwer Academic Publishers, 1998.

[67] T. A. C. M. C. a. W. F. G. Mecklenbrauker, "The wigner distribution: A tool for Time-Frequency signal analysis - part 2: Discrete-Time signals," Philips Journal of Research, vol. 35, pp. 276-350, 1980.

[68] K. M. M. Prabhu and R. Shanmuga Sundaram, "Fast algorithm for pseudodiscrete Wigner-Ville distribution using moving discrete Hartley transform," Vision, Image and Signal Processing, IEE Proceedings -, vol. 143, pp. 383-386, 1996.

[69] "http://www.avtechpulse.com/appnote/general/."