Analysis of Signals and Characterization of Systems

Chih-Ping LinGeotechnical Engineering GroupNational Chiao Tung University

Outline

1. Signals and Systems

2. Time Domain Analysis

3. Frequency Domain Analysis of Signals

4. Frequency Domain Analysis of Systems

5. Discrete Inverse Theory

Signals and Systems

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Signals and Systems

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Classification of Systems

Lumped vs. distributed parameter Causal vs. non-causal Linear vs. non-linear Time-invariant vs. time-variant

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Combination of Systems

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Combination of Systems

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Classification of Signals

Multi-channel and multi-dimensional Continuous time vs. discrete time Continuous-valued vs. discrete-valued Deterministic vs. random Periodic vs. aperiodic

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Elementary Signals

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Decomposition Methods

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Applications

Computation of the output response

Identification of similarities

Noise control

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Time Domain Analysis

Convolution: Computing the output signal (in a LTI system)

Cross-Correlation: Identifying similarities

Noise control in time domain

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Convolution

Decomposition in time domain

Convolution sum

k

knkxnx )()()(

kk

k

knhkxknTkx

knkxTnxTny

),()()()(

)()()()(

)(),( knhknh

)(*)()()()( nhnxknhkxnyk

Time invariant

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Properties of Convolution

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Graphical Computation of Convolution

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Convolution sum in matrix form

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Example of convolution sum

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Determination of Impulse Response

Direct solution of the difference equation

Estimated experimentally by excite the system with a signal of short duration

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Cross-Correlation

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n

xy mnynxmr )()()(

n

xy nymnxmr )()()(

)(*)()()()( mymxmnynxmrn

xy

Definition:or

Computation:

)()( mrmr xyyx Properties:

Cross-Correlation

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Cross-CorrelationTypical example

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Application of Cross-Correlation

Identifying similarities Determination of travel time Identifying replicas in noisy signals Biases: Lossy media, dispersive and

multiple paths

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Application of Cross-Correlation

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Noise control in time domain

StackingIf xi's are independent and identical (i.i.d) Gaussian with mean and variance 2 (i.e. N(, 2) ) and y= mean(x), then y = N(, 2/N).

Smoothing with moving average

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2/)1(

2/)1(

)()()(M

Mk

kixkwiy 1)(kwwhere

)(*)()( nwnxny

Noise control in time domainStacking

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Noise control in time domainMoving average

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Frequency Domain Analysis of Signals

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The concept of frequency in continuous and discrete time

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1

2 tFtAtAtxa ),2cos()cos()(

•Continuous-time sinusoidal signals

a. For every F, xa is periodicb. Xa distinct for distinct Fc. Rate of oscillation increases with F

•Discrete-time sinusoidal signals nfnAnAnx ),2cos()cos()(

a. xa is periodic only when f is a rational numberb. Xa identical when frequencies (f) are separated

by an integer numberc. The highest rate of oscillation is attained when

|f| =1/2

Effect of sampling on frequency

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1

2

F f-Fs/2 Fs/2 -1/2 1/2F1 F3 F2 f1 f3 f2

Fi = fi *Fs

Aliasing

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The sampling theorem

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max2F

Fs

Fs/2 : Nyquist frequency

To avoid aliasing:

F is uniquely defined in –Fs/2 F Fs/2

Types of signal (revisited)

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•Periodic Signal vs. Aperiodic Signal

•Energy Signal vs. Power Signal

dttxE

2)(

n

nxE2

)(

T

TTdttx

TP

2)(

2

1lim

N

NnN

nxN

P2

)(12

1lim

•Energy signal is sufficient to guarantee the existence of Fourier Transform

•Power signal is sufficient to guarantee the existence of Fourier Series expansion

Complex Number

Complex Exponential

Euler reasoned that he needed something with half the strength of a minus sign (or the −1) so when it was squared it would equal a minus sign.

Complex Exponential

Spectral Analysis and Synthesis of Signals

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Orthogonal Expansion

Eigenvalue Problems Orthogonal Vectors Eigenvector expansion

Sturm-Liouville Problems Orthogonal Functions Eigenfunction expansion

Fourier Expansion (Spectral Expansion)

•Fourier Series (for power signals)

•Fourier Transform (for energy signals)

are basically orthogonal expansion of a continuous functionusing sinusoidal functions or complex exponential functions

Fourier Series

Spectral Analysis Sampling in time domain

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1

2

Spectral Analysis Sampling in frequency domain

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1

2

1

0

/2 1,,2,1,0 ,)()(N

n

Nknj NkenxkX

1

0

/2 1,,2,1,0 ,)(1

)(N

k

Nknj NnekXN

nx

•Discrete Fourier Transform (DFT)

•Inverse Discrete Fourier Transform (IDFT)

Duality relation

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1

2X(nt)

t0 T=1/ft

f0 f

X(nf)

TimeWindow

FrequencyWindow

F=1/t

Spectral AnalysisSummary

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2

Sampling of bandlimited signal Aliasing of spectral components

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2

Symmetry Properties of DTFT

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Properties of DTFT

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2

Symmetry Properties of DFT

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Properties of DFT

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Computation of DFT

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Fast Fourier Transform (FFT) algorithmPlots:

Practical considerationFrequency resolution – zero padding

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Practical considerationTruncation leakage - windowing

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Frequency Domain Analysis of Systems

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Does the sinusoid offer any advantage in the study of system?

Impulse Inpulse response

Sinusoid Frequency Response

System

System

The sinusoid is an eigenfunction analogous to eigenvector in matrix form•If x is an eigenvector, then y = x•If x is a sinusoid, then y=H() xAnalytical and experimental determination of H(w)Are made easier because of the eigenfunction property

Convolution

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1

2

Response of a LTI systemto periodic signals

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Response of a LTI systemto periodic signals

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2

Response of a LTI systemto aperiodic signals

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t

x(t)

t

x(t)

Time Periodof Interest

PaddingZeros

Tr

2Tr

Response of a LTI systemto aperiodic signals

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1

2

X(nt)

t0 T=1/ft

f0 f

X(nf)

TimeWindow

FrequencyWindow

F=1/t

tfF

12 max

ftT

12 max

Cross-correlation and cross-spectral density

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1

2)()()(

)}(*)({))({

fYfXfC

nynxDTFTnrDTFT

xy

xy

)()()( fXfXfCxx

)()()()()()()()( fCfHfHfXfXfYfXfC xxyx

)()()()()()()()()(2

______________

fCfHfXfHfXfHfYfYfC xxyy

Cross-spectral density

Auto-spectral density (power spectral density)

Important relations:

Noise control in frequency domain

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1

2)()()(' kWkXkX

Filters

Filters

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Determination of frequency response

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2

)(

)()(

)(

)()()()(

kX

kRkH

kX

kRkXkHkH noisy

)()()(

)()()]())()([(

)()]()()([

)()()(

kCkHkC

kXkRkHkXkX

kXkRkHkX

kXkZkCC

xrxx

xz

)(

)()(

kC

kCkH

xx

xz

)(

)()(

kX

kYkH

Measure of Noise

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4

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1

2Coherence:

)(

)()(2

kC

kCk

ZZ

YY

)()(

)()(

22

kCkC

kCk

ZZXX

XZ

)(

)()(

kC

kCkNSR

YY

NN

)(

)(1)(

2

2

k

kkNSR

Noise-to-signal ration (N/S):

Discrete Inverse Theory

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1

2

ModelParameters

mMODEL g(.)

DataMeasurement

d

Methods of solution

Exact inversion Neural network Inversion based on a forward model

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Neural network

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21

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4

1

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2

1

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3

Input layer Hidden layer Output layer

Inversion based on a forward model

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1

2Method of Solution

Exact InversionInversion Based on a

Forward ModelNeural Network

StatisticalApproach

Linear AlgebraicApproach

ClassicalInterpretation

BayesianInterpretation

Least Squares Solution If the problem y=hx is over-determined and inconsistent, there

is no unique solution, and the estimate of the solution x will be a compromise.

One alternative is to minimize the L2-norm of the vector of residuals e=y-hx. The objective function to be minimized:

The solution is found by setting the derivative of Γw.r.t. x equal to zero. The solution is found as

hxhxyhxhxyyy

hxyhxyTTTTTT

T

)()(

yhhhx TTest

1)(

Inversion based on Optimization

Matlab function fmin() or fminsearch() Examples

2cxbxay

)/(1000 bxay

Bayesian Inversion

Thank You!

Chih-Ping LinGeotechnical Engineering GroupNational Chiao Tung University