Digital Communication VTU unit 1

DIGITAL COMMUNICATION (VTU)-10EC61

UNIT - 1

Basic signal processing operations in digital communication. Sampling Principles: Sampling

Theorem, Quadrature sampling of Band pass signal, Practical aspects of sampling and signal

recovery. 7 Hours

TEXT BOOK:

1. Digital communications, Simon Haykin, John Wiley India Pvt. Ltd, 2008.

REFERENCE BOOKS:

1. Digital and Analog communication systems, Simon Haykin, John Wildy India Lts, 2008

2. An introduction to Analog and Digital Communication, K. Sam Shanmugam, John Wiley

India Pvt. Ltd, 2008.

3. Digital communications - Bernard Sklar: Pearson education 2007

Special Thanks To:

Faculty (Chronological): Arunkumar (STJIT), Raviteja B (GMIT).

Students: Shubham S Dhivagnya (6th sem GMIT)

PREPARED BY:

RAGHUDATHESH G P

Asst Prof

ECE Dept, GMIT

Davangere 577004

Cell: +917411459249

Mail: [email protected]

Quotes:

The fragrance of flowers spreads only in the direction of the wind. But the goodness of a person

spreads in all directions.

Trust is like a paper once it crumbled it cant be perfect again. Learn from the mistakes of others, you can't live long enough to make them all yourselves. As soon as the fear approaches near, attack and destroy it. Once you start a working on something, don't be afraid of failure and don't abandon it. People

who work sincerely are the happiest.

Intoduction & Sampling Theorem Raghudathesh G P Asst Professor

Department of ECE [email protected] Page No -1

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Introduction to Digital Communication

Communication:

Definition1: Communication is the process of establishing a connection or link between

two points for information exchange.

Definition2: Communication is simply the process of conveying message at a distance.

Definition3: Communication is the basic process of exchanging information.

The electronic equipments which are used for communication purpose, are called

communication equipments.

Different communication equipments when assembled together form a communication

system.

Ex: line telephony, line telegraphy, radio telephony, radio broadcasting, point-to-point

communication and mobile communication, computer communication, radar

communication, television broadcasting, radio telemetry, radio aids to navigation, radio

aids to aircraft landing etc.

Sources and Signals:

A source of information generates a message.

Ex: man voice, television picture, teletype data, atmospheric temperature pressure.

In above examples, the message is not electric in nature, thus a transducer is used to

convert it into an electrical waveform called as message signal.

The electrical waveform or message signal is also referred to as a baseband signal.

The message signal can be of an analog or digital type.

An analog signal is one in which both amplitudes and time vary continuously over their

respective intervals.

Ex: A speech signal, a television signal, and a signal representing atmospheric

temperature or pressure at some location.

In a digital signal, on the other hand, both amplitude and time take on discrete values.

Ex: Computer data and telegraph signals.

An analog signal can always be converted into digital form by combining three basic

operations:

1. Sampling

2. Quantizing and

3. Encoding



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The above Figure shown the block diagram of analog to digital conversion logic. In the sampling operation, only sample values of the analog signal at uniformly spaced

discrete instants of time are retained.

In the quantizing operation, each sample value is approximated by the nearest level in a finite set of discrete levels.

In the encoding operation, the selected level is represented by a code word that consists of a prescribed number of code elements.

The analog-to-digital conversion process described in figure above a segment of an analog waveform.

The figure above shows the corresponding digital waveform, based on the use of a binary code.

In the above example, symbols 0 and 1 of the binary code are represented by zero and one volt, respectively.

The code word consists of four binary digits (bits), with the last bit assigned the role of a sign bit that signifies whether the sample value in question is positive or negative.



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The remaining three bits are chosen to provide a numerical representation for the absolute value of a sample in accordance with Table below.

As a result of sampling and quantizing operations, errors are introduced into the digital signal. These errors are nonreversible in that it is not possible to produce an exact replica

of the original analog signal from its digital representation.

However, the errors are under a designer's control. Indeed, by proper selections of the sampling rate and code-word length (i.e., number of quantizing levels), the errors due to

sampling and quantizing can be made so small that the difference between the analog

signal and its digital reconstruction is not discernible by a human observer.



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The 3 Basic Operation/Steps Used To Convert Analog Signal Into Digital

Signal is Pictorically Represented As:



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Advantages Of Digital Communication:

1. The digital communication systems are cheaper compared to analog communication systems because of the advances made in the IC technologies.

2. In digital communication, the speech, video and other data may be merged and transmitted over a common channel using multiplexing.

3. Using data encryption, only permitted receivers may be allowed to detect the transmitted data. This property is most important in military applications.

4. Since the transmission is digital and the channel encoding is used, therefore the noise does not accumulate from repeater to repeater in long distance communications.

5. Since the transmitted signal is digital in nature, therefore a large amount of noise interference may be tolerated.

6. Since in digital communication, channel coding is used, therefore the errors may be detected and corrected in the receivers.

7. Digital communication is adaptive to other advanced branches of data processing such as digital signal processing, image processing and data compression etc.

8. We can avoid signal jamming using spread spectrum techniques.

Disadvantages of Digital Communication:

Although digital communication offers, so many advantages as discussed above, it has some

drawbacks also. However, the advantages of digital communication outweigh disadvantages. The

disadvantages are as under:

1. Large System Bandwidth: - Digital transmission requires a larger system bandwidth to communicate the same information in a digital format as compared to analog format.

2. System Synchronization:- Digital detection requires system synchronization whereas the analog signals generally have no such requirement.

3. System Complexity:- Digital system are more complex as compared to the analog systems.



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Comparison chart For Analog and Digital System:

Analog Digital

Signal

Analog signal is a continuous

signal which represents physical

measurements.

Digital signals are discrete time signals

generated by digital modulation.

Waves Denoted by sine waves Denoted by square waves

Representation Uses continuous range of values

to represent information

Uses discrete or discontinuous values to

represent information

Example Human voice in air, analog

electronic devices.

Computers, CDs, DVDs, and other

digital electronic devices.

Technology Analog technology records

waveforms as they are.

Samples analog waveforms into a

limited set of numbers and records them.

Data

transmissions

Subjected to deterioration by

noise during transmission and

write/read cycle.

Can be noise-immune without

deterioration during transmission and

write/read cycle.

Response to

Noise

More likely to get affected

reducing accuracy

Less affected since noise response are

analog in nature

Flexibility Analog hardware is not flexible. Digital hardware is flexible in

implementation.

Uses

Can be used in analog devices

only. Best suited for audio and

video transmission.

Best suited for Computing and digital

electronics.

Applications Thermometer PCs, PDAs

Bandwidth

Analog signal processing can be

done in real time and consumes

less bandwidth.

There is no guarantee that digital signal

processing can be done in real time and

consumes more bandwidth to carry out

the same information.

Memory Stored in the form of wave signal Stored in the form of binary bit

Power Analog instrument draws large

power

Digital instrument drawS only negligible

power

Cost Low cost and portable Cost is high and not easily portable

Impedance Low High order of 100 megaohm

Errors

Analog instruments usually have a

scale which is cramped at lower

end and give considerable

observational errors.

Digital instruments are free from

observational errors like parallax and

approximation errors.



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Basic Signal Processing Operations In Digital Communications:

The figure shows the block diagram of a digital communication system. In this diagram, three basic signal-processing operations are identified:

1. Source coding 2. Channel coding, and 3. Modulation.

It is assumed that the source of information is digital by nature or converted into it by design.

Source Coding: The encoder maps the digital signal generated at the source output into another

signal in digital form. The mapping is one-to-one, and the objective is to eliminate

or reduce redundancy so as to provide an efficient representation of the source

output.

As the source encoder mapping is one-to-one, the source decoder simply performs the inverse mapping and thereby delivers to the user destination a reproduction of

the original digital source output.

The primary benefit thus gained from the application of source coding is a reduced bandwidth requirement.

Channel Coding: The objective is for the encoder to map the incoming digital signal into a channel

input and for the decoder to map the channel output into an output digital signal in

such a way that the effect of channel noise is minimized.



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That is, the combined role of the channel encoder and decoder is to provide for reliable communication over a noisy channel.

Above provision is satisfied by introducing redundancy in a prescribed fashion in the channel encoder and exploiting it in the decoder to reconstruct the original

encoder input as accurately as possible.

Thus, in source coding, we remove redundancy, whereas in channel coding, we introduce controlled redundancy.

Clearly, we may perform source coding alone, channel coding alone, or the two together. In the latter case, naturally, the source encoding is performed first, followed by channel

encoding in the transmitter as illustrated in figure.

In the receiver, we proceed in the reverse order; channel decoding is performed first, followed by source decoding.

Whichever combination is used, the resulting improvement in system performance is achieved at the cost of increased circuit complexity.

Modulation: It is performed with the purpose of providing for the efficient transmission of

the signal over the channel.

In particular, the modulator (constituting the last stage of the transmitter in the figure) operates by keying shifts in the amplitude, frequency, or phase of a

sinusoidal carrier wave to the channel encoder output.

The digital modulation technique for so doing is referred to as amplitude-shift keying, frequency-shift keying, or phase-shift keying, respectively.

The detector (constituting the first stage of the receiver in the figure) performs demodulation (the inverse of modulation), thereby producing a signal that follows the

time variations in the channel encoder output (except for the effects of noise).

The combination of modulator, channel, and detector, enclosed inside the dashed rectangle shown in the figure, is called a discrete channel. It is so called since both its

input and output signals are in discrete form.

Traditionally, coding and modulation are performed as separate operations, and the introduction of redundant symbols by the channel encoder appears to imply increased

transmission bandwidth.

In some applications, however, these two operations are performed as one function in such a way that the transmission bandwidth need not be increased.



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Basic Concepts:

1. Source Coding:

This is done to reduce the size of the information (data compression) being transmitted and conserve the available bandwidth.

This process reduces redundancy.

Ex. zipping files, video coding (H.264, AVS-China, Dirac) etc.

2. Channel Coding:

This is done to reduce errors during transmission of data along the channel from the source to the destination.

This process adds to the redundancy of data. Ex. Turbo codes, convolution codes etc.

Channels for Digital Communications:

The modulation and coding used in a digital communication system depend on the characteristics of the channel.

The two main characteristics of the channel are BANDWIDTH and POWER. In addition the other characteristics are whether the channel is linear or nonlinear, and how

free the channel is free from the external interference.

Five channels are considered in the digital communication, namely: 1. Telephone channels 2. Coaxial cables 3. Optical fibers 4. Microwave radio 5. Satellite channels.

1. Telephone channel:

It is designed to provide voice grade communication. Also good for data communication over long distances.

The channel has a band-pass characteristic occupying the frequency range 300Hz to 3400hz, a high SNR of about 30db, and approximately linear response.

For the transmission of voice signals the channel provides flat amplitude response. But for the transmission of data and image transmissions, since the phase delay variations

are important an equalizer is used to maintain the flat amplitude response and a linear

phase response over the required frequency band.



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Transmission rates upto16.8 kilobits per second have been achieved over the telephone lines.

2. Coaxial Cable:

The coaxial cable consists of a single wire conductor centered inside an outer conductor, which is insulated from each other by a dielectric.

The main advantages of the coaxial cable are wide bandwidth and low external interference. But closely spaced repeaters are required.

With repeaters spaced at 1km intervals the data rate of 274 megabits per second have been achieved.

3. Optical Fibers:

An optical fiber consists of a very fine inner core made of silica glass, surrounded by a concentric layer called cladding that is also made of glass.

The refractive index of the glass in the core is slightly higher than refractive index of the glass in the

cladding. Hence if a ray of light is launched into an optical fiber at the right oblique

acceptance angle, it is continually refracted into the core by the cladding. That means the

difference between the refractive indices of the core and cladding helps guide the

propagation of the ray of light inside the core of the fiber from one end to the other.

Compared to coaxial cables, optical fibers are smaller in size and they offer higher transmission bandwidths and longer repeater separations.

4. Microwave radio:

A microwave radio, operating on the line-of-sight link, consists basically of a transmitter and a receiver that are equipped with antennas.

The antennas are placed on towers at sufficient height to have the transmitter and receiver in line-of-sight of each other.

The operating frequencies range from 1 to 30 GHz. Under normal atmospheric conditions, a microwave radio channel is very reliable and

provides path for high-speed digital transmission. But during meteorological variations,

a severe degradation occurs in the system performance.

5. Satellite Channel:

A Satellite channel consists of a satellite in geostationary orbit, an uplink from ground station, and a down link to another ground station.



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Both link operate at microwave frequencies, with uplink the uplink frequency higher than the down link frequency.

In general, Satellite can be viewed as repeater in the sky. It permits communication over long distances at higher bandwidths and relatively low cost.

Sampling theorem:

Necessity: There are two types of signals:

1. Continuous time signal and 2. Discrete time signal

Due to advance development in digital technology over the past few decades, the inexpensive, light weight, programmable and easily reproducible discrete time

systems are available. Processing of discrete-time signals is more flexible and is

also preferable to processing of continuous time signals.

For the above purpose we should be able to convert a continuous time signal into discrete-time signal.

This problem is solved by a fundamental mathematical tool known as sampling theorem.

The sampling theorem is extremely important and useful in signal processing. With the help of sampling theorem a continuous time signal may be completely

represented and recovered from the knowledge of sample taken uniformly. This

means that sampling theorem provides a mechanism for representing continuous

time signal by discrete-time signal.

Therefore, sampling theorem may be viewed a bridge between continuous time signals and discrete-time signals.

Definition: The statement of sampling theorem for low pass signal can be given in two parts

as:

1. A band limited signal of finite energy, which has no frequency-component higher than fm Hz, is completely described by its sample values at

uniform intervals less than or equal to second apart.

2. A band limited signal of finite energy, which has no frequency components higher than fm Hz, may be completely recovered from the

knowledge of its samples taken at the rate 2fm samples per second.

The first part represents the representation of the signal in its samples and minimum rate required to represent a continuous time signal into its samples.



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The second part of the theorem represents reconstruction of the original signal from its sample.

It gives sampling rate required for satisfactory reconstruction of signal from its samples.

Combining the two parts the sampling theorem may be stated as under: "A continuous-time signal may be completely represented in its samples and

recovered back if the sampling frequency is fs 2fm.

Here, fs = sampling frequency and fm = the maximum frequency present in the signal.

Classification: Ideal or Instantaneous or Impulse Sampling. Natural or Chopper sampling. Flat top sampling.

Proof of Sampling Theorem For Low Pass (LP) Signals:

Consider an analog signal g(t) that is continuous in time and amplitude. We assume that g(t) has infinite duration, but finite energy. Let the sample values of the signal g(t) at times t = 0, Ts, 2Ts, 3Ts,., be denoted by

the series{g(n Ts), n = 0, 1,2,}.

Here, Ts = sampling period and fs (1/ Ts) = sampling rate

A segment of the signal g(t) is as shown below



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Proof involves two parts: 1. Representation of g(t) interms of its samples. 2. Reconstruction of g(t) from its samples.

Proof: The discreate time signal, , resulted from the sampling process is given as below,

-------------- (1)

In the above equation delta function is located at time t = nTs. Equation 1 can be written as,

--------- (2)

Here, = Dirac comb or ideal sampling function.

Applying Fourier transform to the above equation. Let G(f) and G(f) denotes the fourier transforms of g(t) and g(t) respectively, we get,

--------- (3)

From the above equation we see that G(f) represents a spectrum that is periodic in the frequency f with period fs.



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The process of uniform sampling a signal in time domain results in a periodic spectrum in the frequency domain with a period equal to the sampling rate.

Rewriting the above equation we get,

------------ (4)

Above equation is passed through the lowpass filter we get,

--------- (5)

Applying fourier transform to equation (2) we get,



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--------- (6)

Putting equation (6) in equation (5) and fs = 2W we get,

------- (7)

If the sample values

of the signal are known for all time n, then the Fourier

transform G(f) of the signal is determined uniquely by the equation (7).

Reconstruction Of The Signal g(t) From Its Sequence Of Sample Values

:

The process of reconstructing a continuous time signal g(t) from its samples is called interpolation.

Consider the equation below:

-------- (1)

Appling Inverse Fourier Transform (IFT) to the above equation



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Here the limits of integration can be taken from W to +W, also interchanging the order of integration and summation and combining the exponential terms we get,

But

and here thus,

We know that

, thus above equation is reduced to,

-------------- (2)



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The above Equation is known as Interpolation formula for reconstruction of g(t) from its

sequence of sample values

where each sample is multiplied by a delayed Sinc

function.

Important Points Relating to Sampling:

Effect on Spectrum Due to the selection of Sampling Rate:

Case 1: fs > 2W

Figure above shows the spectrum of sampled signal. the spectrum G(W) or G(f) will repeat periodically without overlapping. The spectrum of sampled signal extends up to infinity and the ideal bandwidth of

sampled signal is infinite.

The original or desired spectrum is centered at W = 0 or f = 0 and is having bandwidth or maximum frequency equal to Wm. The desired spectrum may be recovered by passing

the sampled signal with spectrum G(w) through a low pass filter with cut-off frequency

Wm.

This means that since a low-pass filter allows to pass only low frequencies up to cut-off frequency (Wm) and rejects all other higher frequencies, the original spectrum extended

upto Wm will be selected and all other successive higher frequency cycles in the

sampled-spectrum will be rejected. Therefore, in this way, original spectrum will be

extracted out of spectrum G(w). This original spectrum can now be converted into time-

domain signal.

It may also be observed from figure that for the Case fs > 2fm, the successive cycles of G(w) are not overlapping each other. Hence in this case, there is no problem in

recovering the original spectrum.

In this case there will be guard band between components of the spectrum components.



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Case 2: fs = 2W:

For this case the successive cycles of G(w) are not overlapping each other, but they are touching each other.

In this case also, the original spectrum can be recovered from the sampled spectrum G(w) using a low-pass filter with cut-off frequency Wm ideally.

Practically no filter has a sharp roll-off. Thus, practically it is not possible to recover tha signal g(t).

Case 3: fs < 2W:

The successive cycles, of the sampled spectrum will overlap each other and hence in this case, the original spectrum cannot be extracted out of the spectrum G(w).



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Thus, the output of LPF will have distortion due to the unwanted frequency component. This is known as Aliasing. This type of sampling is known as Under Sampling.

Hence, for reconstruction without distortion, Sampling Frequency must be fs

2fm

Basic Concepts:

3. Nyquist Rate:

When the sampling rate becomes exactly equal to 2 fm samples per second, then it is called Nyquist rate.

Nyquist rate is also called the minimum sampling rate. It is given by

fs 2fm

4. Nyquist Interval:

Maximum sampling interval is called Nyquist interval. It is given by

Effect of Under Sampling (Aliasing):

When a continuous rime band limited signal is sampled at a lower than Nyquist rate fs < 2fm, then the successive cycles of the spectrum G(w) of the sampled signal g(t) overlap

with each other as shown below



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Here the signal is under-sampled and some amount of aliasing is produced in this under-sampling process.

In fact, aliasing is the phenomenon in which a high frequency component in the frequency-spectrum of the signal takes identity of a lower-frequency component in

the spectrum of the sampled signal.

Because of the overlap due to aliasing phenomenon, it is not possible to recover original signal from sampled signal g(t) by low-pass filtering since the spectral components in the

overlap regions add and hence the signal is distorted.

Since any information signal contains a large number of frequencies, so, to decide a sampling frequency is always a problem. Therefore, a signal is first passed through a low-

pass filter. This low-pass filter blocks all the frequencies which are above fm Hz. This

process is known as bandlimiting of the original signal.

This low-pass filter is called prelias filter because it is used to prevent aliasing effect. After band limiting, it becomes easy to decide sampling frequency since the maximum

frequency is fixed at fm Hz.

Steps to avoid aliasing: 1. Prealias filter must be used to limit band of frequencies of the signal to fm Hz. 2. Sampling frequency 'fs must be selected such that fs > 2 fm.

Sampling of Bandpass Signal:

Definition: A bandpass signal say x(t) whose maximum bandwidth is 2W can be completely represented into and recovered from its samples if it is sampled at the

minimum rate twice the bandwidth( i.e, 4W).

Generation of In and Quadrature component of a bans pass signal:



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In this scheme, the band pass signal is converted into lowpass signals by multiplying bandpass signals with two sinusoidal signals which are phase Quadrant with each

other results in into two components, one is in-phase component and other is

quadrature component.

These two components will be lowpass signals and are sampled separately. This form of sampling is called quadrature sampling.

Let g(t) be a band pass signal, of bandwidth 2W centered around the frequency, fc,(fc>W). The in-phase component, gI(t) is obtained by multiplying g(t) with

cos(2fct) and then filtering out the high frequency components.

Parallelly a quadrature phase component is obtained by multiplying g(t) with sin(2fct) and then filtering out the high frequency components..

The band pass signal g(t) can be expressed as,

---------- (1)

The in-phase, gI(t) and quadrature phase gQ(t) signals are lowpass signals, having band limited to (-W < f < W). Accordingly each component may be

sampled at the rate of 2W samples per second as shown.

The spectrum of bandpass signal and inphase and quadreture component is as shown below.

Reconstruction: From the sampled signals gI(nTs) and gQ(nTs), the signals gI(t) and gQ(t) are

obtained.

To reconstruct the original band pass signal, multiply the signals gI(t)and gQ(t) by cos(2fct) and sin(2fct) respectively and then add the results.



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Practical Aspects of Sampling and Signal Recovery:

In the previous section we studied impulse or ideal sampling. The samples were impulses of height equal to modulating signal at sampling instant.

But this type of ideal sampling is not possible in practice. Practical sampling differs from ideal sampling in following aspects:

1. The sampled pulses have finite duration rather than impulses. The amplitude of these pulses is also finite.

2. The practical signal to be sampled are not strictly bandlimited. Therefore, there is always problem in selecting precise sampling frequency.

3. The practical reconstruction filters are not ideal filters. The practical reconstruction filters need a guard band or gap between the spectrums of sampled

signal.

Basically, there are two types of sampling techniques as under: Ideal sampling:

Instantaneous sampling. practical sampling:

Natural sampling or chopper Sampling Flat-top sampling or rectangular pulse sampling

Natural sampling and flat top sampling are practical sampling methods. Flat top normally preferred because of its noise immunity. The signals to be sampled are not strictly bandlimited, there is always chance of aliasing. Aliasing occurs because of some high frequency spurious signals in such cases. To avoid this problem, normally prealias filters are used before sampling. These prealias

filters are normally lowpass bandlimiting filters.



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Natural Sampling or Chopper Sampling:

Drawback of Instantaneous Sampling: 1. The instantaneous sampling results in the samples whose width approaches zero.

Due to this, the power content in the instantaneously sampled pulse is negligible.

2. Thus, this method is not suitable for transmission purpose. Operational Logic:

In natural sampling the pulse has a finite width equal to . Consider an analog continuous-time signal x(t) to be sampled at the rate of fs Hz.

Here it is assumed that fs is higher than Nyquist rate such that sampling theorem

is satisfied.

Consider a sampling function c(t) which is a train of periodic pulses of width and frequency equal to fs Hz.

Figure shows a functional diagram of a natural sampler. With the help of this natural sampler, a sampled signal g(t) is obtained by multiplication of sampling

function c(t) and input signal x(t).

In the figure when c(t) goes high the switch S is closed. Therefore,

g(t) = x(t) when c(t) = A

g(t) = 0 When c(t) = 0

Where A is the amplitude of c(t). The waveforms of signals x(t), c(t) and g(t) have been illustrated in figure (a), (b) and (c)

below respectively.



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Now, the sampled signal g(t) may also be described mathematically as

------- (1)

Here, c(t) = the periodic train of pulse of width and frequency fs.

Exponential Fourier series for any periodic waveform is expressed as

-------- (2)

Also, for the periodic pulse train of c(t). we have

Therefore according to equation (2) for periodic pulse train c(t), we get

-------- (3)

Here

As c(t) is a rectangular pulse train, thus Cn for the this waveform is given as,



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------- (4)

Here, TA = pulse width =

fn = harmonic frequency = nfs.

Thus, equation (4) can be written as,

------ (5)

Substituting equation (5) in (3) we get,

-------- (6)

Substituting equation (6) in (1) we get the required time-domain representation for naturally sampled signal x(t),

---------- (7)

In order to get the frequency- domain representation of the naturally sampled signal g(t), we take the Fourier transform of the above signal

----- (8)

But,

But, fn = harmonic frequency = nfs Thus the spectrum of naturally sampled signal is given as below,



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-------- (9)

Thus, the above equation shows that the spectrum x(t) i.e, X(f) are periodic in fs and are weighted by the sinc function.

Flat Top Sampling or Rectangular Pulse Sampling:

Flat top sampling is a practically possible sampling method. Natural sampling is little complex whereas it is quite easy to get flat top samples. In flat-top sampling or rectangular pulse sampling, the top of the samples remains

constant and is equal to the instantaneous value of the baseband signal x(t) at the start of

sampling.

The duration or width of each sample is and sampling rate is equal to fs = 1/Ts. Figure (a) below shows the functional diagram of a sample and hold circuit which is used

to generate the flat top samples.

Figure (b) below shows the general waveform of flat top samples.



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From figure (b), it may be noted that only starting edge of the pulse represents instantaneous value of the baseband signal x(t).

Also the flat top pulse of g(t) is mathematically equivalent to the convolution of instantaneous sample and a pulse h(t) as depicted in figure (c) below.

This means that the width of the pulse in g(t) is determined by the width of h(t) and the sampling instant is determined by delta function.

In figure (b), the starting edge of the pulse represents the point where baseband signal is sampled and width is determined by function h(t). Therefore g(t) will be expressed as

----- (1)

This equation has been explained in figure (d) below.



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The train of impulses may be represented mathematically as

------ (2)

The signal s(t) is obtained by multiplication of baseband signal x(t) and ,

------ (3)

Now, the sampled signal g(t) is given by equation (1) as,

Thus,

------ (4)

According to the shifting property of delta function,

------- (5)

From equation (4) and (5),

------- (6)



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This equation represents value of g(t) in terms of sampled value x(nTs) and function h(t-nTs) for flat top sampled signal.

Now, again from equation (1), we have,

Taking Fourier transform of both sides of above equation we get,

------ (7)

But S(f) is given as,

------- (8)

Substituting equation (8) in (76) we get,

------- (9)

The above is the equation for the spectrum of flat top sampled signal.

Aperture Effect:

The spectrum of flat top sampled signal is given by,

This equation shows that the signal g(t) is obtained by passing the signal s(t) through a filter having transfer function H(f).

The corresponding impulse response h(t) in time domain is as shown in figure below.



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The above is one pulse of rectangular pulse train where each sample of x(t)(i.e., s(t)) is convolved with this pulse.

The spectrum of one rectangular pulse of h(t) is as shown in figure below.

The spectrum of one rectangular pulse is expressed as,

------ (1)

Seeing figure above, it may be observed that by using flat top samples an amplitude distortion is introduced in the reconstructed signal x(t) from g(t).

In fact, the high frequency roll off H(f) acts like a low-pass filter and thus attenuates the upper portion of message signal spectrum. These high frequencies of x(t) are affected.

This type of effect is known as aperture effect.

The aperture effect can be compensated by: 1. Reducing the pulse width of h(t) very small.

Using equalizer in cascade with the reconstruction filter in the receiver as shown below we can compensate for attenuation,

Equalizer used in cascade with the reconstruction filter has the effect of decreasing the inband loss of the reconstruction filter as the frequency increases in such a way as to

compensate for the aperture effect.



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Also, the transfer function of the equalizer is expressed as,

------ (1)

Here td' is known as the delay introduced by low-pass filter which is equal to .Thus,

------- (2)

Which is the transfer function of an equalizer.

Sample and Hold Circuit for Signal Recovery:

Sampling process may takes the form of: 1. Natural sampling or 2. Flattop sampling.

In both cases, the spectrum of the sampled signal is scaled by the ratio T/Ts . Here,

T = sampling-pulse duration and

Ts = sampling period.

Typically this ratio is quite small, with the result that the signal power at the output of the lowpass reconstruction filter in the receiver is correspondingly small.

Obvious remedy to this situation is the use of amplification, which can be quite large. A more attractive approach, however, is to use a simple sample and hold circuit.

The circuit, shown in the figure above, consists of an amplifier of unity gain and low output impedance, a switch, and a capacitor; it is assumed that the load impedance is

large.



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The switch is timed to close only for the small duration T of each sampling pulse, during which time the capacitor rapidly charges up to a voltage level equal to that of the input

sample.

When the switch is open, the capacitor retains its voltage level until the next closure of the switch.

Thus the sample-and-hold circuit, in its ideal form, produces an output waveform that represents a staircase interpolation of the original analog signal, as shown in figure

below.

With reference to flat-top sampling, the output of a sample-and-hold circuit is defined as,

-------- (1)

Here, h(t) = impulse response representing the action of the sample and hold circuit i.e.,

-------- (2)

Spectrum of the output of the sample-and-hold circuit given by,

------- (3)

Here, G(f) = the Fourier transform of the original analog signal g(t).

H(f) = transfer function of the sample and hold circuit and is given by

-------- (4)



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In order to recover the original signal g(t) Without distortion, we need to pass the output of the sample-and-hold through a low-pass filter designed to remove components of the

spectrum U(f) at multiples of the sampling rate fs and an equalizer whose amplitude

response equals 1/|H( f)|.

These operations arc illustrated by the block diagram shown in figure below



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Comparison of Various Sampling Techniques:

Sl

No

Parameter of

comparison

Ideal or

instantaneous

sampling

Natural sampling Flat top Sampling

1 Sampling

principle

It uses

multiplication

It uses chopping principle It uses sample and

hold circuit

2 Generation

circuit

3 Waveforms

involved

4 Feasibility This is not a

practically

possible method

This method is used practically This method is also

used practically

5 Sampling rate Sampling rate

tends to infinity

Sampling rate satisfies Nyquist

criteria

Sampling rate

satisfies Nyquist

criteria

6 Noise

interference

Noise interference

is maximum

Noise interference is minimum Noise interference

is maximum

7 Time domain

representatio

n

8 Frequency

domain

representatio

n



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Formulas

1) Nyquist rate

2) Sampling period

3) Angular Frequency

4)

5)

6)

7) For Band Pass Signal :Inphase and quadrature component

8)

9)

10)

11) consider 3 frequencies Then



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Problems:

1. Consider the cosine wave . Plot the spectrum of discrete time signal

derived by sampling g(t) at the times =

where n=0, ,and

(i)

(ii)

(iii)

repeat your calculations for the sine wave ,comment on the results.

Solution:

given

Taking FT on both sides of equation (1)

Spectrum of message signal g(t).

W.k.t

i) =



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=

=

2. The signal g(t)=10 cos(20 cos(200 is sampled at the rate of 250 samples per

second.

a) Determine spectrum of resulting sampled signal.

b) Specify the cutoff frequency of ideal reconstruction filter so as to recover g(t)

from its sampled version.

c) What is the Nyquist rate for g(t)?

Solution:

given

Taking FT on both sides of above equation



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The spectrum of signal g(t) is drawn using above expression

Recall; )

Using above equation spectrum is drawn as follows .it should be noted that is

symmetric about vertical axis and is periodic with a period equal to .

(b) the cutoff frequency of ideal LPF should be greater than 110HZ and less than 140HZ for

recovering g(t) from .



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(c) Nyquist rate , =2W, where W is highest frequency in g(t).thus

3. A signal g(t) consists of two frequency components =3.9KHZ and =4.1KHZ in such

a relationship they cancel out each other when g(t) is sampled at the instants

t=0,T,2T,..,where T=125 micro sec. The signal g(t) is defined by

Find the values of A and of the second frequency component.

Solution:

given

Also given in problem that, g(nT)=0

Hence ,

i) Let n=0 Then, equation (1) becomes

A 0, we get =

ii) Let n=1; equation (1) becomes



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The above equation is satisfied only when A = 1 and +ve sign for the second term is taken. The

+ve sign can appear in the above equation, only when

in equation (1). Thus , A=1 and

.

4. If E denotes the energy of a strictly band limited signal g(t), then prove that

Where W is the highest frequency component of g(t).

Solution:

If g(t) is band-limited to , we may write g(t) as

But

Put (1) in (2)

But

Hence equation (3) becomes



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4. The signal (t) = 10 cos(100t) and (t) =10 cos50t are both sampled with = 75Hz.

Show that the two sequences so obtained are identical.

Solution:

given (f) = 5[ (f 50) + (f + 50)]

With = = 75Hz, the spectrum of sampled version of (t) is

(f) =

=

Similarly ,the ft of (t) is

(f) =

Letting m=l 1 in the first term of the equation (2) and m=k 1 for the second term we get

(f) = 375 + 375

(3)

Since l and k are dummy variables ,they can be replaced by m. Consequently equation (3)

becomes

Hence from equation (1) and (3) sampled version of (t) and (t) are identical.



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5. The spectrum of bandpass signal g(t) has bandwidth of 0.6KHz centered around

12KHz.find the Nyquist rate for quadrature sampling of inphase and quadrature

components of the signal g(t).

Solution:

The signal g (t) can be expressed in terms of inphase and quadrature components as

Where (t) and (t) are lowpass signals with a bandwidth of

The Nyquist rate for (t) and (t) is therefore

=2W = 20.3 = 0.6KHz.

6. A signal g(t) = 2cos(400t) + 6cos(640t) is ideally sampled at 500Hz. If the sampling

signal is passed through an ideal lowpass filter with a cutoff frequency of 400Hz ,what

components will appear in filter output?

Solution:

g(t) = 2cos(400t) + 6cos(640t) (1)

From eq (1)

=400 ,hence = 200Hz.

= 640, hence = 320Hz.

taking FT on both sides of the eq (2) we get



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Using eq (3) ,the spectrum of the signal g(t) is drawn and it appears as shown below

ii) W.k.t ; )

)

f- -500n)+(f +200 -500n) ]

+ 3 f- -500n)+(f +320 -500n) ] } (4)

The spectrum of the sampled signal is drawn using eq (4)

The components that are appeared at the filter o/p are 180Hz, 200Hz, 300Hz, 320Hz.

7. An analog signal is expressed by the equation

. Calculate the nyquist rate and nyquist interval for this signal.

Solution:



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eq(1) is in the form

comparing equation (1) and (2)

hence

=25Hz

= 150Hz

= 50Hz.

= max( he e = 150Hz Nyquist rate =2 = 2150Hz

=300Hz

Nyquist interval

= 3.3msec.

8. Find the Nyquist rate and Nyquist interval for the signal

Solution:

Given signal

w.k.t

Equation (2) is in the form of



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=1500Hz

= 2500Hz

= max( . hence = 2500Hz

Nyquist rate =2 = 22500Hz

=5000Hz

Nyquist interval

= 0.2msec.

9. The signal is sampled at rate of 500 samples/sec.

i) Determine the spectrum of the resulting sampled signal .

ii) What is the nyquist rate for g(t).

iii) Specify the cutoff frequency of ideal reconstruction filter .

Solution:

Given

w.k.t

from eq (2)

=198Hz

= 202Hz

= max( he e = 202Hz taking ft on both sides of eq (2) ,we get

spectrum of G(f) is as shown in below figure:



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i) w.k.t )

f- - 500n) + G(f +198 - 500n) ]

+ f- -250n)+G(f +110 -250n)]}

ii) fig

Cutoff frequency of ideal reconstruction filter is 202Hz.

iii) Nyquist rate =2 = 2202Hz =404Hz.



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11. A continuous time is given as

Determine

i) Minimum sampling rate i.e ,Nyquist rate required to avoid aliasing.

ii) If sampling frequency = 400Hz . what is the discrete time signal x[n] or

x[n ] obtained after sampling?

iii) What is the frequency

of the sinusoidal that yields samples

identical to those obtained in part ii).

Solution:

i) The highest frequency component of continuous time signal is f =100Hz.hence minimum sampling rate required to avoid aliasing is called Nyquist rate and is give

Given as Nyquist rate =

=200Hz.

ii) The continuous time signal x(t) is sampled at = 400Hz . The frequency of discrete time signal will be

Then, the discrete time signal is given by

iii) The continuous time signal x(t) is sampled at = 150Hz . the frequency of discrete time signal will be

Then the discrete time signal will be given by

iv) For sampling rate of = 150Hz

Then, the sinusoidal signal will be



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Sampling at = 150Hz, yields identical samples hence f= 100Hz is an aliasing of f= 150Hz.

12. Determine the Nyquist rate for a continuous time signal

Solution:

In general form, any continuous time signal can be written as

and the given signal is

On comparing the signal (1) and (2) ,we obtain the frequencies of given signal as follows

=

=50/2 =25HZ

=

=300/2 =150HZ

=

=100/2 =50HZ

Thus highest frequency component of given message signal will be

= 150Hz

Therefore, Nyquist rate = 2 =2 x 150 =300Hz.

13. Figure shows spectrum of the arbitrary signal x(t) . This signal is sampled at the

Nyquist rate with a periodic train of triangular pulses of duration 50/3. Milliseconds

.Determine the spectrum of the sampled signal for frequencies upto 50Hz giving relevant

expression .

Solution:

from fig it may be observed that signal is bandlimited to 10Hz.

Thus = 10Hz



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So Nyquist rate is = 2 = 210 = 20Hz

Since the signal is sampled at Nyquist rate,the sampling frequency would be

= 20Hz

Given that the rectangular pulses are used for sampling i.e flat top sampling is used.

The spectrum of flat top sampled signal is given by

Value of H(f) is expressed as

Here is the value of rectangular pulse used for sampling .

The given value of rectangular sampling pulse duration is 50/3 milliseconds.

=0.05/3 sec

Substituting the value of in eq(2) we get

Again, putting this value of H(f) and in equation(1) , we get

( =20)

This expression gives the spectrum up to 60Hz (since n = 3) for the sampled signal.

14. A flat top sampling system samples a signal of maximum frequency with 3.5Hz

sampling frequency. The duration of the pulse is 0.2 sec. Compute amplitude distortion due

to aperture effect at the highest signal frequency. Also determine the equal characteristic.

Solution:

Given the sampling frequency = 2.5Hz



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Maximum signal frequency = 1Hz and pulse width = 0.2sec.

We know that the aperture effect is expressed by transfer function H(f) as

The magnitude of this equation will be

Now, aperture effect at the highest frequency will be obtained by putting f = = 1Hz in eq(1)

Also, equaliser characteristic is expessed as

(f)

Putting = 0.2 sec and assuming

K =1, the last equation becomes

(f) =

This equation is plot of (f) versus f and it represents the equalisation characteristic to

overcome aperture effect.



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Things to be Known:

Signals and Their Spectrum:

1. Single Sine Wave: f(x) = Sin (x)

2. Sum of Two Sine Waves f(x) = Sin (x) + Sin (2x)

3. Box Function:



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4. Wider Box Function:

5. Triangular Function:

5. Comb Function:

6. Wider Comb Function:



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Commonly Used Filters for Reconstruction:

1. Constant Filter:

2. Linear Filter:

3. Catmull-Rom Filter:



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4. B-Spline Filter:

5. Truncated Sinc Filter:

6. Full Sinc Filter:



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7. Lanczos Filter:

VTU Questions

1. Explain the sampling theorem for low pass signals and derive the interpolation formula. June

2013 (09 M)

2. With a neat block diagram, explain the scheme for signal reconstruction for practical

sampling. June 2013 (06 M)

3. If E denotes the energy of a strictly band limited signal g(t), then prove that

Where W is the highest frequency component of g(t). June 2013 (05 M)

4. Explain natural sampling with relevant waveforms. Give all necessary time-domain and

frequency domain equations. June 2014 (10 M)

5. What is aliasing error? Give 2 corrective measures to remove the effect of aliasing in practice.

June 2014 (04 M)

6. Consider the analog signal x(t) = 5 cos(2000t) + 10 cos(6000t).

i. What is Nyquest rate and Nyquist interval?

ii. Assume that if we sample the signal using sampling frequency Fs = 5000 Hz, what is

the resulting discrete time signal obtained after sampling?

iii. Draw the spectrum of the sapling signal. June 2014 (04 M)



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Documents

Digital Communication VTU unit 1