introduction to the sampling theorempropagation.ece.gatech.edu/ee3614_vt/files/lecture8.pdfecpe 3614...

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Copyright Gregory D. Durgin 2000VIRGINIA POLYTECHNIC INSTITUTEAND STATE UNIVERSITY

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ECPE 3614 Introduction to Communication Systems

Introduction to the Sampling Theorem

ECPE 3614Lecture 8

Instructor: Gregory D. Durgin


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ECPE 3614 Introduction to Communications Systems

L82

Outline of Lecture

§Review of Time Replication and Frequency Sampling§Duality Result for Time Sampling and

Frequency Replication§Aliasing and the Nyquist Rate§Example 1: Impulsive Sampling§Example 2: Non-Impulsive Sampling§Conclusions

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L83

Review: Homework #3, Problems 1 and 2.

In this problem, a gated exponential function is shifted and copied in time to produce a periodic function, y(t).


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L84

Replication in Time to Make a Periodic Function

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1

t (seconds)

sign

al

Exponential

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1

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t (seconds)

sign

al

Exponential Sawtooth

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L85

Fourier Transform Pairs

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t (seconds)

sign

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Exponential

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2

4

6

8

10

12

14

f (Hz)

spec

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Fourier Transform of Exponential


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L86

Fourier Transform Pairs

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t (seconds)

sign

al

Exponential Sawtooth

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f (Hz)

spec

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Fourier Transform of Exponential Sawtooth

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L87

Replication Problem

s(t) Replicator( )h t

A Replicator can be modeled as a linear time-invariant (LTI) system that takes an input, s(t) and turns it into a periodic function y(t).


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L88

The Operation of Replication

Since the Replicator is an LTI system, it has an impulse response, h(t). The impulse response is just an impulse train of period T:

Thus, we can write

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L89

Show that the Convolution of s(t) and h(t) is y(t):

Is the act of replication a realistic filter? No, because h(t) is not causal. However, it is a powerful way to think about the act of replication both in the time domain and…


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L810

… In the Frequency Domain

In the frequency domain, a convolution becomes a :

The Fourier Transform of an impulse train is an impulse train:

multiplication

impulse train

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L811

Multiplication By an Impulse Train Samples the Spectrum

-8 -6 -4 -2 0 2 4 6 80

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f (Hz)

spec

trum

Fourier Transform of Exponential Sawtooth

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2

4

6

8

10

12

14

f (Hz)

spec

trum

Fourier Transform of Exponential


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L812

Summary of Time Replication

t

Replication = Convolution by an Impulse Traint

t

y t( )

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L813

Summary of Frequency Sampling

f

Sampling = Multiplication by an Impulse Train

Sampling in the Frequency Domain is Replication in the Time Domain


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L814

The $50 Question

If Replication in the time domain is Sampling in the frequency domain, then what frequency-domain operation is equivalent to Sampling in the time domain? Answer: By Duality, we can show that Sampling in the time domain is really Replication in the frequency domain. (In other words, multiplication by an impulse train in the time domain is really convolution of an impulse train in the frequency domain.)

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L815

Time Domain Sampling

t

y nT( ) T

t

y t( )

f

Y( ) f

f

In the Frequency Domain…

In the Time Domain…


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L816

Recovering the Original Baseband Signal

y t( ) y t( )

LPFSampler

t tt

y nT( ) T

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L817

Recovering the Original Baseband Signal

f

f

Y( ) f

H ( )LP f 2B

f

Sample Signal:

Ideal Low-Pass Filter:

Original Signal:


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L818

Mathematical Representation of the Recovery Operation

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L819

Lot’s of Math – Yuck!

We can simplify this equation by performing several operations. The end result on the next page is one of the most famous equations of all time.


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L820

Shannon’s Sampling Theorem

Shannon’s sampling theorem for band-limited signals:

Published by Claude Shannon in 1948 in his famous paper “The Mathematical Theory of Communications”. This equation is the main reason why our technology is becoming more and more digital.

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L821

How Can the Sampled Signal Contain ALL This Information?

t

y t( )

t

y nT( )

In sampling, it would seem as though we have thrown away an infinite amount of information by representing a continuous function segment with a finite number of data points. But this is really the case, provided that a certain criterion is meet…


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L822

Effects of Sampling Interval Size on Spectral Replication

t

y nT( ) T

f

R 1/TS=

The Sampling Period, T, is the spacing between samples in the time domain. The Sampling Rate, RS, is the spacing between replicas in the frequency domain.

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L823

Effects of Sampling Interval Size on Spectral Replication

Decreasing the Sampling Period, T, will increase the Sampling Rate, RS, and the spacing between replicas in the frequency domain.

t

y nT( )T

f

R 1/TS=


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L824

Decreasing Sampling Rate(Increasing Sampling Period)

Does anyone anticipate a problem for small sampling rates?

For small values of RS (large values of T), the spectral replicas begin to overlap. If RS becomes too small, then the replicas overlap and it becomes impossible to filter out the exact copy of the original signal. This effect is called aliasing.

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L825

Aliasing (Imperfect Sampling)

f

R 1/TS=

If the sampling rate is not large enough, then the spectral replicas overlap. Notice that passing the sampled signal through a band-pass filter does not recover the original signal. Aliasing causes the loss of information.


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L826

The Nyquist Rate

f

R 1/TS=

The absolute minimum rate that a band-limited signal may be sampled without losing information to aliasing is the Nyquist rate. The Nyquist rate is equal twice the bandwidth of a baseband signal, 2fm (or simply W if you are measuring total width).

f

fm

W

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L827

Matlab Simulation Procedure

§Generate a Random Band-Limited Signal§Sample the Band-Limited Signal in the time

domain with an impulse train§Pass the sampled signal through an Ideal Low-

Pass Filter (ILPF)§Compare the original signal with the recovered

signal.§Repeat the exercise for decreasing sampling

periods, T.


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L828

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x(t)

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|X(f)

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e 1

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L829

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time, s

Σ δ(

t-nT)

Impulse Train

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frequency, f

1/T

Σ δ(

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L830

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time, s

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Sampled x(t) for T=0.156s

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L831

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xn(t)

Sampled x(t) through ILPF

Recovered x(t)Original x(t)

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L832

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L833

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Impulse Train

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L834

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L835

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L836

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Input Signal

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L837

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time, s

Σ δ(

t-nT)

Impulse Train

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10

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frequency, f

1/T

Σ δ(

t-n/T

)

Cas

e 3


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L838

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L839

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L840

Non-Impulsive Sampling

§Any periodic waveform may be used to sample a band-limited function in the time domain – not just an impulse train.§A periodic waveform in the time domain produces

a non-uniform impulse train in the frequency domain.§Sampling in the time-domain with a non-

impulsive periodic waveform is equivalent to non-uniform replication in the frequency domain.

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L841

Matlab Simulation Procedure

§Generate a Random Band-Limited Signal§Sample the Band-Limited Signal in the time

domain with periodic rectangular pulses§Pass the sampled signal through an Ideal Low-

Pass Filter (ILPF)§Compare the original signal with the recovered

signal.§Repeat the exercise for decreasing sampling

periods, T.


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L842

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frequency, f

|X(f)

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e 1

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L843

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ave

On-and-Off Switching

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L844

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L846

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L847

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L848

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L849

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L850

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L851

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L852

Cas

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L853

Cas

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|Xn(

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L854

Key Points of Lecture

§Replication in the Time Domain is Sampling in the Frequency Domain

§ Sampling in the Time Domain is Replication in the Frequency Domain

§ If a Band-limited signal is sampled at a high enough rate, it may be recovered perfectly by passing the sampled signal through a low-pass filter.

§Aliasing occurs when a signal is sampled less than the Nyquist rate (twice the bandwidth).

§Non-Impulsive periodic waveforms may be used to sample as well – they simply produce non-uniform frequency-domain replicas of the original signal.

introduction to the sampling theorempropagation.ece.gatech.edu/ee3614_vt/files/lecture8.pdfecpe 3614...

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