9.1 Sampling Theorem

9.1 Sampling Theorem

In the sampling procedure, a continuous-time signal is represented by a

sequence of numbers (samples) that represent the signal values at the particular

time instants. The sampling process is performed, in general, at the equidistant

time instants so that the samplingperiod s (the time betweentwo adjacent

samples) is constant. Thesamples of the continuous-time signal are defined

by s . It should be pointed out that “thesampling

processrepresents a very drastic chopping operation on the original signal,

and therefore,it will introduce a lot of spurioushigh-frequency components into

the frequencyspectrum,” Orfanidis,Digital Signal Processing, PrenticeHall, 1996.

Sampling isa mandatory step in preparingsignal data for digital computer

processing. Sincereal physical systems operate in continuous-time, at some point

we must recover thecontinuous-time signal from its discrete-time version (from its

The slides contain the copyrighted material from Linear Dynamic Systems and Signals, Prentice Hall, 2003. Prepared by Professor Zoran Gajic 9–1

sample values).We would like to perform this operation as accurately as possible

and without generatingredundant data thatwill put unnecessary computational

burden. To that end, anatural question tobe asked is:What isthe minimalvalue

for the samplingperiod s such thatthe originalsignal can beuniquely (at alltime

instants) reproducedfrom its discrete-time values? Anothermore general question

would be: Can all continuous-time signals be discretized suchthat a meaningful

(with tolerable errors) recovery procedurecan be performed? The answers tothese

two fundamental questions are givenin the sampling theorem.

First, we give the definitions of bandlimited and time-limited signals.

Definition 9.1: A signal is bandlimited if its magnitude spectrum is equal to

zero for all frequencies greater thanmax. The frequency max max is

called the signal bandwidth frequency.


Definition 9.2: A signal is time-limited if its is different from zeroonly in a

finite time interval, say min max .

It can be shown thata signal can not be both time-limited and bandlimited. The

proof of this interesting signal property is outside of the scope of this course.

A bandlimited signal magnitude spectrum is presented in Figure 9.1.

X(j )ω

0

ω−ω

max

ωmax

Figure 9.1: A bandlimited signal magnitude spectrum

Now, we state the sampling theorem.


Theorem 9.1: Sampling Theorem

A continuous-time bandlimited signal with the bandwidth frequencymax

can be uniquely reconstructed from its sample values s

if the sampling frequencys s satisfies

ss

max

The frequency max is called theNyquist frequency, and the frequencyinterval

max max is called theNyquist interval.

The sampling theoremis often called Shannon’s sampling theorem in honor of

his celebrated paperpublished in 1949. Note thatthe main results stated in the

sampling theoremwere known in mathematics for manyyears before Shannon’s

paper was published. In the engineering literature, the main result of the sampling

theorem can be deduced fromthe paper published in 1928 by Nyquist.


Outline of the Proof of the Sampling Theorem

The proof is not difficult and does not go beyond basic Fourier analysis. It is

given in the textbook and for the interest of time omitted in this presentation. Here,

we give its outline and make some observations (conclusions).

In the proof, the relationship between a continuous-time bandlimited signal

and its samplevalues is derived, showing thatthe bandlimited signal can beuniquely

represented interms of its sample valuesas

1

k=�1 maxmax

The condition that facilitates this unique representation is the sampling theorem

condition s1Ts max. The proof requires that the replicated frequency

domain signal p presented in Figure 9.2 be formed such that

p for max.


ωω

max

−ωmax 0

X (j

)ωp

−3ωmax

3ωmax

. . .

. . .

Figure 9.2: A periodic frequency domain signal magnitude spectrum obtained

by replicating the original bandlimited signal magnitude spectrum

Since p must be periodic, this is possible only under the assumption that

replicated signals do not overlap, that is, whens max. In practice the

sampling frequency is slightly increased leading tos s max.

Important Observations

It can be concluded from the sampling theorem thatsignal sampling makes the

signal frequency spectrum periodic. This means that due to sampling every fre-

quency component of the original bandlimited signal, say, is replicated infinitely


many timesat highfrequencies as s . We haveobserved

thesimilar phenomenonin the sectionon the Fourier series:signal periodicitymakes

the signal frequencyspectrum discrete (linespectra), which impliesthat sampling

the frequency spectrumcorresponds to periodicityin the timedomain.

Aliasing

In the case when the conditions ofthe sampling theorem are notsatisfied

(bandlimited signalsampled with the sampling frequencys s max)

high frequency components of the replicated baseband spectrum can overlap with

the original baseband spectrum and cause problems in recovery of the original

signal. Thealiasing phenomenon of a bandlimited signal, for which the condition

s max is not satisfied,is demonstrated in Figure 9.3. The sampling theorem

condition s max ( s max) eliminates aliasingof frequencies by

providing a frequency guard bandbetween the original frequency spectrum and its


replicas introducedby the sampling process.

ωω

max

−ωmax

2ωmax

ωs

−ωs

ωs

−ωmax0

−ω +s

ωmax

−2ωmax

X (j

)ωp

Figure 9.3: Aliasing phenomenon: baseband signal

spectrum (solid line) and spectrum replicas (dashed lines)

Due to thefact that in practice many signals are not bandlimited, those signals

can be made bandlimited by using an analog prefilter as demonstrated in Figure

9.4. Such filters are calledantialiasing filterssince they either drastically reduce or

completely eliminate the aliasing phenomenon (whens max and assuming

ideal low pass-filtering).


ωmax

ω

|X(j )|ω

0

ω

|X(j ) H(j )|ω ω

−ωmax

0

Prefilter

H(j )ωinput signal output signal

Figure 9.4: Analog signal prefiltering in order to avoid signal aliasing

Analog signal prefiltering also reduces the requirement imposed by the sampling

theorem on the sampling frequency, which is important in cases when the hardware

used in the sampling process imposes limitations on the upper value on the sampling

frequency.

In the following we consider the ideal and practical sampling techniques. The

ideal one is primary used to derive the discrete-time Fourier transform and to draw

some conclusions. The practical sampling procedure indicates what actually can

be done in practice.


9.1.1 Samplingwith an Ideal Sampler and DTFT

We present the sampling operation using an ideal sampler and derive the discrete-

time Fouriertransform (DTFT).

Consider acontinuous-time signal presentedin Figure 9.5a. This signal is

sampled byan ideal sampleras shown in Figure 9.5b. The ideal sampler can

be represented by a periodictrain of impulse delta signals(called the Dirac comb)

Ts

1

k=�1s

where s is the sampling period. The signal sampled by an ideal sampler is given by

� Ts

� is called theideal sampled signal.


x(t)

t

0

t

0

. . .. . .

(t)δT

s

T

s

-T

s

T

s

2 T

s

3 T

s

4T

s

-2T

s

-3

t

0

. . .. . .

T

s

-T

s

T

s

2 T

s

3 T

s

4T

s

-2T

s

-3

x(t) (t)δT

s

.

=x (t)δ

(a)

(b)

( )c

Figure 9.5: A continuous-time signal sampled by an ideal sampler

Since Ts is a periodic signal, its Fourier series coefficients are

s

Ts=2

�Ts=2

�jk!st

s


with the corresponding Fourierseries given by

Ts

1

k=�1s

s

1

k=�1jk!st

ss

Using therelationship thatexists between thecomplex Fourierseries coefficients

and the trigonometric Fourierseries coefficients, we have

0s

ks

k

so that the trigonometric form of the Fourier series for the train of impulse delta

signals is given by

Tss s

1

k=1

s


The ideallysampled signalcan be representedas

� Ts

s s s s

s s s s

1

k=�1s s

Since the Fourier transform of the shifted impulsedelta signal is given by

�jkTs!s , it follows that the Fourier transform ofthe ideal sampled

signal is

� �

1

k=�1s�jkTs!

This formula, in fact, defines thediscrete-time Fourier transform(DTFT).


Using thetrigonometric formof the train of impulse delta signals), it can be

easily shown that

� Ts

ss s s

�s

1

k=�1s s

s

where the Fouriertransform modulation property has beenused in deriving the last

expression. This formulaindicates the expected result thatthefrequency spectrum of

the ideal sampled signal is periodic. In addition, it relates the frequency spectrum

of the original continuous-time signal and the frequency spectrum of the ideal

sampled (discrete-time)signal, that is

s � max


9.1.2 Samplingwith a Physically Realizable Sampler

Using the train of the impulse delta signals is not practically realizable. Instead of

the trainof the impulse delta signals we can practically use any periodic train of

narrow pulsessuch asthe train of rectangular pulses (or a train of triangular pulses)

with a very narrow width, whose waveform is presented in Figure 9.6.

0 T

s

τ

p(t)

t

T

s

2 T

s

3-T

s

T

s

-2

1

. . .. . .

Figure 9.6: A train of rectangular pulses with a narrow width used for practical sampling

For simplicity, we assume that the rectangular pulse height is equal to one, but

any value can be used for the pulse height. For example, we can use the pulse

height equal to so that the pulse area is equal to one.


Also, for simplicity, weassume that thetrain of pulses is an even function so that

its Fourier seriesexpansion contains onlyharmonics corresponding to the cosine

terms. Due to the a rectangularpulse train, itcan be represented by the Fourier

series,whose trigonometricform with complexcoefficients is givenby1

k=�1� s

0

1

k=1

k s s k s ss

The complex Fourier series coefficients are obtained from

0s

Ts2

�Ts2

�0

s

k ss

Ts2

�Ts2

��jk!st

k k


where k k are trigonometricFourier seriescoefficients. The assumption that the

pulse isan evenfunction implies that k so that k s are realfor

every , which further impliesthat k s . It should beemphasized

that this conditionis not crucial for the considered sampling procedure.It only

simplifies derivations.

The sampledsignal is now given by

s

1

k=1

k s s

The constant canbe made equal to oneby choosing the pulse height as s .

The Fourier transform of theabove sampled signal is givenby

s

1

k 6=0; k=�1k s s s


Assuming that is a bandlimited signal,that is, assumingthat its frequency

spectrumis different from zero onlyin the interval max max , it can be

noticedfrom the lastexpression thatthe sampling theoremcondition s max

( s max) will eliminate aliasingof frequencies. Namely,all frequency

componentsin the infinite sum areoutside of the frequency range of the original

signal, max max . Simply, a low-pass filter will be ableto extract the

original signal inthe frequency domain.


9.2 Discrete-Time Fourier Transform (DTFT)

The discrete-time Fourier transform is derived in Section 9.1.2 as

� �

1

k=�1s�jkTs!

It is theFourier transform of acontinuous-time signal sampled by anideal sampler,

� . The signal � is defined by

� Ts

1

k=�1s s

In the studyof the DTFT, it iscustomary to use the concept of thedigital frequency

defined by

s


Employing thenotation thatwe have usedin this textbook for representation of

discrete-timesignals, thatis, s , the discrete-time Fouriertransform

can be represented bythe following infinite summation

� �

1

k=�1s�jkTs!

1

k=�1�jk

Example 9.1: The DTFT of the discrete-time deltaimpulse signal is

1

k=�1�jk

1

k=�1�jk

The shifted impulse delta signal has the DTFT equal to

1

k=�1�jk �jm


Example 9.2: Consider thefollowing discrete-timesignal

The DTFT infinite sum simplifies into a sum of three terms, that is

1

k=�1�jk j �j2

The -periodicity of the DTFT can be also confirmed from the definition

formula since

1

k=�1�jk(+2�)

1

k=�1�jk

where we have used the fact that�jk2� for any integer .


We can observefrom the definitionformula that

�

which indicatesthat themagnitude spectrumof theDTFT is aneven functionand

that thephase spectrum of the DTFTis an odd function, thatis

Due to the above facts, we indeed need to plot the spectrum of the DTFT only in

the frequency interval . It follows that low frequency signal components are

around zerofrequency (and due to periodicity around for some integer )

and highfrequency signal components are aroundfrequency (and

for some integer ).


The definitionformula ofDTFT can beviewed as the frequency domain Fourier

seriesexpansion ofthe –periodic signal with playing therole of

the corresponding Fourier seriescoefficients, . This leads tothe

conclusion that thesignal sample values can beobtained interms of DTFTof

using theformula for the Fourier seriescoefficients, that is

�

��

jk

This formula definesthe inverseDTFT. Thediscrete-time domain signal and

the -domain frequency signal form the correspondingpair that we denote

by .


Existence Condition for DTFT

The DTFT exists if its infinite sum exists. This leads to the following existence

condition

1

k=�1�jk

1

k=�1�jk

1

k=�1�jk

1

k=�1

It follows that if the signal is absolutely summablethen the corresponding

DTFT will exist. Note that this conditionis only a sufficient condition, which

means thatif the condition is satisfiedthan the DTFT exists, but thereverse is not

generally true.


Example 9.3: Consider thesignal definedby

k

It follows that1

k=�1�jk

1

k=0

k �jk1

k=0j

k

j

j

j

Note that thecondition j , which is the consequence of , has

been usedto sum the geometric seriesobtained. Hence, the convergence condition

is satisfied and the corresponding DTFT exists.


Property #1: Linearity

The DTFT is defined by an infinite sum, and it obeys to the linearity property,

which saysthat for pairs i i , andan arbitrary set

of constants i , the following holds

1 1 2 2 n n

1 1 2 2 n n

The proof ofthis property is a consequencethe linearity property of summation.

Property #2: Time Shifting

Let the following pair exist. Then, signaltime shifting implies

the following pair

�jm


It should be emphasizedthat this formulais valid for both positiveand negative

valuesof .

In order to establish the proof we have to use the definition formula and the

change of variables as , that is

1

k=�1�jk

1

n=�1�j(n+m)

�jm1

n=�1�jn �jm

Property #3: Frequency Shifting

Let the following pair exist, then the frequency shifted signal

produces thefollowing pair

j00


This propertycan beproved as follows

1

k=�1�jk(�0)

1

k=�1jk0 jk jk0

Property #4: Time Reversal

Let . It follows from the definition formula thatthe time

reversal producesthe following pair

To prove this property we introduce a change of variables in the definition

formula asfollows


1

k=�1

�jk�1

n=1jn

1

n=�1jn

1

n=�1�jn(�)

Note that inthe above proof wehave changed by .

Property #5: Conjugation

Let , then conjugation produces the following pair

� �

Hence, signal conjugationin the time domain impliesboth conjugation and fre-

quency reversalin the frequency domain.


The proof is as follows

1

k=�1

� �jk1

k=�1

jk

�

1

k=�1

�jk(�)�

�

Property #6: Frequency Differentiation

Let , then thedifferentiation of with respect to

produces

The proof simplyfollows from the definition formula,namely1

k=�1�jk


which implies

Property #7: Modulation

The modulationproperty is a generalization ofthe frequency shifting property.

Let , then themodulation property states the followingresults

0 0 0

and

0 0 0

The modulation property has many applications in digital signal processing and

digital communication systems.


The proofof this property requires firstrepresentation of sine and cosine func-

tions via Euler’s formulas, that is

0j0k �j0k

0j0k �j0k

and thenfollows the proof of the frequency shift property.

Property #8: Time Convolution

Let 1 1 and 2 2 . Then, thetime domain

convolution ofthese two signals corresponds toa product in the frequencydomain

1 2 1 2

This property isuseful for deriving theoretical results aboutthe response of

linear-time invariantsystems. The convolution proof is similar to the corresponding

convolution property proof from Chapter 5.


Property #9: Periodic FrequencyDomain Convolution

Let 1 1 and 2 2 . Then, the time domain product

of these two signals corresponds to the frequency domain like convolution called

the periodic convolutionand defined by

1 2

�

��1 2

The proof of this property uses both thedefinitions of the DTFT andinverse

DTFT as follows


1 2

1

k=�11 2

�jk

1

k=�1

�

��1

jk�2

�jk

�

��1

1

k=�12

�jk(��)

�

��1 2

Note that for , this formula gives the expression for the discrete-time

signal total energy (Parseval theoremfor discrete-time signals)

11

k=�1

2

�

��

2


Generalized DTFT

We have observed in Chapter 3 that signals that do not satisfy the existence

condition of the Fourier transform, more precisely, signals for which the Fourier

integral does not exists in terms of regular functions, still can be transformed into

the frequencydomain by usinggeneralized functions (continuous-time impulse delta

function, ). Similarly, we can define the generalized DTFTby using the

continuous-timeimpulse delta signal (the signal thataccounts for infinitely large

values). Thisprocedure is demonstrated in severalexamples.

Example 9.4: Consider now the discrete-time signalthat does not satisfy the

DTFT existencecondition given by


It is obvious thatfor this signal,the DTFT sum is equal to infinity. It follows that1

k=�1

�jk1

k=�1

�jk

This infinite sum can be evaluated using the results established in Example 3.101

k=�1�jkT0!

0

1

k=�10 0

0

Using the digitalfrequency as 0 and the propertyof the continuous-time

delta impulsefunction ( ), the above expression implies1

k=�1

�jk1

k=�10

0

This establishes thefollowing generalized DTFT pair1

k=�1


Example 9.5: Using theresults establishedin the previousexample, we can find

the DTFT of sine and cosinesignals as follows. By modulating the signal

using thecosine function and evoking the modulationproperty, we have

0

1

k=�10

1

k=�10

or

0

1

k=�10

1

k=�10


Similarly, by modulating thesignal by thesine functionwe obtain

0

1

k=�10

1

k=�10

9.2.1 DTFT in Linear Systems

The DTFT can beused in analysis and design ofdiscrete-time linear systems with

constant coefficients in the sameway the Fourier transform isused in continuous-

time linear systems with constant coefficients. Hence, the DTFT can be used for

finding the zero-state response of discrete-time linear time invariant systems.

Such a discrete-time system is defined in Section 5.3, as

n�1 1 0

m m�1 1 0

where i i are constant coefficients and is the system order.


Let and . Applying the DTFTand using the

time shifting property,we have

jnn�1 j(n�1)

1j

0

mjm

m�1 j(m�1)1j

0

from which we obtain

mjm

m�1 j(m�1)1j

0

jnn�1 j(n�1)

1j

0

where the quantity

mjm

m�1 j(m�1)1j

0

jnn�1 j(n�1)

1j

0

defines thediscrete-time system digital frequency transfer function.


Assuming thatthe inputsignal is equalto the discrete-time impulse delta function

, whoseDTFT is equalto one,we see that the system impulse response in the

–frequencydomain is equalto . It can be

concluded that is the DTFT of the discrete-time systemimpulse response

so that we have the followingpair

From the time convolution property of the DTFT, it follows that thelinear

discrete-time system response to any excitation is given by the convolution of

that inputsignal and the system impulse response, that is


Linear System Response toSinusoidal Inputs

Finding the linear discrete-time system response due to sinusoidal inputs can be

done employing the same technique as in Section 5.4.1, which leads to the same

result with j!T being replaced by . Let the system input be given

by 0 , thenby replacing j!T by 0 , thesystem

output is

0 0 0

where 0 is the system transfer function evaluated at the digital frequency

0. It shouldbe emphasized that this result is valid at steady state.


Discrete-time domain s -domain

1 1 2 2 1 1 2 2

�jm

j0 0

� �

dX(j)d

12 0 0

j2 0 0

1 2 1 2

1 2 12�

�

��1 2

x1x2 1�2

Table 9.1: Properties of the DTFTThe slides contain the copyrighted material from Linear Dynamic Systems and Signals, Prentice Hall, 2003. Prepared by Professor Zoran Gajic 9–42

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9.1 Sampling Theorem