lti discrete-time systems in the transform...

1Copyright © 2001, S. K. Mitra

LTI Discrete-Time Systems inLTI Discrete-Time Systems inthe Transform Domainthe Transform Domain

• An LTI discrete-time system is completelycharacterized in the time-domain by itsimpulse response sequence {h[n]}

• Thus, the transform-domain representationof a discrete-time signal can also be equallyapplied to the transform-domainrepresentation of an LTI discrete-timesystem


LTI Discrete-Time Systems inLTI Discrete-Time Systems inthe Transform Domainthe Transform Domain

• Such transform-domain representationsprovide additional insight into the behaviorof such systems

• It is easier to design and implement thesesystems in the transform-domain for certainapplications

• We consider now the use of the DTFT andthe z-transform in developing the transform-domain representations of an LTI system


Finite-Dimensional LTIFinite-Dimensional LTIDiscrete-Time SystemsDiscrete-Time Systems

• In this course we shall be concerned withLTI discrete-time systems characterized bylinear constant coefficient differenceequations of the form:

∑∑==

−=−M

kk

N

kk knxpknyd

00][][



• Applying the DTFT to the differenceequation and making use of the linearity andthe time-invariance properties of Table 3.2we arrive at the input-output relation in thetransform-domain as

where and are the DTFTs ofy[n] and x[n], respectively

)()(00

ω

=

ω−ω

=

ω− ∑=∑ jM

k

kjk

jN

k

kjk eXepeYed

)( ωjeY )( ωjeX



• In developing the transform-domainrepresentation of the difference equation, ithas been tacitly assumed that and

exist• The previous equation can be alternately

written as

)( ωjeY)( ωjeX

)()(00

ω

=

ω−ω

=

ω−

∑=

∑ jM

k

kjk

jN

k

kjk eXepeYed



• Applying the z-transform to both sides ofthe difference equation and making use ofthe linearity and the time-invarianceproperties of Table 3.9 we arrive at

where Y(z) and X(z) denote the z-transformsof y[n] and x[n] with associated ROCs,respectively

)()( zXzpzYzdM

k

kk

N

k

kk ∑∑

=

−

=

− =00



• A more convenient form of the z-domainrepresentation of the difference equation isgiven by

)()( zXzpzYzdM

k

kk

N

k

kk

=

∑∑=

−

=

−

00


The Frequency ResponseThe Frequency Response• Most discrete-time signals encountered in

practice can be represented as a linearcombination of a very large, maybe infinite,number of sinusoidal discrete-time signalsof different angular frequencies

• Thus, knowing the response of the LTIsystem to a single sinusoidal signal, we candetermine its response to more complicatedsignals by making use of the superpositionproperty


The Frequency ResponseThe Frequency Response

• An important property of an LTI system isthat for certain types of input signals, calledeigen functions, the output signal is theinput signal multiplied by a complexconstant

• We consider here one such eigen functionas the input



• Consider the LTI discrete-time system withan impulse response {h[n]} shown below

• Its input-output relationship in the time-domain is given by the convolution sum

x[n] h[n] y[n]

∑∞

−∞=−=

kknxkhny ][][][



• If the input is of the form

then it follows that the output is given by

• Let

∞<<∞−= ω nenx nj ,][

nj

k

kj

k

knj eekhekhny ω∞

−∞=

ω−∞

−∞=

−ω

∑=∑= ][][][ )(

∑=∞

−∞=

ω−ω

k

kjj ekheH ][)(



• Then we can write

• Thus for a complex exponential input signal , the output of an LTI discrete-time

system is also a complex exponential signalof the same frequency multiplied by acomplex constant

• Thus is an eigen function of the system

njj eeHny ωω= )(][

)( ωjeH

nje ω

nje ω



• The quantity is called the frequencyresponse of the LTI discrete-time system

• provides a frequency-domaindescription of the system

• is precisely the DTFT of the impulseresponse {h[n]} of the system

)( ωjeH

)( ωjeH

)( ωjeH


The Frequency ResponseThe Frequency Response• , in general, is a complex function

of ω with a period 2π• It can be expressed in terms of its real and

imaginary parts

or, in terms of its magnitude and phase,

where

)( ωjeH

)()()( ωωω += jim

jre

j eHjeHeH

)()()( ωθωω = jjj eeHeH

)(arg)( ω=ωθ jeH



• The function is called themagnitude response and the functionis called the phase response of the LTIdiscrete-time system

• Design specifications for the LTI discrete-time system, in many applications, aregiven in terms of the magnitude response orthe phase response or both

)( ωjeH)(ωθ



• In some cases, the magnitude function isspecified in decibels as

where G(ω) is called the gain function• The negative of the gain function

is called the attenuation or loss function

dBeH j )(log20)( 10ω=ωG

)()( ω−=ω GA


The Frequency ResponseThe Frequency Response• Note: Magnitude and phase functions are

real functions of ω, whereas the frequencyresponse is a complex function of ω

• If the impulse response h[n] is real then itfollows from Table 3.4 that the magnitudefunction is an even function of ω:

and the phase function is an odd function ofω:

)()( ω−ω = jj eHeH

)()( ω−θ−=ωθ


The Frequency ResponseThe Frequency Response• Likewise, for a real impulse response h[n],

is even and is odd• Example - Consider the M-point moving

average filter with an impulse responsegiven by

• Its frequency response is then given by

)( ωjre eH )( ωj

im eH

otherwise0101

,,/ −≤≤ MnM

=][nh

∑=−

=

ω−ω 1

0

1)(M

n

njM

j eeH



• Or,

∑−∑=

∞

=

ω−∞

=

ω−ω

Mn

nj

n

njM

j eeeH0

1)(

( ) ω−

ω−ω−∞

=

ω−

−−⋅=−

∑= j

jM

MjM

n

njM e

eee1

11 1

0

1

2/)1()2/sin()2/sin(1 ω−−

ωω⋅= MjeM

M



• Thus, the magnitude response of the M-pointmoving average filter is given by

and the phase response is given by)2/sin()2/sin()( 1

ωω⋅=ω MeH

Mj

∑µπ+ω−−=ωθ=02

)1()(k

MM

kπ−ω 2( )M/2


Frequency ResponseFrequency ResponseComputation Using MATLABComputation Using MATLAB

• The function freqz(h,w) can be used todetermine the values of the frequencyresponse vector h at a set of givenfrequency points w

• From h, the real and imaginary parts can becomputed using the functions real andimag, and the magnitude and phasefunctions using the functions abs andangle



• Example - Program 4_1 can be used togenerate the magnitude and gain responsesof an M-point moving average filter asshown below

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Mag

nitu

de

ω/π

M=5M=14

0 0.2 0.4 0.6 0.8 1-200

-150

-100

-50

0

50

100

Pha

se, d

egre

es

ω/π

M=5M=14



• The phase response of a discrete-timesystem when determined by a computermay exhibit jumps by an amount 2π causedby the way the arctangent function iscomputed

• The phase response can be made acontinuous function of ω by unwrapping thephase response across the jumps



• To this end the function unwrap can beused, provided the computed phase is inradians

• The jumps by the amount of 2π should notbe confused with the jumps caused by thezeros of the frequency response as indicatedin the phase response of the moving averagefilter


Steady-State ResponseSteady-State Response• Note that the frequency response also

determines the steady-state response of anLTI discrete-time system to a sinusoidalinput

• Example - Determine the steady-stateoutput y[n] of a real coefficient LTIdiscrete-time system with a frequencyresponse for an input

∞<<∞−φ+ω= nnAnx o ),cos(][

)( ωjeH


Steady-State ResponseSteady-State Response

• We can express the input x[n] as

where

• Now the output of the system for an input is simply

][*][][ ngngnx +=

njj oeAeng ωφ=21][

nj oe ω

njj oo eeH ωω )(



• Because of linearity, the response v[n] to aninput g[n] is given by

• Likewise, the output v*[n] to the input g*[n]is

njjj oo eeHAenv ωωφ= )(][21

njjj oo eeHAenv ω−ω−φ−= )(][*21



• Combining the last two equations we get

njjjnjjj oooo eeHAeeeHAe ω−ω−φ−ωωφ += )()(21

21

][*][][ nvnvny +=

{ }nojjojnojjojoj eeeeeeeHA ω−φ−ωθ−ωφωθω += )()()(21

))(cos()(21 φ+ωθ+ωω= oooj neHA



• Thus, the output y[n] has the same sinusoidalwaveform as the input with two differences:(1) the amplitude is multiplied by ,the value of the magnitude function at(2) the output has a phase lag relative to theinput by an amount , the value of thephase function at

)( ojeH ω

)( oωθ

oω=ω

oω=ω


Response to a CausalResponse to a CausalExponential SequenceExponential Sequence

• The expression for the steady-state responsedeveloped earlier assumes that the system isinitially relaxed before the application ofthe input x[n]

• In practice, excitation x[n] to a discrete-timesystem is usually a right-sided sequenceapplied at some sample index

• We develop the expression for the outputfor such an input

onn =



• Without any loss of generality, assumefor n < 0

• From the input-output relation

we observe that for an input

the output is given by

∑∞−∞= −= k knxkhny ][][][

][][][0

)( nekhnyn

k

knj µ

= ∑

=

−ω

][][ nenx nj µ= ω

0][ =nx



• Or,

• The output for n < 0 is y[n] = 0• The output for is given by

][][][0

neekhny njn

k

kj µ

= ω

=

ω−∑

nj

nk

kjnj

k

kj eekheekh ω∞

+=

ω−ω∞

=

ω−

−

= ∑∑

10][][

0≥nnj

n

k

kj eekhny ω

=

ω−

= ∑

0][][



• Or,

• The first term on the RHS is the same asthat obtained when the input is applied atn = 0 to an initially relaxed system and isthe steady-state response:

nj

nk

kjnjj eekheeHny ω∞

+=

ω−ωω

−= ∑

1][)(][

njjsr eeHny ωω= )(][



• The second term on the RHS is called thetransient response:

• To determine the effect of the above termon the total output response, we observe

nj

nk

kjtr eekhny ω

∞

+=

ω−

−= ∑

1][][

∑∑∑∞

=

∞

+=

∞

+=

−ω− ≤≤=011

)( ][][][][knknk

nkjtr khkhekhny



• For a causal, stable LTI IIR discrete-timesystem, h[n] is absolutely summable

• As a result, the transient response is abounded sequence

• Moreover, as ,

and hence, the transient response decays tozero as n gets very large

][nytr

∞→n0][1 →∑∞

+=nk kh



• For a causal FIR LTI discrete-time systemwith an impulse response h[n] of lengthN + 1, h[n] = 0 for n > N

• Hence, for• Here the output reaches the steady-state

value at n = N

0][ =nytr 1−> Nn

njjsr eeHny ωω= )(][


The Concept of FilteringThe Concept of Filtering

• One application of an LTI discrete-timesystem is to pass certain frequencycomponents in an input sequence withoutany distortion (if possible) and to blockother frequency components

• Such systems are called digital filters andone of the main subjects of discussion inthis course



• The key to the filtering process is

• It expresses an arbitrary input as a linearweighted sum of an infinite number ofexponential sequences, or equivalently, as alinear weighted sum of sinusoidal sequences

∫ ω=π

π−

ωωπ

deeXnx njj )(][21



• Thus, by appropriately choosing the valuesof the magnitude function of theLTI digital filter at frequenciescorresponding to the frequencies of thesinusoidal components of the input, some ofthese components can be selectively heavilyattenuated or filtered with respect to theothers

)( ωjeH



• To understand the mechanism behind thedesign of frequency-selective filters,consider a real-coefficient LTI discrete-timesystem characterized by a magnitudefunction

≅ω)( jeH

π≤ω<ωω≤ω

cc

,0,1



• We apply an input

to this system• Because of linearity, the output of this

system is of the form

π<ω<ω<ω<ω+ω= 2121 0,coscos][ cnBnAnx

( ))(cos)(][ 111 ωθ+ω= ω neHAny j

( ))(cos)( 222 ωθ+ω+ ω neHB j



• As

the output reduces to

• Thus, the system acts like a lowpass filter• In the following example, we consider the

design of a very simple digital filter

0)(,1)( 21 ≅≅ ωω jj eHeH

( ))(cos)(][ 111 ωθ+ω≅ ω neHAny j


The Concept of FilteringThe Concept of Filtering• Example - The input consists of a sum of two

sinusoidal sequences of angular frequencies0.1 rad/sample and 0.4 rad/sample

• We need to design a highpass filter that willpass the high-frequency component of theinput but block the low-frequency component

• For simplicity, assume the filter to be an FIRfilter of length 3 with an impulse response:

h[0] = h[2] = α, h[1] = β


The Concept of FilteringThe Concept of Filtering• The convolution sum description of this

filter is then given by

• y[n] and x[n] are, respectively, the outputand the input sequences

• Design Objective: Choose suitable valuesof α and β so that the output is a sinusoidalsequence with a frequency 0.4 rad/sample

]2[]2[]1[]1[][]0[][ −+−+= nxhnxhnxhny]2[]1[][ −α+−β+α= nxnxnx



• Now, the frequency response of the FIRfilter is given by

ω−ω−ω ++= 2]2[]1[]0[)( jjj ehehheHω−ω− β++α= jj ee )1( 2

ω−ω−ω−ω

β+

+α= jjjj

eeee2

2

ω−β+ωα= je)cos2(


The Concept of FilteringThe Concept of Filtering• The magnitude and phase functions are

• In order to block the low-frequencycomponent, the magnitude function atω = 0.1 should be equal to zero

• Likewise, to pass the high-frequencycomponent, the magnitude function atω = 0.4 should be equal to one

β+ωα=ω cos2)( jeHω−=ωθ )(



• Thus, the two conditions that must besatisfied are

• Solving the above two equations we get

0)1.0cos(2)( 1.0 =β+α=jeH

1)4.0cos(2)( 4.0 =β+α=jeH

76195.6−=α456335.13=β



• Thus the output-input relation of the FIRfilter is given by

where the input is

• Program 4_2 can be used to verify thefiltering action of the above system

( ) ]1[456335.13]2[][76195.6][ −+−+−= nxnxnxny

][)}4.0cos()1.0{cos(][ nnnnx µ+=



• Figure below shows the plots generated byrunning this program

0 20 40 60 80 100-1

0

1

2

3

4

Am

plitu

de

Time index n

y[n]x

2[n]

x1[n]



• The first seven samples of the output areshown below


The Concept of FilteringThe Concept of Filtering• From this table, it can be seen that,

neglecting the least significant digit,

• Computation of the present value of theoutput requires the knowledge of thepresent and two previous input samples

• Hence, the first two output samples, y[0]and y[1], are the result of assumed zeroinput sample values at and

2for))1(4.0cos(][ ≥−= nnny

1−=n 2−=n



• Therefore, first two output samplesconstitute the transient part of the output

• Since the impulse response is of length 3,the steady-state is reached at n = N = 2

• Note also that the output is delayed versionof the high-frequency component cos(0.4n)of the input, and the delay is one sampleperiod

lti discrete-time systems in the transform...

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