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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
2Copyright © 2001, S. K. Mitra
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
3Copyright © 2001, S. K. Mitra
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][][
4Copyright © 2001, S. K. Mitra
Finite-Dimensional LTIFinite-Dimensional LTIDiscrete-Time SystemsDiscrete-Time Systems
• 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
5Copyright © 2001, S. K. Mitra
Finite-Dimensional LTIFinite-Dimensional LTIDiscrete-Time SystemsDiscrete-Time Systems
• 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
6Copyright © 2001, S. K. Mitra
Finite-Dimensional LTIFinite-Dimensional LTIDiscrete-Time SystemsDiscrete-Time Systems
• 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
7Copyright © 2001, S. K. Mitra
Finite-Dimensional LTIFinite-Dimensional LTIDiscrete-Time SystemsDiscrete-Time Systems
• A more convenient form of the z-domainrepresentation of the difference equation isgiven by
)()( zXzpzYzdM
k
kk
N
k
kk
=
∑∑=
−
=
−
00
8Copyright © 2001, S. K. Mitra
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
9Copyright © 2001, S. K. Mitra
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
10Copyright © 2001, S. K. Mitra
The Frequency ResponseThe Frequency Response
• 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 ][][][
11Copyright © 2001, S. K. Mitra
The Frequency ResponseThe Frequency Response
• 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 ][)(
12Copyright © 2001, S. K. Mitra
The Frequency ResponseThe Frequency Response
• 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
13Copyright © 2001, S. K. Mitra
The Frequency ResponseThe Frequency Response
• 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
14Copyright © 2001, S. K. Mitra
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
15Copyright © 2001, S. K. Mitra
The Frequency ResponseThe Frequency Response
• 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
16Copyright © 2001, S. K. Mitra
The Frequency ResponseThe Frequency Response
• 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
17Copyright © 2001, S. K. Mitra
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
18Copyright © 2001, S. K. Mitra
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
19Copyright © 2001, S. K. Mitra
The Frequency ResponseThe Frequency Response
• 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
20Copyright © 2001, S. K. Mitra
The Frequency ResponseThe Frequency Response
• 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
21Copyright © 2001, S. K. Mitra
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
22Copyright © 2001, S. K. Mitra
Frequency ResponseFrequency ResponseComputation Using MATLABComputation Using MATLAB
• 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
23Copyright © 2001, S. K. Mitra
Frequency ResponseFrequency ResponseComputation Using MATLABComputation Using MATLAB
• 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
24Copyright © 2001, S. K. Mitra
Frequency ResponseFrequency ResponseComputation Using MATLABComputation Using MATLAB
• 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
25Copyright © 2001, S. K. Mitra
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
26Copyright © 2001, S. K. Mitra
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 )(
27Copyright © 2001, S. K. Mitra
Steady-State ResponseSteady-State Response
• 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
28Copyright © 2001, S. K. Mitra
Steady-State ResponseSteady-State Response
• 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
29Copyright © 2001, S. K. Mitra
Steady-State ResponseSteady-State Response
• 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
30Copyright © 2001, S. K. Mitra
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 =
31Copyright © 2001, S. K. Mitra
Response to a CausalResponse to a CausalExponential SequenceExponential Sequence
• 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
32Copyright © 2001, S. K. Mitra
Response to a CausalResponse to a CausalExponential SequenceExponential Sequence
• 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][][
33Copyright © 2001, S. K. Mitra
Response to a CausalResponse to a CausalExponential SequenceExponential Sequence
• 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= )(][
34Copyright © 2001, S. K. Mitra
Response to a CausalResponse to a CausalExponential SequenceExponential Sequence
• 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
35Copyright © 2001, S. K. Mitra
Response to a CausalResponse to a CausalExponential SequenceExponential Sequence
• 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
36Copyright © 2001, S. K. Mitra
Response to a CausalResponse to a CausalExponential SequenceExponential Sequence
• 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= )(][
37Copyright © 2001, S. K. Mitra
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
38Copyright © 2001, S. K. Mitra
The Concept of FilteringThe Concept of Filtering
• 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
39Copyright © 2001, S. K. Mitra
The Concept of FilteringThe Concept of Filtering
• 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
40Copyright © 2001, S. K. Mitra
The Concept of FilteringThe Concept of Filtering
• 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
41Copyright © 2001, S. K. Mitra
The Concept of FilteringThe Concept of Filtering
• 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
42Copyright © 2001, S. K. Mitra
The Concept of FilteringThe Concept of Filtering
• 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
43Copyright © 2001, S. K. Mitra
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
44Copyright © 2001, S. K. Mitra
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
45Copyright © 2001, S. K. Mitra
The Concept of FilteringThe Concept of Filtering
• 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(
46Copyright © 2001, S. K. Mitra
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 )(
47Copyright © 2001, S. K. Mitra
The Concept of FilteringThe Concept of Filtering
• 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
48Copyright © 2001, S. K. Mitra
The Concept of FilteringThe Concept of Filtering
• 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+=
49Copyright © 2001, S. K. Mitra
The Concept of FilteringThe Concept of Filtering
• 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]
50Copyright © 2001, S. K. Mitra
The Concept of FilteringThe Concept of Filtering
• The first seven samples of the output areshown below
51Copyright © 2001, S. K. Mitra
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
52Copyright © 2001, S. K. Mitra
The Concept of FilteringThe Concept of Filtering
• 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