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UNIT IUNIT I
SIGNALSIGNAL
►►Signal is a physical quantity that varies with Signal is a physical quantity that varies with
respect to time , space or any other respect to time , space or any other
independent variableindependent variable
EgEg x(tx(t)= sin t.)= sin t.
►►the major classifications of the signal the major classifications of the signal
are:are:
(i) Discrete time signal (i) Discrete time signal
(ii) (ii) Continuous time signal Continuous time signal
Unit Step &Unit ImpulseUnit Step &Unit Impulse
�� Discrete time Unit impulse is defined asDiscrete time Unit impulse is defined as
δδ [n]= {0, n[n]= {0, n≠≠ 00
{1, n=0{1, n=0
Unit impulse is also known as unit sample.Unit impulse is also known as unit sample.
Discrete time unit step signal is defined by Discrete time unit step signal is defined by
U[nU[n]={0,n=0]={0,n=0
{1,n>= 0{1,n>= 0
Continuous time unit impulse is defined as Continuous time unit impulse is defined as
δδ (t)={1, t=0(t)={1, t=0
{0, t {0, t ≠≠ 00
Continuous time Unit step signal is defined as Continuous time Unit step signal is defined as
U(t)={0, t<0U(t)={0, t<0
{1, t{1, t≥≥00
►► Periodic Signal & Periodic Signal & AperiodicAperiodic SignalSignal
�� A signal is said to be periodic ,if it exhibits A signal is said to be periodic ,if it exhibits periodicity.i.eperiodicity.i.e., ., X(tX(t +T)=+T)=x(tx(t), for all values of t. Periodic signal has the ), for all values of t. Periodic signal has the property that it is unchanged by a time shift of T. A signal property that it is unchanged by a time shift of T. A signal that does not satisfy the above periodicity property is that does not satisfy the above periodicity property is called an called an aperiodicaperiodic signalsignal
►► even and odd signal ?even and odd signal ?�� A discrete time signal is said to be even when, A discrete time signal is said to be even when, x[x[--nn]=]=x[nx[n]. ].
The continuous time signal is said to be even when, The continuous time signal is said to be even when, x(x(--tt)= )= x(tx(t) For ) For example,Cosexample,Cosωωnn is an even signal.is an even signal.
SIGNALSIGNAL
Energy and power signalEnergy and power signal
►► A signal is said to be energy signal if it A signal is said to be energy signal if it
have finite energy and zero power.have finite energy and zero power.
►► A signal is said to be power signal if it A signal is said to be power signal if it
have infinite energy and finite power.have infinite energy and finite power.
►► If the above two conditions are not If the above two conditions are not
satisfied then the signal is said to be satisfied then the signal is said to be
neigtherneigther energy nor power signal energy nor power signal
Fourier SeriesFourier SeriesThe Fourier series represents a periodic signal in terms of The Fourier series represents a periodic signal in terms of
frequency components:frequency components:
We get the Fourier series coefficients as followsWe get the Fourier series coefficients as follows::
The complex exponential Fourier coefficients are a sequence of The complex exponential Fourier coefficients are a sequence of
complex numbers representing the frequency component complex numbers representing the frequency component ωω00k.k.
∫ω−
=
p
0
tikk dte)t(x
p
1X 0∑
−
=
ω−=
1p
0n
nikk
0e)n(xp
1X
∑−
=
ω=
1p
0k
nikk
0eX)n(x ∑∞
−∞=
ω=
k
tikk
0eX)t(x
Fourier seriesFourier series
►► Fourier series: a complicated waveform analyzed into a Fourier series: a complicated waveform analyzed into a number of harmonically related sine and cosine functionsnumber of harmonically related sine and cosine functions
►► A continuous periodic signal A continuous periodic signal x(tx(t) with a period T0 may be ) with a period T0 may be represented by: represented by: �� X(tX(t)=)=ΣΣ∞∞
kk=1=1 ((AAkk coscos kkωω t + Bt + Bkk sin sin kkωω t)+ t)+ AA00
►► DirichletDirichlet conditions conditions must be placed on must be placed on x(tx(t) ) for the series for the series to be valid: the integral of the magnitude of to be valid: the integral of the magnitude of x(tx(t) ) over a over a complete period must be finite, and the signal can only complete period must be finite, and the signal can only have a finite number of discontinuities in any finite have a finite number of discontinuities in any finite intervalinterval
Trigonometric form for Fourier seriesTrigonometric form for Fourier series
►►If the two fundamental components of a If the two fundamental components of a
periodic signal areB1cosperiodic signal areB1cosωω0t and C1sin0t and C1sinωω0t, 0t,
then their sum is expressed by trigonometric then their sum is expressed by trigonometric
identities:identities:
►►X(tX(t)= )= AA00 + + ΣΣ∞∞kk=1 =1 (( BBk k
22++ AAk k 22))1/21/2 (C(Ckk coscos kkωω tt--
φφkk) or ) or
►►X(tX(t)= )= AA00 + + ΣΣ∞∞kk=1 =1 (( BBk k
22++ AAk k 22))1/21/2 (C(Ckk sin sin kkωω t+ t+
φφkk))
UNIT IIUNIT II
Fourier TransformFourier Transform
►► Viewed periodic functions in terms of frequency components (FourViewed periodic functions in terms of frequency components (Fourier ier
series) as well as ordinary functions of timeseries) as well as ordinary functions of time
►► Viewed LTI systems in terms of what they do to frequency Viewed LTI systems in terms of what they do to frequency
components (frequency response)components (frequency response)
►► Viewed LTI systems in terms of what they do to timeViewed LTI systems in terms of what they do to time--domain signals domain signals
(convolution with impulse response)(convolution with impulse response)
►► View View aperiodicaperiodic functions in terms of frequency components via functions in terms of frequency components via
Fourier transformFourier transform
►► Define (continuousDefine (continuous--time) Fourier transform and DTFT time) Fourier transform and DTFT
►► Gain insight into the meaning of Fourier transform through Gain insight into the meaning of Fourier transform through
comparison with Fourier seriescomparison with Fourier series
The Fourier TransformThe Fourier Transform
►►A transform takes one function (or signal) A transform takes one function (or signal)
and turns it into another function (or signal)and turns it into another function (or signal)
►►Continuous Fourier Transform:Continuous Fourier Transform:
( ) ( )
( ) ( )∫
∫∞
∞−
−
∞
∞−
=
=
dfefHth
dtethfH
ift
ift
π
π
2
2
Continuous Time Fourier TransformContinuous Time Fourier Transform
We can extend the formula for continuousWe can extend the formula for continuous--time Fourier series time Fourier series
coefficients for a periodic signalcoefficients for a periodic signal
to to aperiodicaperiodic signals as well. The continuoussignals as well. The continuous--time Fourier time Fourier
series is not defined for series is not defined for aperiodicaperiodic signals, but we call the signals, but we call the
formulaformula
the (continuous time)the (continuous time)
Fourier transformFourier transform..
∫∫−
ω−ω−==
2/p
2/p
tikp
0
tikk dte)t(x
p
1dte)t(x
p
1X 00
∫∞
∞−
ω−=ω dte)t(x)(X ti
Inverse TransformsInverse TransformsIf we have the full sequence of Fourier coefficients for a perioIf we have the full sequence of Fourier coefficients for a periodic dic
signal, we can reconstruct it by multiplying the complex signal, we can reconstruct it by multiplying the complex
sinusoids of frequency sinusoids of frequency ωω00k by the weights k by the weights XXkk and summing:and summing:
We can perform a similar reconstruction for We can perform a similar reconstruction for aperiodicaperiodic signalssignals
These are called the These are called the inverse transformsinverse transforms..
∑−
=
ω=
1p
0k
nikk
0eX)n(x ∑∞
−∞=
ω=
k
tikk
0eX)t(x
∫∞
∞−
ω ωωπ
= de)(X2
1)t(x ti
∫π
π−
ω ωωπ
= de)(X2
1)n(x ni
Fourier Transform of Impulse FunctionsFourier Transform of Impulse FunctionsFind the Fourier transform of the Find the Fourier transform of the DiracDirac delta function:delta function:
Find the DTFT of the Find the DTFT of the KroneckerKronecker delta function:delta function:
The delta functions contain all frequencies at equal amplitudes.The delta functions contain all frequencies at equal amplitudes.
Roughly speaking, thatRoughly speaking, that’’s why the system response to an impulse s why the system response to an impulse
input is important: it tests the system at all frequencies.input is important: it tests the system at all frequencies.
1edte)t(dte)t(x)(X 0ititi ==δ==ω ω−∞
∞−
ω−∞
∞−
ω−∫∫
1ee)n(e)n(x)(X 0i
n
ni
n
ni ==δ==ω ω−∞
−∞=
ω−∞
−∞=
ω− ∑∑
LaplaceLaplace TransformTransform►► LapalceLapalce transform is a generalization of the Fourier transform in the stransform is a generalization of the Fourier transform in the sense ense
that it allows that it allows ““complex frequencycomplex frequency”” whereas Fourier analysis can only whereas Fourier analysis can only handle handle ““real frequencyreal frequency””. Like Fourier transform, . Like Fourier transform, LapalceLapalce transform allows transform allows us to analyze a us to analyze a ““linear circuitlinear circuit”” problem, no matter how complicated the problem, no matter how complicated the circuit is, in the frequency domain in stead of in he time domaicircuit is, in the frequency domain in stead of in he time domain.n.
►► Mathematically, it produces the benefit of converting a set of dMathematically, it produces the benefit of converting a set of differential ifferential equations into a corresponding set of algebraic equations, whichequations into a corresponding set of algebraic equations, which are much are much easier to solve. Physically, it produces more insight of the cireasier to solve. Physically, it produces more insight of the circuit and cuit and allows us to know the bandwidth, phase, and transfer characterisallows us to know the bandwidth, phase, and transfer characteristics tics important for circuit analysis and design.important for circuit analysis and design.
►► Most importantly, Most importantly, LaplaceLaplace transform lifts the limit of Fourier analysis to transform lifts the limit of Fourier analysis to allow us to find both the steadyallow us to find both the steady--state and state and ““transienttransient”” responses of a linear responses of a linear circuit. Using Fourier transform, one can only deal with he steacircuit. Using Fourier transform, one can only deal with he steady state dy state behavior (i.e. circuit response under indefinite sinusoidal excibehavior (i.e. circuit response under indefinite sinusoidal excitation). tation).
►► Using Using LaplaceLaplace transform, one can find the response under any types of transform, one can find the response under any types of excitation (e.g. switching on and off at any given excitation (e.g. switching on and off at any given time(stime(s), sinusoidal, ), sinusoidal, impulse, square wave excitations, etcimpulse, square wave excitations, etc..
LaplaceLaplace TransformTransform
Application of Application of LaplaceLaplace Transform to Transform to
Circuit AnalysisCircuit Analysis
system
►• A system is an operation that transforms input signal x into output signal y.
LTI Digital Systems
►Linear Time Invariant
• Linearity/Superposition:
►If a system has an input that can be expressed as a sum of signals, then the response of the system can be expressed as a sum of the individual responses to the respective systems.
►LTI
Time-Invariance &Causality
► If you delay the input, response is just a delayed version of original response.
► X(n-k) y(n-k)
► Causality could also be loosely defined by “there is no output signal as long as there is no input signal” or “output at current time does not depend on future values of the input”.
Convolution
►The input and output signals for LTI systems have special relationship in terms of convolution sum and integrals.
►Y(t)=x(t)*h(t) Y[n]=x[n]*h[n]
UNIT IIIUNIT III
Sampling theory
► The theory of taking discrete sample values (grid of color pixels) from functions defined over continuous domains (incident radiance defined over the film plane) and then using those samples to reconstruct new functions that are similar to the original (reconstruction).
► Sampler: selects sample points on the image plane► Filter: blends multiple samples together
Sampling theory
►For band limited function, we can just increase the sampling rate
►• However, few of interesting functions in computer graphics are band limited, in particular, functions with discontinuities.
►• It is because the discontinuity always falls between two samples and the samples provides no information of the discontinuity.
Sampling theory
Aliasing
ZZ--transformstransforms
►►For discreteFor discrete--time systems, time systems, zz--transforms play transforms play
the same role of the same role of LaplaceLaplace transforms do in transforms do in
continuouscontinuous--time systemstime systems
►►As with the As with the LaplaceLaplace transform, we compute transform, we compute
forward and inverse forward and inverse zz--transforms by use of transforms by use of
transforms pairs and propertiestransforms pairs and properties
[ ]∑∞
−∞=
−=n
nznhzH ][
Bilateral Forward z-transform
∫+−=
R
n dzzzHj
nh 1 ][
2
1][
π
Bilateral Inverse z-transform
Region of ConvergenceRegion of Convergence
►► Region of the complex Region of the complex
zz--plane for which plane for which
forward forward zz--transform transform
convergesconverges Im{z}
Re{z}
Entire
plane
Im{z}
Re{z}
Complement
of a disk
Im{z}
Re{z}
Disk
Im{z}
Re{z}
Intersection
of a disk and
complement
of a disk
►► Four possibilities (Four possibilities (zz=0 =0
is a special case and is a special case and
may or may not be may or may not be
included)included)
ZZ--transform Pairstransform Pairs
►►hh[[nn] = ] = δδ[[nn]]
Region of convergence: Region of convergence:
entire entire zz--planeplane
►►hh[[nn] = ] = δδ[[nn--11]]
Region of convergence: Region of convergence:
entire entire zz--planeplane
hh[[nn--1] 1] ⇔⇔ zz--1 1 HH[[zz]]
[ ] [ ] 1 ][0
0
=== ∑∑=
−∞
−∞=
−
n
n
n
nznznzH δδ
[ ] [ ] 11
1
1 1][−
=
−∞
−∞=
− =−=−= ∑∑ zznznzHn
n
n
n δδ
[ ]
1 if
1
1
][
00
<
−
=
==
=
∑∑
∑∞
=
∞
=
−
∞
−∞=
−
z
a
z
a
z
aza
znuazH
n
n
n
nn
n
nn
►►hh[[nn] = ] = aan n uu[[nn]]
Region of Region of convergence: |convergence: |zz| | > |> |aa| which is | which is the complement the complement of a diskof a disk
[ ] azza
nuaZ
n >−
↔−
for 1
11
StabilityStability
►►Rule #1: For a causal sequence, poles are Rule #1: For a causal sequence, poles are inside the unit circle (applies to zinside the unit circle (applies to z--transform transform functions that are ratios of two polynomials)functions that are ratios of two polynomials)
►►Rule #2: More generally, unit circle is Rule #2: More generally, unit circle is included in region of convergence. (In included in region of convergence. (In continuouscontinuous--time, the imaginary axis would time, the imaginary axis would be in the region of convergence of the be in the region of convergence of the LaplaceLaplace transform.)transform.)
�� This is stable if |This is stable if |aa| < 1 by rule #1.| < 1 by rule #1.
�� It is stable if |It is stable if |zz| > || > |aa| and || and |aa| < 1 by rule #2.| < 1 by rule #2.
Inverse Inverse zz--transformtransform
►►Yuk! Using the definition requires a contour Yuk! Using the definition requires a contour
integration in the complex integration in the complex zz--plane.plane.
►►Fortunately, we tend to be interested in only Fortunately, we tend to be interested in only
a few basic signals (pulse, step, etc.)a few basic signals (pulse, step, etc.)
�� Virtually all of the signals weVirtually all of the signals we’’ll see can be built ll see can be built
up from these basic signals. up from these basic signals.
�� For these common signals, the For these common signals, the zz--transform pairs transform pairs
have been tabulated (see have been tabulated (see LathiLathi, Table 5.1), Table 5.1)
[ ] [ ] dzzzFj
nfn
jc
jc
1
2
1 −
∞+
∞−
∫=π
ExampleExample
►► Ratio of polynomial zRatio of polynomial z--
domain functionsdomain functions
►► Divide through by the Divide through by the
highest power of zhighest power of z
►► Factor denominator into Factor denominator into
firstfirst--order factorsorder factors
►► Use partial fraction Use partial fraction
decomposition to get decomposition to get
firstfirst--order termsorder terms
2
1
2
3
12][
2
2
+−
++=
zz
zzzX
21
21
2
1
2
31
21][
−−
−−
+−
++=
zz
zzzX
( )11
21
12
11
21][
−−
−−
−
−
++=
zz
zzzX
1
2
1
10
1
2
11
][−
− −+
−
+=z
A
z
ABzX
Example (Example (concon’’tt))
►►Find Find BB00 by by
polynomial divisionpolynomial division
►►Express in terms of Express in terms of
BB00
►►Solve for Solve for AA11 and and AA22
15
23
2
1212
3
2
1
1
12
1212
−
+−
+++−
−
−−
−−−−
z
zz
zzzz
( )11
1
12
11
512][
−−
−
−
−
+−+=
zz
zzX
8
2
1
121
2
11
21
921
441
1
21
1
1
21
2
2
1
21
1
1
1
=++
=
−
++=
−=−
++=
−
++=
=
−
−−
=
−
−−
−
−
z
z
z
zzA
z
zzA
Example (Example (concon’’tt))
►►Express Express XX[[zz]] in terms of in terms of BB00, , AA11, and , and AA22
►►Use table to obtain inverse Use table to obtain inverse zz--transformtransform
►►With the unilateral With the unilateral zz--transform, or the transform, or the
bilateral bilateral zz--transform with region of transform with region of
convergence, the inverse convergence, the inverse zz--transform is transform is
uniqueunique
11 1
8
2
11
92][
−− −
+
−
−=z
z
zX
[ ] [ ] [ ] [ ]nununnx
n
82
1 9 2 +
−= δ
ZZ--transform Propertiestransform Properties
►►LinearityLinearity
►►Right shift (delay)Right shift (delay)
[ ] [ ] [ ] [ ]zFazFanfanfa 22112211 +⇔+
[ ] [ ] [ ]zFzmnumnf m−⇔−−
[ ] [ ] [ ] [ ]
−+⇔− ∑
=
−−−m
n
nmmznfzzFznumnf
1
ZZ--transform Propertiestransform Properties
[ ] [ ] [ ] [ ]
[ ] [ ]{ } [ ] [ ]
[ ] [ ]
[ ] [ ]
[ ] [ ] ( )
[ ] [ ]
[ ] [ ]zFzF
zrfzmf
zrfmf
zmnfmf
zmnfmf
mnfmfZnfnfZ
mnfmfnfnf
r
rm
m
m r
mr
m n
n
n
n
m
m
m
21
21
21
21
21
2121
2121
=
=
=
−=
−=
−=∗
−=∗
∑∑
∑ ∑
∑ ∑
∑ ∑
∑
∑
∞
−∞=
−−∞
−∞=
∞
−∞=
∞
−∞=
+−
∞
−∞=
∞
−∞=
−
∞
−∞=
−∞
−∞=
∞
−∞=
∞
−∞=
►► Convolution definitionConvolution definition
►► Take Take zz--transformtransform
►► ZZ--transform definitiontransform definition
►► Interchange summationInterchange summation
►► SubstituteSubstitute rr = = nn -- mm
►► ZZ--transform definitiontransform definition
UNIT IVUNIT IV
IntroductionIntroduction
►► Impulse responseImpulse response hh[n[n] can fully characterize a LTI ] can fully characterize a LTI
system, and we can have the output of LTI system assystem, and we can have the output of LTI system as
►► The zThe z--transform of impulse response is called transform of impulse response is called transfer or transfer or
system functionsystem function HH((zz).).
►► Frequency responseFrequency response at is valid if at is valid if
ROC includes and ROC includes and
[ ] [ ] [ ]nhnxny ∗=
( ) ( ) ( ).zHzXzY =
( ) ( )1=
=z
jzHeH
ω
,1=z
( ) ( ) ( )ωωω jjjeHeXeY =
5.1 Frequency Response of LIT 5.1 Frequency Response of LIT
SystemSystem
►► Consider and Consider and
, then, then
�� magnitudemagnitude
�� phasephase
►► We will model and analyze LTI systems based on the We will model and analyze LTI systems based on the
magnitude and phase responses. magnitude and phase responses.
)()()(ωωω j
eXjjjeeXeX
∠=)()()(
ωωω jeHjjjeeHeH
∠=
)()()( ωωω jjjeHeXeY =
)()()( ωωω jjjeHeXeY ∠+∠=∠
System FunctionSystem Function
►►General form of LCCDEGeneral form of LCCDE
►►Compute the zCompute the z--transformtransform
[ ] [ ]knxbknyaM
k
k
N
k
k −=− ∑∑== 00
( )zXzbzYzak
M
k
k
N
k
k
k
−
==
−
∑∑ =00
)(
( )( )( )
∑
∑
=
−
−
===N
k
k
k
kM
k
k
za
zb
zX
zYzH
0
0
System Function: Pole/zero System Function: Pole/zero
FactorizationFactorization
►►Stability requirement can be verified.Stability requirement can be verified.
►►Choice of ROC determines causality.Choice of ROC determines causality.
►►Location of zeros and poles determines the Location of zeros and poles determines the
frequency response and phasefrequency response and phase
( )( )
( )∏
∏
=
−
=
−
−
−
=N
k
k
M
k
k
zd
zc
a
bzH
1
1
1
1
0
0
1
1 .,...,,:zeros 21 Mccc
.,...,,:poles 21 Nddd
SecondSecond--order Systemorder System
►► Suppose the system function of a LTI system isSuppose the system function of a LTI system is
►► To find the difference equation that is satisfied by To find the difference equation that is satisfied by
the input and out of this systemthe input and out of this system
►► Can we know the impulse response? Can we know the impulse response?
.
)4
31)(
2
11(
)1()(
11
21
−−
−
+−
+=
zz
zzH
)(
)(
8
3
4
11
21
)4
31)(
2
11(
)1()(
21
21
11
21
zX
zY
zz
zz
zz
zzH =
−+
++=
+−
+=
−−
−−
−−
−
]2[2]1[2][]2[8
3]1[
4
1][ −+−+=−−−+ nxnxnxnynyny
System Function: StabilitySystem Function: Stability
►►Stability of LTI system:Stability of LTI system:
►►This condition is identical to the condition This condition is identical to the condition
that that
�� The stability condition is equivalent to the The stability condition is equivalent to the
condition that the ROC of condition that the ROC of HH((zz) includes the unit ) includes the unit
circle.circle.
∑∞
−∞=
∞<n
nh ][
.1 when][ =∞<∑∞
−∞=
−zznh
n
n
System Function: CausalitySystem Function: Causality
►► If the system is causal, it follows that If the system is causal, it follows that hh[[nn] must be a right] must be a right--
sided sequence. The ROC of sided sequence. The ROC of HH((zz) must be outside the ) must be outside the
outermostoutermost pole.pole.
►► If the system is antiIf the system is anti--causal, it follows that causal, it follows that hh[[nn] must be a ] must be a
leftleft--sided sequence. The ROC of sided sequence. The ROC of HH((zz) must be inside the ) must be inside the
innermostinnermost pole.pole.
1a
Im
Re1a
Im
Reba
Im
Re
Right-sided(causal)
Left-sided(anti-causal)
Two-sided(non-causal)
Determining the ROCDetermining the ROC
►►Consider the LTI systemConsider the LTI system
►►The system function is obtained asThe system function is obtained as
][]2[]1[2
5][ nxnynyny =−+−−
)21)(2
11(
1
2
51
1)(
11
21
−−
−−
−−
=
+−
=
zz
zz
zH
System Function: Inverse SystemsSystem Function: Inverse Systems
►► is an inverse system for , ifis an inverse system for , if
►► The The ROCsROCs of must overlap.of must overlap.
►► Useful for canceling the effects of another systemUseful for canceling the effects of another system
►► See the discussion in Sec.5.2.2 regarding ROCSee the discussion in Sec.5.2.2 regarding ROC
( )zH i( )zH
1)()()( == zHzHzG i
)(
1)(
zHzH i =
)(
1)(
ω
ω
j
j
ieH
eH =⇔
[ ] [ ] [ ] [ ]nnhnhng i δ=∗=⇔
)( and )( zHzH i
AllAll--pass Systempass System
►►A system of the form (or cascade of these)A system of the form (or cascade of these)
( )1
1
1 −
∗−
−
−=
az
azZH Ap
( ) 1=ωjAp eH
( )ω
ωω
ω
ωω
j
jj
j
jj
Apae
eae
ae
aeeH
−
−
−
∗−
−
−=
−
−=
1
*1
1
θ
θ
j
j
era
rea
1*/1 :zero
:pole
−=
=
AllAll--pass System: General Formpass System: General Form
►►In general, all pass systems have formIn general, all pass systems have form
( ) ∏∏=
−−
−−
=−
−
−−
−−
−
−=
cr M
k kk
kk
M
k k
kAp
zeze
ezez
zd
dzzH
11*1
1*1
11
1
)1)(1(
))((
1
Causal/stable: 1, <kk de
real poles complex poles
AllAll--Pass System ExamplePass System Example
0.8
0.5
z-plane
Unit
circle
4
3−
3
4− 2
Re
Im
1 and 2 == cr MM
zeros. and poles 42 has system pass-all This =+== rc MMNM
θθ jj erre 1conjugate & reciprocal :zero:pole − →
MinimumMinimum--Phase SystemPhase System
►► MinimumMinimum--phase system:phase system: all zeros and all poles are all zeros and all poles are
inside the unit circle.inside the unit circle.
►► The name The name minimumminimum--phasephase comes from a property of the comes from a property of the
phase response (minimum phasephase response (minimum phase--lag/grouplag/group--delay).delay).
►► MinimumMinimum--phase systems have some special properties.phase systems have some special properties.
►► When we design a filter, we may have multiple choices to When we design a filter, we may have multiple choices to
satisfy the certain requirements. Usually, we prefer the satisfy the certain requirements. Usually, we prefer the
minimum phase which is unique.minimum phase which is unique.
►► All systems can be represented as a minimumAll systems can be represented as a minimum--phase phase
system and an allsystem and an all--pass system.pass system.
UNIT VUNIT V
ExampleExample
►►Block diagram representation ofBlock diagram representation of
[ ] [ ] [ ] [ ]nxb2nya1nyany 021 +−+−=
Block Diagram RepresentationBlock Diagram Representation
►►LTI systems with LTI systems with
rational system rational system
function can be function can be
represented as represented as
constantconstant--coefficient coefficient
difference equationdifference equation
►►The implementation of The implementation of
difference equations difference equations
requires delayed requires delayed
values of thevalues of the
�� inputinput
Direct Form IDirect Form I
►►General form of difference equationGeneral form of difference equation
►►Alternative equivalent formAlternative equivalent form
[ ] [ ]∑∑==
−=−M
0kk
N
0kk knxbknya
[ ] [ ] [ ]∑∑==
−=−−M
0kk
N
1kk knxbknyany
Direct Form IDirect Form I
►►Transfer function can be written asTransfer function can be written as
►►Direct Form I RepresentsDirect Form I Represents
( )∑
∑
=
−
=
−
−
=N
1k
kk
M
0k
kk
za1
zb
zH
( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )zV
za1
1zVzHzY
zXzbzXzHzV
zb
za1
1zHzHzH
N
1k
kk
2
M
0k
kk1
M
0k
kkN
1k
kk
12
−
==
==
−
==
∑
∑
∑∑
=
−
=
−
=
−
=
−
[ ] [ ]
[ ] [ ] [ ]nvknyany
knxbnv
N
1kk
M
0kk
+−=
−=
∑
∑
=
=
Alternative RepresentationAlternative Representation
►►Replace order of cascade LTI systemsReplace order of cascade LTI systems( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )zWzbzWzHzY
zX
za1
1zXzHzW
za1
1zbzHzHzH
M
0k
kk1
N
1k
kk
2
N
1k
kk
M
0k
kk21
==
−
==
−
==
∑
∑
∑∑
=
−
=
−
=
−=
−
[ ] [ ] [ ]
[ ] [ ]∑
∑
=
=
−=
+−=
M
0kk
N
1kk
knwbny
nxknwanw
Alternative Block DiagramAlternative Block Diagram
►►We can change the order of the cascade We can change the order of the cascade
systemssystems
[ ] [ ] [ ]
[ ] [ ]∑
∑
=
=
−=
+−=
M
0kk
N
1kk
knwbny
nxknwanw
Direct Form IIDirect Form II
►► No need to store the same data No need to store the same data
twice in previous systemtwice in previous system
►► So we can collapse the delay So we can collapse the delay
elements into one chainelements into one chain
►► This is called Direct Form II or This is called Direct Form II or
the Canonical Formthe Canonical Form
►► Theoretically no difference Theoretically no difference
between Direct Form I and IIbetween Direct Form I and II
►► Implementation wise Implementation wise
�� Less memory in Direct IILess memory in Direct II
�� Difference when using Difference when using
finitefinite--precision arithmeticprecision arithmetic
Signal Flow Graph RepresentationSignal Flow Graph Representation
►►Similar to block diagram representationSimilar to block diagram representation
�� Notational differencesNotational differences
►►A network of directed branches connected A network of directed branches connected
at nodesat nodes
ExampleExample
►►Representation of Direct Form II with signal Representation of Direct Form II with signal
flow graphsflow graphs [ ] [ ] [ ][ ] [ ][ ] [ ] [ ][ ] [ ][ ] [ ]nwny
1nwnw
nwbnwbnw
nwnw
nxnawnw
3
24
41203
12
41
=
−=
+=
=
+=
[ ] [ ] [ ][ ] [ ] [ ]1nwbnwbny
nx1nawnw
1110
11
−+=
+−=
Determination of System Determination of System
Function from Flow GraphFunction from Flow Graph[ ] [ ] [ ][ ] [ ][ ] [ ] [ ][ ] [ ][ ] [ ] [ ]nwnwny
1nwnw
nxnwnw
nwnw
nxnwnw
42
34
23
12
41
+=
−=
+=
α=
−=
( ) ( ) ( )( ) ( )( ) ( ) ( )
( ) ( )( ) ( ) ( )zWzWzY
zzWzW
zXzWzW
zWzW
zXzWzW
42
134
23
12
41
+=
=
+=
α=
−=
−
( ) ( )( )
( ) ( ) ( )
( ) ( ) ( )zWzWzY
z1
1zzXzW
z1
1zzXzW
42
1
1
4
1
1
2
+=
α−
α−=
α−
−α=
−
−
−
−
( ) ( )( )
[ ] [ ] [ ]nu1nunh
z1
z
zX
zYzH
1n1n
1
1
+−
−
−
α−−α=
α−
α−==
Basic Structures for IIR Systems: Basic Structures for IIR Systems:
Direct Form IDirect Form I
Basic Structures for IIR Systems: Basic Structures for IIR Systems:
Direct Form IIDirect Form II
Basic Structures for IIR Systems: Basic Structures for IIR Systems:
Cascade FormCascade Form►► General form for cascade implementationGeneral form for cascade implementation
►► More practical form in 2More practical form in 2ndnd order systemsorder systems
( )( ) ( )( )
( ) ( )( )∏∏
∏∏
=
−∗−
=
−
=
−∗−
=
−
−−−
−−−
=21
21
N
1k
1k
1k
N
1k
1k
M
1k
1k
1k
M
1k
1k
zd1zd1zc1
zg1zg1zf1
AzH
( ) ∏=
−−
−−
−−
−+=
1M
1k2
k21
k1
2k2
1k1k0
zaza1
zbzbbzH
ExampleExample
►► Cascade of Direct Form I subsectionsCascade of Direct Form I subsections
►► Cascade of Direct Form II subsectionsCascade of Direct Form II subsections
( ) ( )( )( )( )
( )( )
( )( )1
1
1
1
11
11
21
21
z25.01
z1
z5.01
z1
z25.01z5.01
z1z1
z125.0z75.01
zz21zH
−
−
−
−
−−
−−
−−
−−
−
+
−
+=
−−
++=
+−
++=
Basic Structures for IIR Systems: Basic Structures for IIR Systems:
Parallel FormParallel Form►► Represent system function using partial fraction expansionRepresent system function using partial fraction expansion
►► Or by Or by pairingthepairingthe real polesreal poles
( ) ( )( )( )∑ ∑∑
= =−∗−
−
−=
−
−−
−+
−+=
P PP N
1k
N
1k1
k1
k
1kk
1k
kN
0k
kk
zd1zd1
ze1B
zc1
AzCzH
( ) ∑∑=
−−
−
=
−
−−
++=
SP N
1k2
k21
k1
1k1k0
N
0k
kk
zaza1
zeezCzH
ExampleExample►►Partial Fraction ExpansionPartial Fraction Expansion
►►Combine poles to getCombine poles to get
( )( ) ( )1121
21
z25.01
25
z5.01
188
z125.0z75.01
zz21zH
−−−−
−−
−−
−+=
+−
++=
( )21
1
z125.0z75.01
z878zH
−−
−
+−
+−+=
Transposed FormsTransposed Forms
►►Linear signal flow graph property:Linear signal flow graph property:
�� Transposing doesnTransposing doesn’’t change the inputt change the input--output output
relationrelation
►►Transposing:Transposing:
�� Reverse directions of all branchesReverse directions of all branches
�� Interchange input and output nodesInterchange input and output nodes
►►Example:Example:
( )1az1
1zH
−−=
ExampleExample
Transpose
►►Both have the same system function or Both have the same system function or
difference equationdifference equation
[ ] [ ] [ ] [ ] [ ] [ ]2nxb1nxbnxb2nya1nyany 21021 −+−++−+−=
Basic Structures for FIR Systems: Direct FormBasic Structures for FIR Systems: Direct Form
►►Special cases of IIR direct form structuresSpecial cases of IIR direct form structures
►► Transpose of direct form I gives direct form II Transpose of direct form I gives direct form II
►► Both forms are equal for FIR systemsBoth forms are equal for FIR systems
►►Tapped delay lineTapped delay line
Basic Structures for FIR Systems: Basic Structures for FIR Systems:
Cascade FormCascade Form
►►Obtained by factoring the polynomial Obtained by factoring the polynomial
system functionsystem function
( ) [ ] ( )∑ ∏= =
−−− ++==M
0n
M
1k
2k2
1k1k0
nS
zbzbbznhzH
Structures for LinearStructures for Linear--Phase FIR Phase FIR
SystemsSystems►► Causal FIR system with generalized linear phase are Causal FIR system with generalized linear phase are
symmetricsymmetric::
►► Symmetry means we can half the number of multiplicationsSymmetry means we can half the number of multiplications
►► Example: For even M and type I or type III systemsExample: For even M and type I or type III systems::
[ ] [ ][ ] [ ] IV)or II (type M0,1,...,n nhnMh
III)or I (type M0,1,...,n nhnMh
=−=−
==−
[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]
[ ] [ ] [ ] [ ] [ ] [ ]
[ ] [ ] [ ]( ) [ ] [ ]2/Mnx2/MhkMnxknxkh
kMnxkMh2/Mnx2/Mhknxkh
knxkh2/Mnx2/Mhknxkhknxkhny
12/M
0k
12/M
0k
12/M
0k
M
12/Mk
12/M
0k
M
0k
−++−+−=
+−−+−+−=
−+−+−=−=
∑
∑∑
∑∑∑
−
=
−
=
−
=
+=
−
==
Structures for LinearStructures for Linear--Phase FIR Phase FIR
SystemsSystems
►►Structure for even MStructure for even M
►►Structure for odd MStructure for odd M
