2. SIGNALSIGNAL
►Signal is a physical quantity that varies withSignal is a physical quantity that varies with
respect to time , space or any otherrespect to time , space or any other
independent variableindependent variable
Eg x(t)= sin t.Eg x(t)= sin t.
►the major classifications of the signalthe major classifications of the signal
are:are:
(i) Discrete time signal(i) Discrete time signal
(ii)(ii) Continuous time signalContinuous time signal
3. Unit Step &Unit ImpulseUnit Step &Unit Impulse
Discrete time Unit impulse is defined asDiscrete time Unit impulse is defined as
δ [n]= {0, n≠ 0δ [n]= {0, n≠ 0
{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 byDiscrete time unit step signal is defined by
U[n]={0,n=0U[n]={0,n=0
{1,n>= 0{1,n>= 0
Continuous time unit impulse is defined asContinuous time unit impulse is defined as
δ (t)={1, t=0δ (t)={1, t=0
{0, t ≠ 0{0, t ≠ 0
Continuous time Unit step signal is defined asContinuous time Unit step signal is defined as
U(t)={0, t<0U(t)={0, t<0
{1, t≥0{1, t≥0
4. ► Periodic Signal & Aperiodic SignalPeriodic Signal & Aperiodic Signal
A signal is said to be periodic ,if it exhibits periodicity.i.e.,A signal is said to be periodic ,if it exhibits periodicity.i.e.,
X(t +T)=x(t), for all values of t. Periodic signal has theX(t +T)=x(t), for all values of t. Periodic signal has the
property that it is unchanged by a time shift of T. A signalproperty that it is unchanged by a time shift of T. A signal
that does not satisfy the above periodicity property isthat does not satisfy the above periodicity property is
called an aperiodic signalcalled an aperiodic signal
► even and odd signal ?even and odd signal ?
A discrete time signal is said to be even when, x[-n]=x[n].A discrete time signal is said to be even when, x[-n]=x[n].
The continuous time signal is said to be even when, x(-t)=The continuous time signal is said to be even when, x(-t)=
x(t) For example,Cosωn is an even signal.x(t) For example,Cosωn is an even signal.
SIGNALSIGNAL
5. Energy and power signalEnergy and power signal
► A signal is said to be energy signal if itA 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 itA 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 notIf the above two conditions are not
satisfied then the signal is said to besatisfied then the signal is said to be
neigther energy nor power signalneigther energy nor power signal
6. Fourier SeriesFourier Series
The Fourier series represents a periodic signal in terms ofThe 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 ofThe complex exponential Fourier coefficients are a sequence of
complex numbers representing the frequency componentcomplex numbers representing the frequency component ωω00k.k.
∫
ω−
=
p
0
tik
k dte)t(x
p
1
X 0
∑
−
=
ω−
=
1p
0n
nik
k
0
e)n(x
p
1
X
∑
−
=
ω
=
1p
0k
nik
k
0
eX)n(x ∑
∞
−∞=
ω
=
k
tik
k
0
eX)t(x
7. Fourier seriesFourier series
► Fourier series: a complicated waveform analyzed into aFourier 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 x(t) with a period T0 may beA continuous periodic signal x(t) with a period T0 may be
represented by:represented by:
X(t)=ΣX(t)=Σ∞∞
k=1k=1 ((AAkk coscos kkωω t + Bt + Bkk sinsin kkωω t)+t)+ AA00
► Dirichlet conditionsDirichlet conditions must be placed onmust be placed on x(t)x(t) for the series tofor the series to
be valid: the integral of the magnitude ofbe valid: the integral of the magnitude of x(t)x(t) over aover a
complete period must be finite, and the signal can onlycomplete period must be finite, and the signal can only
have a finite number of discontinuities in any finite intervalhave a finite number of discontinuities in any finite interval
8. Trigonometric form for Fourier seriesTrigonometric form for Fourier series
►If the two fundamental components of aIf the two fundamental components of a
periodic signal areB1cosω0t and C1sinω0t,periodic signal areB1cosω0t and C1sinω0t,
then their sum is expressed by trigonometricthen their sum is expressed by trigonometric
identities:identities:
►X(t)=X(t)= AA00 ++ ΣΣ∞∞
k=1k=1 ((BBkk
22
++AAkk
22
))1/21/2
(C(Ckk coscos kkωω t-t- φφkk) or) or
►X(t)=X(t)= AA00 ++ ΣΣ∞∞
k=1k=1 ((BBkk
22
++AAkk
22
))1/21/2
(C(Ckk sin ksin kωω t+t+ φφkk))
10. Fourier TransformFourier Transform
► Viewed periodic functions in terms of frequency componentsViewed periodic functions in terms of frequency components
(Fourier series) as well as ordinary functions of time(Fourier series) as well as ordinary functions of time
► Viewed LTI systems in terms of what they do to frequencyViewed 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 time-domainViewed LTI systems in terms of what they do to time-domain
signals (convolution with impulse response)signals (convolution with impulse response)
► View aperiodic functions in terms of frequency components viaView aperiodic functions in terms of frequency components via
Fourier transformFourier transform
► Define (continuous-time) Fourier transform and DTFTDefine (continuous-time) Fourier transform and DTFT
► Gain insight into the meaning of Fourier transform throughGain insight into the meaning of Fourier transform through
comparison with Fourier seriescomparison with Fourier series
11. 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
12. Continuous Time Fourier TransformContinuous Time Fourier Transform
We can extend the formula for continuous-time Fourier seriesWe can extend the formula for continuous-time Fourier series
coefficients for a periodic signalcoefficients for a periodic signal
to aperiodic signals as well. The continuous-time Fourierto aperiodic signals as well. The continuous-time Fourier
series is not defined for aperiodic signals, but we call theseries is not defined for aperiodic signals, but we call the
formulaformula
the (continuous time)the (continuous time)
Fourier transformFourier transform ..
∫∫
−
ω−ω−
==
2/p
2/p
tik
p
0
tik
k dte)t(x
p
1
dte)t(x
p
1
X 00
∫
∞
∞−
ω−=ω dte)t(x)(X ti
13. Inverse TransformsInverse Transforms
If we have the full sequence of Fourier coefficients for a periodicIf we have the full sequence of Fourier coefficients for a periodic
signal, we can reconstruct it by multiplying the complex sinusoidssignal, we can reconstruct it by multiplying the complex sinusoids
of frequencyof frequency ωω00k by the weights Xk by the weights Xkk and summing:and summing:
We can perform a similar reconstruction for aperiodic signalsWe can perform a similar reconstruction for aperiodic signals
These are called theThese are called the inverse transformsinverse transforms ..
∑
−
=
ω
=
1p
0k
nik
k
0
eX)n(x ∑
∞
−∞=
ω
=
k
tik
k
0
eX)t(x
∫
∞
∞−
ω
ωω
π
= de)(X
2
1
)t(x ti
∫
π
π−
ω ωω
π
= de)(X
2
1
)n(x ni
14. Fourier Transform of Impulse FunctionsFourier Transform of Impulse Functions
Find the Fourier transform of the Dirac delta function:Find the Fourier transform of the Dirac delta function:
Find the DTFT of the Kronecker delta function:Find the DTFT of the Kronecker delta function:
The delta functions contain all frequencies at equal amplitudes.The delta functions contain all frequencies at equal amplitudes.
Roughly speaking, that’s why the system response to an impulseRoughly speaking, that’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−
∑∑
15. Laplace TransformLaplace Transform
► Lapalce transform is a generalization of the Fourier transform in theLapalce transform is a generalization of the Fourier transform in the
sense that it allows “complex frequency” whereas Fourier analysis cansense that it allows “complex frequency” whereas Fourier analysis can
only handle “real frequency”. Like Fourier transform, Lapalce transformonly handle “real frequency”. Like Fourier transform, Lapalce transform
allows us to analyze a “linear circuit” problem, no matter howallows us to analyze a “linear circuit” problem, no matter how
complicated the circuit is, in the frequency domain in stead of in hecomplicated the circuit is, in the frequency domain in stead of in he
time domain.time domain.
► Mathematically, it produces the benefit of converting a set ofMathematically, it produces the benefit of converting a set of
differential equations into a corresponding set of algebraic equations,differential equations into a corresponding set of algebraic equations,
which are much easier to solve. Physically, it produces more insight ofwhich are much easier to solve. Physically, it produces more insight of
the circuit and allows us to know the bandwidth, phase, and transferthe circuit and allows us to know the bandwidth, phase, and transfer
characteristics important for circuit analysis and design.characteristics important for circuit analysis and design.
► Most importantly, Laplace transform lifts the limit of Fourier analysis toMost importantly, Laplace transform lifts the limit of Fourier analysis to
allow us to find both the steady-state and “transient” responses of aallow us to find both the steady-state and “transient” responses of a
linear circuit. Using Fourier transform, one can only deal with helinear circuit. Using Fourier transform, one can only deal with he
steady state behavior (i.e. circuit response under indefinite sinusoidalsteady state behavior (i.e. circuit response under indefinite sinusoidal
excitation).excitation).
► Using Laplace transform, one can find the response under any types ofUsing Laplace transform, one can find the response under any types of
excitation (e.g. switching on and off at any given time(s), sinusoidal,excitation (e.g. switching on and off at any given time(s), sinusoidal,
impulse, square wave excitations, etcimpulse, square wave excitations, etc ..
17. Application of Laplace Transform toApplication of Laplace Transform to
Circuit AnalysisCircuit Analysis
18. system
►• A system is an operation that transforms
input signal x into output signal y.
19. 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
20. 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”.
21. 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]
23. 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
24. 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.
27. ZZ-transforms-transforms
►For discrete-time systems,For discrete-time systems, zz-transforms play-transforms play
the same role of Laplace transforms do inthe same role of Laplace transforms do in
continuous-time systemscontinuous-time systems
►As with the Laplace transform, we computeAs with the Laplace transform, we compute
forward and inverseforward and inverse zz-transforms by use of-transforms by use of
transforms pairs and propertiestransforms pairs and properties
[ ]∑
∞
−∞=
−
=
n
n
znhzH ][
Bilateral Forward z-transform
∫
+−
=
R
n
dzzzH
j
nh 1
][
2
1
][
π
Bilateral Inverse z-transform
28. Region of ConvergenceRegion of Convergence
► Region of the complexRegion of the complex
zz-plane for which-plane for which
forwardforward 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 andis a special case and
may or may not bemay or may not be
included)included)
29. ZZ-transform Pairs-transform Pairs
►hh[[nn] =] = δδ[[nn]]
Region of convergence:Region of convergence:
entireentire zz-plane-plane
►hh[[nn] =] = δδ[[n-1n-1]]
Region of convergence:Region of convergence:
entireentire zz-plane-plane
hh[[nn-1]-1] ⇔⇔ zz-1-1
HH[[zz]]
[ ] [ ] 1][
0
0
=== ∑∑ =
−
∞
−∞=
−
n
n
n
n
znznzH δδ
[ ] [ ] 1
1
1
11][ −
=
−
∞
−∞=
−
=−=−= ∑∑ zznznzH
n
n
n
n
δδ
[ ]
1if
1
1
][
00
<
−
=
==
=
∑∑
∑
∞
=
∞
=
−
∞
−∞=
−
z
a
z
a
z
a
za
znuazH
n
n
n
nn
n
nn
►hh[[nn] =] = aann
uu[[nn]]
Region ofRegion of
convergence: |convergence: |zz||
> |> |aa| which is the| which is the
complement of acomplement of a
30. [ ] az
za
nua
Z
n
>
−
↔ −
for
1
1
1
StabilityStability
►Rule #1: For a causal sequence, poles areRule #1: For a causal sequence, poles are
inside the unit circle (applies to z-transforminside the unit circle (applies to z-transform
functions that are ratios of two polynomials)functions that are ratios of two polynomials)
►Rule #2: More generally, unit circle isRule #2: More generally, unit circle is
included in region of convergence. (Inincluded in region of convergence. (In
continuous-time, the imaginary axis wouldcontinuous-time, the imaginary axis would
be in the region of convergence of thebe in the region of convergence of the
Laplace transform.)Laplace transform.)
This is stable if |This is stable if |aa| < 1 by rule #1.| < 1 by rule #1.
31. InverseInverse zz-transform-transform
►Yuk! Using the definition requires a contourYuk! Using the definition requires a contour
integration in the complexintegration in the complex zz-plane.-plane.
►Fortunately, we tend to be interested in onlyFortunately, 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 we’ll see can be builtVirtually all of the signals we’ll see can be built
up from these basic signals.up from these basic signals.
For these common signals, theFor these common signals, the zz-transform-transform
pairs have been tabulated (see Lathi, Table 5.1)pairs have been tabulated (see Lathi, Table 5.1)
[ ] [ ] dzzzF
j
nf n
jc
jc
1
2
1 −
∞+
∞−
∫=
π
32. ExampleExample
► Ratio of polynomial z-Ratio of polynomial z-
domain functionsdomain functions
► Divide through by theDivide through by the
highest power of zhighest power of z
► Factor denominator intoFactor denominator into
first-order factorsfirst-order factors
► Use partial fractionUse partial fraction
decomposition to getdecomposition to get
first-order termsfirst-order terms
2
1
2
3
12
][
2
2
+−
++
=
zz
zz
zX
21
21
2
1
2
3
1
21
][
−−
−−
+−
++
=
zz
zz
zX
( )11
21
1
2
1
1
21
][
−−
−−
−
−
++
=
zz
zz
zX
1
2
1
1
0
1
2
1
1
][ −
− −
+
−
+=
z
A
z
A
BzX
33. Example (con’t)Example (con’t)
►FindFind BB00 byby
polynomial divisionpolynomial division
►Express in terms ofExpress in terms of
BB00
►Solve forSolve for AA11 andand AA22
15
23
2
121
2
3
2
1
1
12
1212
−
+−
+++−
−
−−
−−−−
z
zz
zzzz
( )11
1
1
2
1
1
51
2][
−−
−
−
−
+−
+=
zz
z
zX
8
2
1
121
2
1
1
21
9
21
441
1
21
1
1
21
2
2
1
21
1
1
1
=
++
=
−
++
=
−=
−
++
=
−
++
=
=
−
−−
=
−
−−
−
−
z
z
z
zz
A
z
zz
A
34. Example (con’t)Example (con’t)
►ExpressExpress XX[[zz]] in terms ofin terms of BB00 ,, AA11 , and, and AA22
►Use table to obtain inverseUse table to obtain inverse zz-transform-transform
►With the unilateralWith the unilateral zz-transform, or the-transform, or the
bilateralbilateral zz-transform with region of-transform with region of
convergence, the inverseconvergence, the inverse zz-transform is-transform is
uniqueunique
1
1 1
8
2
1
1
9
2][ −
− −
+
−
−=
zz
zX
[ ] [ ] [ ] [ ]nununnx
n
8
2
1
92 +
−= δ
38. IntroductionIntroduction
► Impulse responseImpulse response hh[n] can fully characterize a LTI[n] 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 z-transform of impulse response is calledThe z-transform of impulse response is called transfer ortransfer or
system functionsystem function HH((zz).).
► Frequency responseFrequency response at is valid if ROCat is valid if ROC
includes andincludes and
[ ] [ ] [ ]nhnxny ∗=
( ) ( ) ( ).zHzXzY =
( ) ( ) 1=
= z
j
zHeH ω
,1=z
( ) ( ) ( )ωωω jjj
eHeXeY =
39. 5.1 Frequency Response of LIT5.1 Frequency Response of LIT
SystemSystem
► Consider and ,Consider and ,
thenthen
magnitudemagnitude
phasephase
► We will model and analyze LTI systems based on theWe will model and analyze LTI systems based on the
magnitude and phase responses.magnitude and phase responses.
)(
)()(
ω
ωω j
eXjjj
eeXeX ∠
= )(
)()(
ω
ωω j
eHjjj
eeHeH ∠
=
)()()( ωωω jjj
eHeXeY =
)()()( ωωω jjj
eHeXeY ∠+∠=∠
40. System FunctionSystem Function
►General form of LCCDEGeneral form of LCCDE
►Compute the z-transformCompute the z-transform
[ ] [ ]knxbknya
M
k
k
N
k
k −=− ∑∑ == 00
( )zXzbzYza k
M
k
k
N
k
k
k
−
==
−
∑∑ =
00
)(
( ) ( )
( )
∑
∑
=
−
−
=
== N
k
k
k
k
M
k
k
za
zb
zX
zY
zH
0
0
41. System Function: Pole/zeroSystem 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 theLocation of zeros and poles determines the
frequency response and phasefrequency response and phase
( )
( )
( )∏
∏
=
−
=
−
−
−
= N
k
k
M
k
k
zd
zc
a
b
zH
1
1
1
1
0
0
1
1 .,...,,:zeros 21 Mccc
.,...,,:poles 21 Nddd
42. Second-order SystemSecond-order 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 byTo 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
3
1)(
2
1
1(
)1(
)(
11
21
−−
−
+−
+
=
zz
z
zH
)(
)(
8
3
4
1
1
21
)
4
3
1)(
2
1
1(
)1(
)(
21
21
11
21
zX
zY
zz
zz
zz
z
zH =
−+
++
=
+−
+
=
−−
−−
−−
−
]2[2]1[2][]2[
8
3
]1[
4
1
][ −+−+=−−−+ nxnxnxnynyny
43. System Function: StabilitySystem Function: Stability
►Stability of LTI system:Stability of LTI system:
►This condition is identical to the conditionThis condition is identical to the condition
thatthat
The stability condition is equivalent to theThe stability condition is equivalent to the
condition that the ROC ofcondition that the ROC of HH((zz) includes the unit) includes the unit
circle.circle.
∑
∞
−∞=
∞<
n
nh ][
.1when][ =∞<∑
∞
−∞=
−
zznh
n
n
44. System Function: CausalitySystem Function: Causality
► If the system is causal, it follows thatIf the system is causal, it follows that hh[[nn] must be a right-] must be a right-
sided sequence. The ROC ofsided sequence. The ROC of HH((zz) must be outside the) must be outside the
outermostoutermost pole.pole.
► If the system is anti-causal, it follows thatIf the system is anti-causal, it follows that hh[[nn] must be a] must be a
left-sided sequence. The ROC ofleft-sided sequence. The ROC of HH((zz) must be inside the) must be inside the
innermostinnermost pole.pole.
1a
Im
Re 1a
Im
Re ba
Im
Re
Right-sided
(causal)
Left-sided
(anti-causal)
Two-sided
(non-causal)
45. 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
1
1(
1
2
5
1
1
)(
11
21
−−
−−
−−
=
+−
=
zz
zz
zH
46. System Function: Inverse SystemsSystem Function: Inverse Systems
► is an inverse system for , ifis an inverse system for , if
► The ROCs of must overlap.The ROCs 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
( )zHi
( )zH
1)()()( == zHzHzG i
)(
1
)(
zH
zHi =
)(
1
)( ω
ω
j
j
i
eH
eH =⇔
[ ] [ ] [ ] [ ]nnhnhng i δ=∗=⇔
)(and)( zHzH i
47. All-pass SystemAll-pass System
►A system of the form (or cascade of these)A system of the form (or cascade of these)
( ) 1
1
1 −
∗−
−
−
=
az
az
ZHAp
( ) 1=ωj
Ap eH
( ) ω
ω
ω
ω
ω
ω
j
j
j
j
j
j
Ap
ae
ea
e
ae
ae
eH −
−
−
∗−
−
−
=
−
−
=
1
*1
1
θ
θ
j
j
era
rea
1
*/1:zero
:pole
−
=
=
48. All-pass System: General FormAll-pass System: General Form
►In general, all pass systems have formIn general, all pass systems have form
( ) ∏∏ =
−−
−−
=
−
−
−−
−−
−
−
=
cr M
k kk
kk
M
k k
k
Ap
zeze
ezez
zd
dz
zH
1
1*1
1*1
1
1
1
)1)(1(
))((
1
Causal/stable: 1, <kk de
real poles complex poles
49. All-Pass System ExampleAll-Pass System Example
0.8
0.5
z-plane
Unit
circle
4
3
−
3
4
− 2
Re
Im
1and2 == cr MM
zeros.andpoles42hassystempass-allThis =+== rc MMNM
θθ jj
erre 1conjugate&reciprocal
:zero:pole −
→
50. Minimum-Phase SystemMinimum-Phase System
► Minimum-phase system:Minimum-phase system: all zeros and all poles areall zeros and all poles are
inside the unit circle.inside the unit circle.
► The nameThe name minimum-phaseminimum-phase comes from a property of thecomes from a property of the
phase response (minimum phase-lag/group-delay).phase response (minimum phase-lag/group-delay).
► Minimum-phase systems have some special properties.Minimum-phase systems have some special properties.
► When we design a filter, we may have multiple choices toWhen we design a filter, we may have multiple choices to
satisfy the certain requirements. Usually, we prefer thesatisfy the certain requirements. Usually, we prefer the
minimum phase which is unique.minimum phase which is unique.
► All systems can be represented as a minimum-phaseAll systems can be represented as a minimum-phase
system and an all-pass system.system and an all-pass system.
53. Block Diagram RepresentationBlock Diagram Representation
►LTI systems withLTI systems with
rational systemrational system
function can befunction can be
represented asrepresented as
constant-coefficientconstant-coefficient
difference equationdifference equation
►The implementation ofThe implementation of
difference equationsdifference equations
requires delayedrequires delayed
values of thevalues of the
54. Direct Form IDirect Form I
►General form of difference equationGeneral form of difference equation
►Alternative equivalent formAlternative equivalent form
[ ] [ ]∑∑ ==
−=−
M
0k
k
N
0k
k knxbˆknyaˆ
[ ] [ ] [ ]∑∑ ==
−=−−
M
0k
k
N
1k
k knxbknyany
55. 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
k
k
M
0k
k
k
za1
zb
zH
( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )zV
za1
1
zVzHzY
zXzbzXzHzV
zb
za1
1
zHzHzH
N
1k
k
k
2
M
0k
k
k1
M
0k
k
kN
1k
k
k
12
−
==
==
−
==
∑
∑
∑
∑
=
−
=
−
=
−
=
−
[ ] [ ]
[ ] [ ] [ ]nvknyany
knxbnv
N
1k
k
M
0k
k
+−=
−=
∑
∑
=
=
56. Alternative RepresentationAlternative Representation
►Replace order of cascade LTI systemsReplace order of cascade LTI systems
( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )zWzbzWzHzY
zX
za1
1
zXzHzW
za1
1
zbzHzHzH
M
0k
k
k1
N
1k
k
k
2
N
1k
k
k
M
0k
k
k21
==
−
==
−
==
∑
∑
∑
∑
=
−
=
−
=
−=
−
[ ] [ ] [ ]
[ ] [ ]∑
∑
=
=
−=
+−=
M
0k
k
N
1k
k
knwbny
nxknwanw
57. Alternative Block DiagramAlternative Block Diagram
►We can change the order of the cascadeWe can change the order of the cascade
systemssystems
[ ] [ ] [ ]
[ ] [ ]∑
∑
=
=
−=
+−=
M
0k
k
N
1k
k
knwbny
nxknwanw
58. Direct Form IIDirect Form II
► No need to store the same dataNo need to store the same data
twice in previous systemtwice in previous system
► So we can collapse the delaySo we can collapse the delay
elements into one chainelements into one chain
► This is called Direct Form II orThis is called Direct Form II or
the Canonical Formthe Canonical Form
► Theoretically no differenceTheoretically no difference
between Direct Form I and IIbetween Direct Form I and II
► Implementation wiseImplementation wise
Less memory in Direct IILess memory in Direct II
Difference when using finite-Difference when using finite-
precision arithmeticprecision arithmetic
59. Signal Flow GraphSignal Flow Graph
RepresentationRepresentation
►Similar to block diagram representationSimilar to block diagram representation
Notational differencesNotational differences
►A network of directed branches connectedA network of directed branches connected
at nodesat nodes
60. ExampleExample
►Representation of Direct Form II with signalRepresentation of Direct Form II with signal
flow graphsflow graphs [ ] [ ] [ ]
[ ] [ ]
[ ] [ ] [ ]
[ ] [ ]
[ ] [ ]nwny
1nwnw
nwbnwbnw
nwnw
nxnawnw
3
24
41203
12
41
=
−=
+=
=
+=
[ ] [ ] [ ]
[ ] [ ] [ ]1nwbnwbny
nx1nawnw
1110
11
−+=
+−=
62. Basic Structures for IIR Systems:Basic Structures for IIR Systems:
Direct Form IDirect Form I
63. Basic Structures for IIR Systems:Basic Structures for IIR Systems:
Direct Form IIDirect Form II
64. 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
1
k
1
k
N
1k
1
k
M
1k
1
k
1
k
M
1k
1
k
zd1zd1zc1
zg1zg1zf1
AzH
( ) ∏=
−−
−−
−−
−+
=
1M
1k
2
k2
1
k1
2
k2
1
k1k0
zaza1
zbzbb
zH
65. 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
zz21
zH
−
−
−
−
−−
−−
−−
−−
−
+
−
+
=
−−
++
=
+−
++
=
66. 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 pairingthe real polesOr by pairingthe real poles
( ) ( )
( )( )∑ ∑∑ = =
−∗−
−
−
=
−
−−
−
+
−
+=
P PP N
1k
N
1k
1
k
1
k
1
kk
1
k
k
N
0k
k
k
zd1zd1
ze1B
zc1
A
zCzH
( ) ∑∑ =
−−
−
=
−
−−
+
+=
SP N
1k
2
k2
1
k1
1
k1k0
N
0k
k
k
zaza1
zee
zCzH
67. ExampleExample
►Partial Fraction ExpansionPartial Fraction Expansion
►Combine poles to getCombine poles to get
( )
( ) ( )1121
21
z25.01
25
z5.01
18
8
z125.0z75.01
zz21
zH −−−−
−−
−
−
−
+=
+−
++
=
( ) 21
1
z125.0z75.01
z87
8zH −−
−
+−
+−
+=
68. Transposed FormsTransposed Forms
►Linear signal flow graph property:Linear signal flow graph property:
Transposing doesn’t change the input-outputTransposing doesn’t change the input-output
relationrelation
►Transposing:Transposing:
Reverse directions of all branchesReverse directions of all branches
Interchange input and output nodesInterchange input and output nodes
►Example:Example:
( ) 1
az1
1
zH −
−
=
69. ExampleExample
Transpose
►Both have the same system function orBoth have the same system function or
difference equationdifference equation
[ ] [ ] [ ] [ ] [ ] [ ]2nxb1nxbnxb2nya1nyany 21021 −+−++−+−=
70. 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 IITranspose 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
71. Basic Structures for FIRBasic Structures for FIR
Systems: Cascade FormSystems: Cascade Form
►Obtained by factoring the polynomialObtained by factoring the polynomial
system functionsystem function
( ) [ ] ( )∑ ∏= =
−−−
++==
M
0n
M
1k
2
k2
1
k1k0
n
S
zbzbbznhzH
72. Structures for Linear-Phase FIRStructures for Linear-Phase FIR
SystemsSystems
► Causal FIR system with generalized linear phase areCausal 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)orII(typeM0,1,...,nnhnMh
III)orI(typeM0,1,...,nnhnMh
=−=−
==−
[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]
[ ] [ ] [ ] [ ] [ ] [ ]
[ ] [ ] [ ]( ) [ ] [ ]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
−++−+−=
+−−+−+−=
−+−+−=−=
∑
∑∑
∑∑∑
−
=
−
=
−
=
+=
−
==
73. Structures for Linear-Phase FIRStructures for Linear-Phase FIR
SystemsSystems
►Structure for even MStructure for even M
►Structure for odd MStructure for odd M
