This document discusses signals and systems. It defines signals as physical quantities that vary with respect to time, space, or another independent variable. Signals can be classified as discrete time or continuous time. It also defines unit impulse and unit step functions for discrete and continuous time. Periodic and aperiodic signals are discussed. The Fourier series and Fourier transform are introduced as ways to represent signals in the frequency domain. The Laplace transform, which generalizes the Fourier transform, is also mentioned. Key properties of linear time-invariant systems like superposition, time-invariance, and convolution are covered. Finally, sampling theory and the z-transform, which is analogous to the Laplace transform for discrete-time systems, are summarized at a high level
Fast Fourier transform is an extension of discrete Fourier transform, It is based on divide and conquer algorithm,it is of two types, decimation in time and decimation in frequency algorithm
z-Transform is for the analysis and synthesis of discrete-time control systems.The z transform in discrete-time systems play a similar role as the Laplace transform in continuous-time systems
The presentation covers sampling theorem, ideal sampling, flat top sampling, natural sampling, reconstruction of signals from samples, aliasing effect, zero order hold, upsampling, downsampling, and discrete time processing of continuous time signals.
Fast Fourier transform is an extension of discrete Fourier transform, It is based on divide and conquer algorithm,it is of two types, decimation in time and decimation in frequency algorithm
z-Transform is for the analysis and synthesis of discrete-time control systems.The z transform in discrete-time systems play a similar role as the Laplace transform in continuous-time systems
The presentation covers sampling theorem, ideal sampling, flat top sampling, natural sampling, reconstruction of signals from samples, aliasing effect, zero order hold, upsampling, downsampling, and discrete time processing of continuous time signals.
A signal is a pattern of variation that carry information.
Signals are represented mathematically as a function of one or more independent variable
basic concept of signals
types of signals
system concepts
I am Bing Jr. I am a Signal Processing Assignment Expert at matlabassignmentexperts.com. I hold a Master's in Matlab Deakin University, Australia. I have been helping students with their assignments for the past 9 years. I solve assignments related to Signal Processing.
Visit matlabassignmentexperts.com or email info@matlabassignmentexperts.com. You can also call on +1 678 648 4277 for any assistance with Signal Processing Assignments.
Instructions for Submissions thorugh G- Classroom.pptxJheel Barad
This presentation provides a briefing on how to upload submissions and documents in Google Classroom. It was prepared as part of an orientation for new Sainik School in-service teacher trainees. As a training officer, my goal is to ensure that you are comfortable and proficient with this essential tool for managing assignments and fostering student engagement.
How to Make a Field invisible in Odoo 17Celine George
It is possible to hide or invisible some fields in odoo. Commonly using “invisible” attribute in the field definition to invisible the fields. This slide will show how to make a field invisible in odoo 17.
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
We all have good and bad thoughts from time to time and situation to situation. We are bombarded daily with spiraling thoughts(both negative and positive) creating all-consuming feel , making us difficult to manage with associated suffering. Good thoughts are like our Mob Signal (Positive thought) amidst noise(negative thought) in the atmosphere. Negative thoughts like noise outweigh positive thoughts. These thoughts often create unwanted confusion, trouble, stress and frustration in our mind as well as chaos in our physical world. Negative thoughts are also known as “distorted thinking”.
Palestine last event orientationfvgnh .pptxRaedMohamed3
An EFL lesson about the current events in Palestine. It is intended to be for intermediate students who wish to increase their listening skills through a short lesson in power point.
How to Create Map Views in the Odoo 17 ERPCeline George
The map views are useful for providing a geographical representation of data. They allow users to visualize and analyze the data in a more intuitive manner.
The Art Pastor's Guide to Sabbath | Steve ThomasonSteve Thomason
What is the purpose of the Sabbath Law in the Torah. It is interesting to compare how the context of the law shifts from Exodus to Deuteronomy. Who gets to rest, and why?
How to Split Bills in the Odoo 17 POS ModuleCeline George
Bills have a main role in point of sale procedure. It will help to track sales, handling payments and giving receipts to customers. Bill splitting also has an important role in POS. For example, If some friends come together for dinner and if they want to divide the bill then it is possible by POS bill splitting. This slide will show how to split bills in odoo 17 POS.
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
Model Attribute Check Company Auto PropertyCeline George
In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
2. SIGNAL
► Signal is a physical quantity that varies with
respect to time , space or any other
independent variable
Eg x(t)= sin t.
► the major classifications of the signal
are:
(i) Discrete time signal
(ii) Continuous time signal
3. Unit Step &Unit Impulse
Discrete time Unit impulse is defined as
δ [n]= {0, n≠ 0
{1, n=0
Unit impulse is also known as unit sample.
Discrete time unit step signal is defined by
U[n]={0,n=0
{1,n>= 0
Continuous time unit impulse is defined as
δ (t)={1, t=0
{0, t ≠ 0
Continuous time Unit step signal is defined as
U(t)={0, t<0
{1, t≥0
4. SIGNAL
► Periodic Signal & Aperiodic Signal
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 the
property that it is unchanged by a time shift of T. A
signal that does not satisfy the above periodicity property
is called an aperiodic signal
► even and odd signal ?
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)= x(t) For example,Cosωn is an even signal.
5. Energy and power signal
► A signal is said to be energy signal if it
have finite energy and zero power.
► A signal is said to be power signal if it
have infinite energy and finite power.
► If the above two conditions are not
satisfied then the signal is said to be
neigther energy nor power signal
6. Fourier Series
The Fourier series represents a periodic signal in terms of
frequency components:
p −1 ∞
ikω0n ikω0 t
x(n) = ∑ Xk e x( t ) = ∑ Xk e
k =0 k = −∞
We get the Fourier series coefficients as follows:
1 p −1 1p −ikω0 t
Xk = ∑ x(n)e −ikω0n Xk = ∫ x( t )e dt
p n =0 p0
The complex exponential Fourier coefficients are a sequence of
complex numbers representing the frequency component ω0k.
7. Fourier series
► Fourier series: a complicated waveform analyzed into a
number of harmonically related sine and cosine functions
► A continuous periodic signal x(t) with a period T0 may be
represented by:
X(t)=Σ∞k=1 (Ak cos kω t + Bk sin kω t)+ A0
► Dirichlet conditions must be placed on x(t) for the series
to be valid: the integral of the magnitude of x(t) over a
complete period must be finite, and the signal can only
have a finite number of discontinuities in any finite
interval
8. Trigonometric form for Fourier series
► If the two fundamental components of a
periodic signal areB1cosω0t and C1sinω0t,
then their sum is expressed by trigonometric
identities:
► X(t)= A0 + Σ∞k=1 ( Bk 2+ Ak 2)1/2 (Ck cos kω t- φk)
or
► X(t)= A0 + Σ∞k=1 ( Bk 2+ Ak 2)1/2 (Ck sin kω t+ φk)
10. Fourier Transform
► Viewed periodic functions in terms of frequency components (Fourier
series) as well as ordinary functions of time
► Viewed LTI systems in terms of what they do to frequency
components (frequency response)
► Viewed LTI systems in terms of what they do to time-domain signals
(convolution with impulse response)
► View aperiodic functions in terms of frequency components via
Fourier transform
► Define (continuous-time) Fourier transform and DTFT
► Gain insight into the meaning of Fourier transform through
comparison with Fourier series
11. The Fourier Transform
►A transform takes one function (or signal)
and turns it into another function (or signal)
► Continuous Fourier Transform:
∞
H ( f ) = ∫ h ( t ) e 2πift dt
−∞
∞
h ( t ) = ∫ H ( f ) e −2πift df
−∞
12. Continuous Time Fourier Transform
We can extend the formula for continuous-time Fourier series
coefficients for a periodic signal
1p −ikω0 t 1 p/2
Xk = ∫ x( t )e dt = ∫ x( t )e −ikω0 t dt
p0 p −p / 2
to aperiodic signals as well. The continuous-time Fourier
series is not defined for aperiodic signals, but we call the
formula
∞
X(ω) = ∫ x( t )e −iωt dt
the (continuous time)
−∞ Fourier transform.
13. Inverse Transforms
If we have the full sequence of Fourier coefficients for a periodic
signal, we can reconstruct it by multiplying the complex
sinusoids of frequency ω0k by the weights Xk and summing:
p −1 ∞
ikω0n ikω0 t
x(n) = ∑ Xk e x( t ) = ∑ Xk e
k =0 k = −∞
We can perform a similar reconstruction for aperiodic signals
1 π 1 ∞
x(n) = ∫ X(ω)eiωn dω x( t ) = ∫ X(ω)eiωt dω
2π − π 2π − ∞
These are called the inverse transforms.
14. Fourier Transform of Impulse Functions
Find the Fourier transform of the Dirac delta function:
∞ ∞
X(ω) = ∫ x( t )e −iωt dt = ∫ δ( t )e −iωt dt = e −iω0 = 1
−∞ −∞
Find the DTFT of the Kronecker delta function:
∞ ∞
−iωn −iωn −iω0
X( ω) = ∑ x(n)e = ∑ δ(n)e =e =1
n = −∞ n = −∞
The delta functions contain all frequencies at equal amplitudes.
Roughly speaking, that’s why the system response to an impulse
input is important: it tests the system at all frequencies.
15. Laplace Transform
► Lapalce transform is a generalization of the Fourier transform in the sense
that it allows “complex frequency” whereas Fourier analysis can only
handle “real frequency”. Like Fourier transform, Lapalce transform allows
us to analyze a “linear circuit” problem, no matter how complicated the
circuit is, in the frequency domain in stead of in he time domain.
► Mathematically, it produces the benefit of converting a set of differential
equations into a corresponding set of algebraic equations, which are much
easier to solve. Physically, it produces more insight of the circuit and
allows us to know the bandwidth, phase, and transfer characteristics
important for circuit analysis and design.
► Most importantly, Laplace transform lifts the limit of Fourier analysis to
allow us to find both the steady-state and “transient” responses of a linear
circuit. Using Fourier transform, one can only deal with he steady state
behavior (i.e. circuit response under indefinite sinusoidal excitation).
► Using Laplace transform, one can find the response under any types of
excitation (e.g. switching on and off at any given time(s), sinusoidal,
impulse, square wave excitations, etc.
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
► Ifyou 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. Z-transforms
► Fordiscrete-time systems, z-transforms play
the same role of Laplace transforms do in
continuous-time systems
Bilateral Forward z-transform Bilateral Inverse z-transform
∞
1
∑ h[ n] z − n
2 π j ∫R
H [ z] = h[n] = H [ z ] z − n +1dz
n = −∞
► Aswith the Laplace transform, we compute
forward and inverse z-transforms by use of
transforms pairs and properties
28. Region of Convergence
► Region of the complex ► Four possibilities (z=0
z-plane for which is a special case and
forward z-transform may or may not be
converges Im{z} included) Im{z}
Entire Disk
plane Re{z} Re{z}
Im{z} Im{z}
Intersection
Complement of a disk and
of a disk Re{z} complement Re{z}
of a disk
29. Z-transform Pairs
► h[n] = δ[n] ► h[n] = an u[n]
∞ 0
∑ δ [ n] z = ∑ δ [ n] z − n = 1
∞
∑ a u[ n] z
−n
H [ z] = H [ z] = n −n
n = −∞ n =0
n = −∞
Region of convergence: ∞
a ∞ n
= ∑ a z = ∑ n −n
entire z-plane n =0 n =0 z
1 a
= if <1
► h[n] ∞= δ[n-1]1 1−
a z
H [ z ] = ∑ δ [ n − 1] z −n = ∑ δ [ n − 1] z −n = z −1 z
n = −∞ n =1
Region of convergence: Region of
entire z-plane convergence: |z|
> |a| which is
h[n-1] ⇔ z-1 H[z]
the complement
30. Stability
► Rule #1: For a causal sequence, poles are
inside the unit circle (applies to z-transform
functions that are ratios of two polynomials)
► Rule #2: More generally, unit circle is
included in region of convergence. (In
continuous-time, the imaginary axis would
be in the region of convergence of the
Z 1
a u[ n] ↔
Laplace transform.)a z for z > a
n
−1
1−
This is stable if |a| < 1 by rule #1.
31. Inverse z-transform
c + j∞
1
f [ n] = F [ z ] z n −1dz
2πj c∫ j∞
−
► Yuk! Using the definition requires a contour
integration in the complex z-plane.
► Fortunately, we tend to be interested in only
a few basic signals (pulse, step, etc.)
Virtually all of the signals we’ll see can be built
up from these basic signals.
For these common signals, the z-transform pairs
have been tabulated (see Lathi, Table 5.1)
32. Example
z 2 + 2z +1 ► Ratio of polynomial z-
X [ z] =
3 1
z2 − z +
2 2
domain functions
1 + 2 z −1 + z −2
► Divide through by the
X [ z] =
3 1
1 − z −1 + z − 2 highest power of z
2 2
► Factor denominator into
1 + 2 z −1 + z −2
X [ z] =
first-order factors
1 − z (1 − z )
1 −1 −1
2 ► Use partial fraction
X [ z ] = B0 +
A1
+
A2 decomposition to get
1 −1 1 − z −1
1− z
2
first-order terms
33. Example (con’t)
2
1 − 2 3 −1
z − z + 1 z − 2 + 2 z −1 + 1 ► FindB0 by
2 2
z − 2 − 3 z −1 + 2 polynomial division
5 z −1 − 1
− 1 + 5 z −1
X [ z] = 2 +
► Express in terms of
1 −1
1 − z 1 − z
−1
( )
2
B0
1 + 2 z −1 + z −2 1+ 4 + 4
A1 = = = −9
1 − z −1 z −1 = 2
1− 2
1 + 2 z −1 + z − 2 1+ 2 +1
► Solve for A1 and A2
A2 = = =8
1 1
1 − z −1
2 z −1 =1 2
34. Example (con’t)
► Express X[z] in terms of B0, A1, and A2
9 8
X [ z] = 2 − +
1 −1 1 − z −1
1− z
2
► Use table to obtain inverse z-transform
n
1
x[ n] = 2 δ [ n] − 9 u[ n] + 8 u[ n]
2
► With the unilateral z-transform, or the
bilateral z-transform with region of
convergence, the inverse z-transform is
unique
35. Z-transform Properties
► Linearity
a1 f1 [ n] + a2 f 2 [ n] ⇔ a1 F1 [ z ] + a2 F2 [ z ]
► Right shift (delay)
f [ n − m] u[ n − m] ⇔ z − m F [ z ]
m
f [ n − m] u[ n] ⇔ z F [ z ] + z ∑ f [ − n] z − n
−m −m
n =1
36. Z-transform Properties
► Convolution definition
∞
f1 [ n] ∗ f 2 [ n] = ∑ f [ m] f [ n − m]
m = −∞
1 2
► Take z-transform
∞
Z { f1 [ n] ∗ f 2 [ n]} = Z ∑ f1 [ m] f 2 [ n − m]
m = −∞
► Z-transform definition
∞
∞
= ∑ ∑ f1 [ m] f 2 [ n − m] z − n ► Interchange summation
n = −∞ m = −∞
∞ ∞
= ∑ f [ m] ∑ f [ n − m] z
1 2
−n ► Substitute r = n - m
m = −∞ n = −∞
∞ ∞
f1 [ m] ∑ f 2 [ r ] z −( r + m )
► Z-transform definition
= ∑
m = −∞ r = −∞
∞
− m
∞
= ∑ f1 [ m]z ∑ f 2 [ r ] z − r
m = −∞ r = −∞
= F1 [ z ] F2 [ z ]
38. Introduction
► Impulse response h[n] can fully characterize a LTI
system, and we can have the output of LTI system as
y[ n] = x[ n] ∗ h[ n]
► The z-transform of impulse response is called transfer or
Y ( z ) = X ( z ) H ( z ).
system function H(z).
► ( )
Frequency response at H e = H ( z )
jω
z =1
is valid if
z =
ROC includes 1, and
Y ( e jω ) = X ( e jω ) H ( e jω )
39. 5.1 Frequency Response of LIT
System
jω jω j∠X ( e jω ) jω jω j∠H ( e jω )
► Consider X (e ) = X (e ) e and H (e ) = H (e ) e
, then
magnitude
Y ( e jω ) = X ( e jω ) H ( e jω )
phase
jω jω jω
∠Y ( e ) = ∠X ( e ) + ∠H ( e )
► We will model and analyze LTI systems based on the
magnitude and phase responses.
40. System Function
► General formNof LCCDE M
∑ a y [ n − k ] = ∑ b x[ n − k ]
k =0
k
k =0
k
N M
∑ ak z Y ( z ) = ∑ bk z −k X ( z )
−k
k =0 k =0
► Compute the z-transform
M
Y ( z) ∑ bk z −k
H ( z) = = k =0
X ( z) N
∑ ak z − k
k =0
41. System Function: Pole/zero
Factorization
► Stability requirement can be verified.
► Choice of ROC determines causality.
► Location of zeros and poles determines the
frequency response1 )and phase : c , c ,..., c .
M
(1 − c z −
b0 ∏ k
zeros 1 2 M
H ( z) = k =1
∏ (1 − d z )
N
a0 −1 poles : d1 , d 2 ,..., d N .
k
k =1
42. Second-order System
► Suppose the system function of a LTI system is
(1 + z −1 )2
H ( z) = .
1 −1 3 −1
(1 − z )(1 + z )
2 4
► Tofind the difference equation that is satisfied by
the input and out of this system
(1 + z −1 )2 1 + 2 z −1 + z −2 Y ( z)
H ( z) = = =
1 3 1 3
(1 − z −1 )(1 + z −1 ) 1 + z −1 − z −2 X ( z )
2 4 4 8
1 3
y[n ] + y[n − 1] − y[n − 2] = x[n ] + 2 x[n − 1] + 2 x[n − 2]
4 8
► Can we know the impulse response?
43. System Function: Stability
► Stability of LTI system:
∞
∑ h[n] < ∞
n = −∞
► This condition is identical to the condition
∞
that ∑ h[n ]z −n < ∞ when z = 1.
n = −∞
The stability condition is equivalent to the
condition that the ROC of H(z) includes the unit
circle.
44. System Function: Causality
► If the system is causal, it follows that h[n] must be a right-
sided sequence. The ROC of H(z) must be outside the
outermost pole.
► If the system is anti-causal, it follows that h[n] must be a
left-sided sequence. The ROC of H(z) must be inside the
innermost pole.
Im Im Im
a 1 a 1 Re a b
Re Re
Right-sided Left-sided Two-sided
(causal) (anti-causal) (non-causal)
45. Determining the ROC
► Consider the LTI system
5
y[n ] − y[n − 1] + y[n − 2] = x[n ]
2
► The system1 function is obtained as
H ( z) =
5 −1 −2
1− z +z
2
1
=
1 −1
(1 − z )(1 − 2 z −1 )
2
46. System Function: Inverse Systems
H i ( z) H ( z)
► is an inverse system for , if
G ( z ) = H ( z ) H i ( z ) = 1 ⇔ g [ n ] = h[ n ] ∗ hi [ n ] = δ [ n ]
1 1
Hi ( z) = jω
⇔ H i (e ) =
H ( z) H ( e jω )
► The ROCs of H ( z ) and H i ( z ) overlap.
must
► Useful for canceling the effects of another system
► See the discussion in Sec.5.2.2 regarding ROC
47. All-pass System
►A system of the form ∗(or cascade of these)
−1
z −a
H Ap ( Z ) = pole : a = re jθ
1 − az −1 zero : 1 / a* = r −1e jθ
− jω ∗ jω
e −a − jω 1 − a * e
H Ap ( e ) =
jω
− jω
=e − jω
1 − ae 1 − ae
H Ap ( e jω ) = 1
48. All-pass System: General Form
► In general, all pass systems have form
Mr
z −1 − d k M c ( z −1 − ek )( z −1 − ek )
*
H Ap ( z ) = ∏ −1 ∏ −1 * −1
k =1 1 − d k z k =1 (1 − ek z )(1 − ek z )
real poles complex poles
Causal/stable: ek , d k < 1
49. All-Pass System Example
Im
Unit
circle z-plane
M r = 2 and M c = 1
0.8
Re
4 3 0.5
− − 2
3 4
pole : re jθ reciprocal & conjugate → zero : r −1e jθ
This all - pass system has M = N = 2 M c + M r = 4 poles and zeros.
50. Minimum-Phase System
► Minimum-phase system: all zeros and all poles are
inside the unit circle.
► The name minimum-phase comes from a property of the
phase response (minimum phase-lag/group-delay).
► Minimum-phase systems have some special properties.
► When we design a filter, we may have multiple choices to
satisfy the certain requirements. Usually, we prefer the
minimum phase which is unique.
► All systems can be represented as a minimum-phase
system and an all-pass system.
52. Example
y[n] = a1y[n − 1] + a2 y[n − 2] + b0x[n]
► Block diagram representation of
53. Block Diagram Representation
► LTI systems with
rational system
function can be
represented as
constant-coefficient
difference equation
► The implementation of
difference equations
requires delayed
values of the
54. Direct Form I
N M
∑ ˆ y[n − k ] = ∑ b x[n − k ]
a
k =0
k
ˆ
k =0
k
► General form of difference equation
N M
y[n] − ∑ a y[n − k ] = ∑ b x[n − k ]
k k
k =1 k =0
► Alternative equivalent form
55. Direct Form I M
∑ bk z −k
H( z ) = k =0
N
1 − ∑ ak z −k
► Transfer function can be written as
k =1
M
H( z ) = H2 ( z )H1 ( z ) = 1 b z −k
∑
N k = 0 k
−k
1 − ∑ ak z
k =1
M −k
V ( z ) = H1 ( z ) X( z ) = ∑ bk z X( z ) M
k =0
v[n] = ∑ b x[n − k ]
k
► Direct Form I Represents
k =0
N
1
y[n] = ∑ a y[n − k ] + v[n]
Y ( z ) = H2 ( z ) V ( z ) = V ( z ) k
k =1
N
1 − ∑ ak z
−k
k =1
56. Alternative Representation
► Replace order of cascade LTI systems
M
1
H( z ) = H ( z )H ( z ) = ∑ b z
−k
1 2
k
1− N
k =0
∑a z
k =1
k
−k
N
w[n] = ∑ a w[n − k ] + x[n]
k
k =1
1
W( z ) = H2 ( z ) X( z ) = X ( z ) M
N y[n] = ∑ b w[n − k ]
k
1 − ∑ ak z
−k
k =0
k =1
M
Y ( z ) = H1 ( z ) W( z ) = ∑ bk z −k W( z )
k =0
57. Alternative Block Diagram
► We can change the order of the cascade
systems
N
w[n] = ∑ a w[n − k ] + x[n]
k
k =1
M
y[n] = ∑ b w[n − k ]
k
k =0
58. Direct Form II
► No need to store the same data
twice in previous system
► So we can collapse the delay
elements into one chain
► This is called Direct Form II or
the Canonical Form
► Theoretically no difference
between Direct Form I and II
► Implementation wise
Less memory in Direct II
Difference when using
finite-precision arithmetic
59. Signal Flow Graph Representation
► Similar to block diagram representation
Notational differences
►A network of directed branches connected
at nodes
60. Example
► Representation of Direct Form II with signal
flow graphs w1 [n] = aw4 [n] + x[n]
w2 [n] = w1 [n]
w3 [n] = b0w2 [n] + b1w4 [n]
w4 [n] = w2 [n − 1]
y[n] = w3 [n]
w1 [n] = aw1 [n − 1] + x[n]
y[n] = b0w1 [n] + b1w1 [n − 1]
61. Determination of System
Function from Flow Graph
w1 [n] = w4 [n] − x[n]
w2 [n] = αw1 [n]
w3 [n] = w2 [n] + x[n]
w4 [n] = w3 [n − 1]
y[n] = w2 [n] + w4 [n]
W1 ( z ) = W4 ( z ) − X( z ) (
αX( z ) z −1 − 1 )
W2 ( z ) =
W2 ( z ) = αW1 ( z ) 1 − αz −1 Y ( z) z −1 − α
H( z ) = =
W3 ( z ) = W2 ( z ) + X( z ) X( z ) z −1 (1 − α ) X( z ) 1 − αz −1
W4 ( z ) =
W4 ( z ) = W3 ( z ) z −1 1 − αz −1 h[n] = αn −1u[n − 1] − αn +1u[n]
Y ( z ) = W2 ( z ) + W4 ( z ) Y ( z ) = W2 ( z ) + W4 ( z )
64. Basic Structures for IIR Systems:
Cascade Form
► General form for cascade implementation
M1 M2
∏ (1 − f z )∏ (1 − g z )(1 − g z )
k
−1
k
−1 ∗
k
−1
H( z ) = A k =1
N1
k =1
N2
∏ (1 − c z )∏ (1 − d z )(1 − d z )
k =1
k
−1
k =1
k
−1 ∗
k
−1
M1
b0k + b1k z −1 − b2k z −2
H( z ) = ∏
k =1 1 − a1k z −1 − a2k z −2
► More practical form in 2nd order systems
65. Example
H( z ) =
1 + 2z + z −1
(1 + z
−2
=
−1
)(1 + z )
−1
1 − 0.75z −1
+ 0.125z −2
(1 − 0.5z −1
)(1 − 0.25z )
−1
=
(1 + z )
−1
(1 + z )
−1
(1 − 0.5z ) (1 − 0.25z )
−1 −1
► Cascade of Direct Form I subsections
► Cascade of Direct Form II subsections
66. Basic Structures for IIR Systems:
Parallel Form
► Represent system function using partial fraction expansion
H( z ) =
NP NP
Ak NP
(
Bk 1 − ek z −1 )
∑ Ck z − k
k =0
+∑
k =1 1 − ck z
−1
+∑
(
k =1 1 − dk z
−1
)(
1 − dk z −1
∗
)
NS
NP
e0k + e1k z −1
H( z ) = ∑C z k
−k
+∑
k =0 k =1 1 − a1k z −1 − a2k z −2
► Or by pairingthe real poles
68. Transposed Forms
► Linear signal flow graph property:
Transposing doesn’t change the input-output
relation 1
H( z ) = −1
► Transposing:1 − az
Reverse directions of all branches
Interchange input and output nodes
► Example:
69. Example
Transpose
y[n] = a1y[n − 1] + a2y[n − 2] + b0x[n] + b1x[n − 1] + b2x[n − 2]
► Both have the same system function or
difference equation
70. Basic Structures for FIR Systems: Direct Form
► Special cases of IIR direct form structures
► Transpose of direct form I gives direct form II
► Both forms are equal for FIR systems
► Tapped delay line
71. Basic Structures for FIR Systems:
Cascade Form
► Obtainedby factoring the polynomial
system function
MS
∏ (b )
M
H( z ) = ∑ h[n]z
n=0
−n
=
k =1
0k + b1k z −1 + b2k z −2
72. Structures for Linear-Phase FIR
Systems
► Causal FIR system with generalized linear phase are
symmetric:
h[M − n] = h[n] n = 0,1,..., M (type I or III)
h[M − n] = −h[n] n = 0,1,..., M (type II or IV)
► Symmetry means we can half the number of
multiplications
► Example: For even M and type I or type III systems:
M M / 2 −1 M
y[n] = ∑ h[k ]x[n − k ] = ∑ h[k ]x[n − k ] + h[M / 2]x[n − M / 2] + ∑ h[k ]x[n − k ]
k =0 k =0 k = M / 2 +1
M / 2 −1 M / 2 −1
= ∑ h[k ]x[n − k ] + h[M / 2]x[n − M / 2] + ∑ h[M − k ]x[n − M + k ]
k =0 k =0
M / 2 −1
= ∑ h[k ]( x[n − k ] + x[n − M + k ] ) + h[M / 2]x[n − M / 2]
k =0