Realizations of discrete time systems 1 unit

Realizations of Discrete-
Time Systems
Mr. HIMANSHU DIWAKAR
Assistant Professor
JETGI
Mr. HIMANSHU DIWAKAR JETGI 1

What is Digital Signal Processing?
• Digital: operating by the use of discrete signals to represent data in the form
of numbers
• Signal: a parameter (electrical quantity or effect) that can be varied in such
a way as to convey information
• Processing: a series operations performed according to programmed
instructions
Changing or analyzing information
which is measured as discrete
sequences of numbers

Realization of Discrete Systems
• We will study the following realization topologies:
• 1) Direct Form I (for FIR & IIR).
(named as transversal for FIR)
• 2) Direct Form II – Canonical Form. (for IIR).
• 3) Cascaded Realization (IIR).
• 4) Parallel Realization (IIR).

Basic structure of IIR filter
𝑥1(𝑛)
𝑥2(𝑛)
𝑥1(𝑛) +𝑥2(𝑛)y(𝑛)=
x(𝑛) a.x(𝑛)a
x(𝑛 − 1)x(𝑛) 𝑍−1
1. Adder
2. Multiplier
3. Delay

FIR Direct or Transversal form

Direct form I Realization

Direct Form I – cont.

Direct Form II
For a discrete-time system described by
ak y n  k 
k0
N
  bk x n  k 
k0
M

the transfer function is of the form
H z 
Y z 
X z 

b0  b1z1
L  bN z N
a0  a1z1
L  aN z N

b0zN
 b1zN 1
L  bN
a0zN
 a1zN 1
L  aN
Here the order of the numerator and denominator are both
indicated as N. If the order of the numerator is actually less
than N, some of the b’s will be zero. But a0 must not be zero.











Direct Form II
H z  H1 z H2 z 
where
H1 z 
Y1 z 
X z 

1
a0zN
 a1zN1
L  aN
H2 z 
Y z 
Y1 z 
 b0zN
 b1zN 1
L  bN
Rearranging H1 z ,
zN
Y1 z  1/ a0  X z  a1zN1
Y1 z L  aN Y1 z   

Direct Form II
zN
Y1 z  1/ a0  X z  a1zN1
Y1 z L  aN Y1 z   

Direct Form II

Direct Form II
For the special case of FIR filters.
(The number of delays has been
changed to M - 1 to conform to
conventions in the DSP literature.)
h n  bk n  k 
k0
M1

This type of filter has M - 1 finite
zeroes and M - 1 poles at z = 0.

Direct Form II
One desirable characteristic of an FIR filter is that it can have linear phase
in its pass band.
The impulse response is
h n  h 0  n  h 1  n 1 L  h M 1  n  M 1  
and its z transform is
H z  h 0  h 1 z1
L  h M 1 z M1 
and its frequency response is
H ej
  h 0  h 1 e j
L  h M 1 e j M1 

Direct Form II
The response y n to an excitation x n is
y n  b0 x n  b1 x n 1 L  bM1 x n  M 1  .
Let M be even and let the filter coefficients be chosen such that
h 0  h M 1  , h 1  h M  2  , L , h M / 2 1  h M / 2 

Direct Form II
Then the frequency response is
H e j
 
h 0  h 0 e j M 1 
 h 1 e j
 h 1 e j M 2 
L
 h M / 2 1 e j M /21 
 h M / 2 1 e j M /2 








or
H e j
  e
 j
M 1
2



 
h 0  e
j M 1 /2 
 e
 j M 1 /2 
 
 h 1  e
j M 3 /2 
 e
 j M 3 /2 
 L
 h M / 2 1  e j
 e j
 














or
H e j
  2e
 j
M 1
2



  h 0 cos
M 1
2



 



  h 1 cos
M  3
2



 



 L
 h M / 2 1 cos  











Direct Form II
In the frequency response
H ej
  2e
 j
M1
2



 h 0 cos
M 1
2



 



  h 1 cos
M  3
2



 



 L
 h M / 21 cos  










there is a factor e
 j M1 /2 
which has a linear phase and the rest of the
frequency response is purely real.

Direct Form II
The recursion relation is
y n  b0 x n  x n  M 1   
b1 x n 1  x n  M  2   L
bM /21 x n  M / 2 1  x M / 2   
which can be realized in this form
that reduces the number of multiplications
by half.

Direct Form II
It can be shown that symmetric or anti-symmetric, even or odd impulse
responses yield linear phase shift in the frequency response.

Cascade Realization
Direct Form II is no the only form of realization. There are several other
forms. Two other important forms are the cascade form and the parallel
form.
The cascade form is realized by first factoring the transfer function
H z  A
z  z1
z  p1
z  z2
z  p2
L
z  zM
z  pM
1
z  pM1
1
z  pM2
L
1
z  pN
Each individual factor is realized as a small Direct Form II subsystem
and the subsystems are then cascaded.

Parallel Realization
The parallel form is realized by first
expressing the transfer function
in partial-fraction form
H z 
K1
z  p1

K2
z  p2
L 
KN
z  pN
Each individual term is realized
as a small Direct Form II subsystem
and the subsystems are then
paralleled.

Complex Poles and Zeroes
In either the cascade or parallel realization, the first-order subsystems may
have complex poles and/or zeroes. In such a case two first-order subsystems
should be combined into one second-order subsystem to avoid the problem of
complex coefficients in the first-order subsystems. Also, for reasons we will
soon see, it is common to do cascade and parallel realizations with second-
order subsystems even when the poles and/or zeroes are real.

1st Order vs 2nd Order
In the case of FIR filters the second-order subsystems take
this form.
Compare this two first-order cascaded stages.
2 delays
3 multiplications
2 additions
2 delays
4 multiplications
2 additions

If the FIR filter has linear phase, a fourth-order structure reduces number of
multiplications even further compared with cascaded first-order or cascaded
second-order subsystems.
1st Order vs 2nd Order

Lattice IIR Structure

A Direct Form II system with
N finite poles and N zeroes at
z= 0.

FIR IIR

Lattice-Ladder IIR Structure
Modify the FIR lattice structure as illustrated below. Reverse the
arrows on all the "f" signals. Reverse the lattice and apply x(n) to the
previous output and take y from the previous input. Also reverse the
signs of the signals arriving from the bottom. This is now a recursive
or feedback structure which can implement an IIR filter.

Lattice-Ladder Example

Thank you

Realizations of discrete time systems 1 unit

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