Digital Signal Processing

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Digital Signal Processing
PALESTINE TECHNICAL COLLEGE
Eng. Akram Abu Garad

• www.ptcdb.edu.ps

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Digital Signal Processing
Chapter 6 Implementation of Discrete-Time
Systems

• Introduction.
• Structures for FIR Systems.
• Structures for IIR Systems.

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Implementation of Discrete-Time Systems
Digital Signal Processing
6.1. Introduction

• In earlier chapters we studied the theory of
  discrete systems in both the time and frequency
  domains.
• We will now use this theory for the processing of
  digital signals.
• To process signals, we have to design and
  implement systems called filters.
• The filter design issue is influenced by such
  factors as
• The type of the filter IIR or FIR
• The form of its implementation structures
• Different filter structures dictate different
  design strategies.

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Implementation of Discrete-Time Systems
Digital Signal Processing
6.1. Introduction

• Since our filters are LTI systems, we need the
  following three elements to describe digital
  filter structures.
• Adder
• Multiplier (Gain)
• Delay element (shift or memory)

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Implementation of Discrete-Time Systems
Digital Signal Processing
6.2. Structures for IIR Systems
The system function of an IIR filter is given by
The difference equation representation of an IIR
filter is expressed as
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Implementation of Discrete-Time Systems
Digital Signal Processing
6.2. Structures for IIR Systems
Three different structures can be used to
implement an IIR filter

• Direct Form
• In this form, there are two parts to this filter,
  the moving average part and the recursive part
  (or the numerator and denominator parts)
• Two version direct form I and direct form II
• Cascade Form
• The system function H(z) is factored into smaller
  second-order sections, called biquads. H(z) is
  then represented as a product of these biquads.
• Each biquad is implemented in a direct form, and
  the entire system function is implemented as a
  cascade of biquad sections.
• Parallel Form
• H(z) is represented as a sum of smaller
  second-order sections.
• Each section is again implemented in a direct
  form.
• The entire system function is implemented as a
  parallel network of sections.

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Implementation of Discrete-Time Systems
Digital Signal Processing
6.2. Structures for IIR Systems

• Direct Form I Structure

Consists of the zeros of H(z)
Consists of the poles of H(z)
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Implementation of Discrete-Time Systems
Digital Signal Processing
6.2. Structures for IIR Systems

• Direct Form I Structure

Direct Form I Structure
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Implementation of Discrete-Time Systems
Digital Signal Processing
6.2. Structures for IIR Systems

• Direct Form I Structure

• As the name suggests, the difference equation is
  implemented as given using delays, multipliers,
  and adders.
• For the purpose of illustration, Let M N 2,

Direct Form I structure
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Implementation of Discrete-Time Systems
Digital Signal Processing
6.2. Structures for IIR Systems
Given a LTI system with a rational transfer
function H(z)
LCCDE
Direct Form I structure
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Implementation of Discrete-Time Systems
Digital Signal Processing
6.2. Structures for IIR Systems
We can implement the system with the following
pair of coupled difference equations
The commutative law of the convolution
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Implementation of Discrete-Time Systems
Digital Signal Processing
6.3. Structures for IIR Systems

• Direct Form II Structure

Direct Form II Structure (M N)
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Implementation of Discrete-Time Systems
Digital Signal Processing
6.2. Structures for IIR Systems

• Direct Form II Structure

The commutative law of the convolution
Direct Form II structure
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Implementation of Discrete-Time Systems
Digital Signal Processing
6.2. Structures for IIR Systems

• Cascade Form

• In this form the system function H(z) is written
  as a product of second-order section with real
  coefficients.
• This is done by factoring the numerator and
  denominator polynomials into their respective
  roots and then combining either a complex
  conjugate root pair or any two real roots into
  second-order polynomials.

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Implementation of Discrete-Time Systems
Digital Signal Processing
6.2. Structures for IIR Systems

• Cascade Form

• We assume that N is an even integer. Then

Where, K is equal to N/2, and Bk,1, Bk,2, Ak,1,
Ak,2 are real numbers representing the
coefficients of second-order section.
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Implementation of Discrete-Time Systems
Digital Signal Processing
6.2. Structures for IIR Systems
Biquad Section
Is called the k-th biquad section. The input to
the k-th biquad section is the output from the
(k-1)-th section, while the output from the k-th
biquad is the input to the (k1)-th biquad. Each
biquad section can be implemented in direct form
II.
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Implementation of Discrete-Time Systems
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6.2. Structures for IIR Systems
Example
Cascade form structure for N4
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Implementation of Discrete-Time Systems
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6.2. Structures for IIR Systems
Example
Determine the cascade structure the following
transfer function
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Implementation of Discrete-Time Systems
Digital Signal Processing
6.2. Structures for IIR Systems
Example
Determine the cascade structure the following
transfer function
Cascade of Direct Form I
Cascade of Direct Form II
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Implementation of Discrete-Time Systems
Digital Signal Processing
6.2. Structures for IIR Systems

• Parallel Form

• In this form the system function H(z) is written
  as a sum of second-order section using partial
  fraction expansion.

KN/2, and B,A are real numbers
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Implementation of Discrete-Time Systems
Digital Signal Processing
6.2. Structures for IIR Systems
The second-order Section
Is the k-th proper rational biquad section. The
filter input is available to all biquad section
as well as to the polynomial section if MgtN
(which is an FIR part) The output from these
sections is summed to form the filter
output. Each biquad section can be implemented in
direct form II.
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Implementation of Discrete-Time Systems
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6.2. Structures for IIR Systems
Parallel form structure for N4 (MN4)
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Implementation of Discrete-Time Systems
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6.2. Structures for IIR Systems
Example
Determine the parallel structure the following
transfer function
H(z) must be expanded in partial fractions
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Implementation of Discrete-Time Systems
Digital Signal Processing
6.2. Structures for IIR Systems
Example (Cont.)
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Implementation of Discrete-Time Systems
Digital Signal Processing
6.2. Structures for IIR Systems
Example
Determine the parallel structure the following
transfer function
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Implementation of Discrete-Time Systems
Digital Signal Processing
6.3. Structures for FIR Systems
A finite-duration impulse response (FIR) filter
has a Transfer Function of the form
Or, equivalently by the Difference Equation
Which is a linear convolution of finite support.
Furthermore, the Impulse Response of the FIR
system is identical to the coefficients bk,
that is,
The order of the filter is M-1, while the length
of the filter is M.
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Implementation of Discrete-Time Systems
Digital Signal Processing
6.3. Structures for FIR Systems
Three different structures can be used to
implement an FIR filter

• Direct Form.
• Cascade Form.
• Linear Phase Form.
• Frequency Sampling Form.
• Lattice Form.

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Implementation of Discrete-Time Systems
Digital Signal Processing
6.3. Structures for FIR Systems

• Direct Form

• The difference equation is implemented as a
  tapped delay line since there are no feedback
  paths.
• Note that since the denominator is equal to
  unity, there is only one direct form structure.

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Implementation of Discrete-Time Systems
Digital Signal Processing
6.3. Structures for FIR Systems
Example
Determine the Direct form structure for the
following FIR Filter
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Implementation of Discrete-Time Systems
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6.3. Structures for FIR Systems

• Cascade Form

Cascade form FIR structure for 6-order FIR filter
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Implementation of Discrete-Time Systems
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6.3. Structures for FIR Systems
Linear Phase System
A LTI system is said to have Linear Phase if the
frequency response has the form
where a real is a real number
A system is said to have Generalized Linear Phase
if the frequency response has the form
where A(ejw) is a real-valued function of ?, and
ß is a constant
Often the term linear phase is used to denote a
system that has either linear or generalized
linear phase.
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Implementation of Discrete-Time Systems
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6.3. Structures for FIR Systems
Linear Phase System
For a causal FIR filter with M Length and impulse
over 0,M-1 interval, the linear-phase conditions
For an FIR filter with a real-valued impulse
response of length M, a sufficient condition for
this filter to have generalized linear phase is
that the h(n) be whether symmetric or anti
symmetric.
The length of the impulse response of the FIR
filter (M) can be even or odd.
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Implementation of Discrete-Time Systems
Digital Signal Processing
6.3. Structures for FIR Systems
Linear Phase System
Linear phase filters may be classified into 4
types, depending upon whether h(n) is symmetric
or anti symmetric and whether M is even or odd.

1. Type I Linear Phase Symmetrical and M even

1. Type II Linear Phase Symmetrical and M odd

1. Type III Linear Phase Anti-symmetrical and M
   even

1. Type IV Linear Phase Anti-symmetrical and M odd

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Implementation of Discrete-Time Systems
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6.3. Structures for FIR Systems
Type I Linear Phase
Symmetric and M even
 Symmetrical Impulse Response, M Even
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Implementation of Discrete-Time Systems
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6.3. Structures for FIR Systems
Type II Linear Phase
H(ejw)
Symmetric and M odd
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Implementation of Discrete-Time Systems
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6.3. Structures for FIR Systems
Type III Linear Phase
H(ejw)
Anti Symmetric and M even
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Implementation of Discrete-Time Systems
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6.3. Structures for FIR Systems
Type IV Linear Phase
H(ejw)
Anti Symmetric and M odd
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Implementation of Discrete-Time Systems
Digital Signal Processing
6.3. Structures for FIR Systems

• Linear Phase Form

Consider the difference equation with a symmetric
impulse response.
The linear-phase structure is essentially a
direct form draw differently to save on
multiplications.
For M odd
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Implementation of Discrete-Time Systems
Digital Signal Processing
6.3. Structures for FIR Systems
Example
Determine the Linear phase structure for the
following FIR Filter
The length of the filter is 7
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Implementation of Discrete-Time Systems
Digital Signal Processing
6.3. Structures for FIR Systems
Example
Determine the Linear phase structure for the
following FIR Filter
The length of the filter is 8
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Implementation of Discrete-Time Systems
Digital Signal Processing
6.4. Signal Flow Graph Representation
A network of directed branches connected at nodes.
Example representation of a difference equation
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Implementation of Discrete-Time Systems
Digital Signal Processing
6.4. Signal Flow Graph Representation
Example
Representation of Direct Form II with signal flow
graphs
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Implementation of Discrete-Time Systems
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6.4. Signal Flow Graph Representation
Example
Find impulse response from the following flow
graph