structure/implementation of discrete-time system · structure/implementation ... lattice and...
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![Page 1: Structure/implementation of discrete-time system · Structure/implementation ... Lattice and lattice-ladder structures (all-pole IIR order 1) ... Given that a higher-order IIR filter](https://reader031.vdocument.in/reader031/viewer/2022030704/5aefcea37f8b9aa9168d1b9d/html5/thumbnails/1.jpg)
Structure/implementation
of discrete-time system
Prasanta Kumar Ghosh
Sep 21, 2017
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Consider the class of LTI DT systems characterized by the general
linear constant-coefficient difference equation
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Special case
Infinite impulse response; not possible to implement it using discrete
convolution! But
Recursive computation (with initial rest condition)
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Basic elements
Adder
Multiplier
Memory
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Example
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Example
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Example
Direct Form I
Implementation
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Example (contd.)
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Example (contd.)
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Example (contd.)
Less delay elements!!
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Example (contd.)
Direct Form II
Implementation
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Signal Flow Graph
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Example
Direct Form II
Implementation
Direct Form I
Implementation
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Cascade Form
Ns second-order system Ns! Pairings of poles with zeros & Ns! orderings of the
resulting second-order sections (Ns!)2 different pairings and orderings
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Example
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Parallel Form
Alternatively,
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Parallel Form
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Example
Using second-order system Using first-order system
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Transposed Form
Reverse directions of all branches in the network while keeping the branch
transmittances as they were and reversing the roles of the input and output so that source
nodes become sink nodes and vice versa.
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Repeating different forms for FIR case
Direct Form
Transposed Form
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Cascade Form
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If FIR is linear-phase FIR phase?
M is even M is odd
I
II
III
IV
I
II
III
IV
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Type-I linear-phase FIR phase?
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Type-III linear-phase FIR phase?
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Frequency-sampling structures
Specify desired frequency response at
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Frequency-sampling structures
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Frequency-sampling structures
With zeros
Parallel bank of
single pole filters
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Frequency-sampling structures
For narrowband
filter it results in
efficient
implementation
With symmetry
the
implementation
can be even more
efficient
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Lattice structures
Lets begin with
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Lattice structures
Direct-form structures of the FIR filter
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Lattice structures
Suppose
Single stage-lattice filter
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Lattice structures
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Lattice structures
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Lattice structures
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Lattice structures
In general
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Lattice structures
In general
Forward
predictor
Backward
predictor
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Lattice structures
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for IIR systems
Lattice and lattice-ladder structures (all-pole IIR order 1)
Lattice structures
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Lattice and lattice-ladder structures (all-pole IIR order 2)
for IIR systems Lattice structures
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Lattice and lattice-ladder structures (pole zero system)
Lattice structures
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Lattice and lattice-ladder structures (pole zero system)
Lattice structures
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Quantization of filter
coefficients
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With quantized coefficients
poles
Pole perturbation
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Relate perturbation in poles to perturbation to coefficients
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Relate perturbation in poles to perturbation to coefficients
Similar results can be derived for zeros
If poles are clustered, the length between poles are small leading to large perturbation
error
Error can be minimized by maximizing the length
One way can be to combine complex valued poles
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Lets consider a two-pole filter section
With finite precision of the
coefficients, the pole
positions are finite
When b bits are used, there
are at most (2b-1)2 possible
pole positions for the poles in
each quadrant, excluding zero
coefficients case
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Lets consider a two-pole filter section
For b = 4 there are 169 unique pole positions
Non-uniformity is due to
quantizing r2
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Lets consider a two-pole filter section
For b = 4 there are 169 unique pole positions
Non-uniformity is due to
quantizing r2
Sparse poles near theta = 0,
unfavorable for low pass
filter; similarly for high pass
filter
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Since there are various ways in which one can realize a second-order filter
section, there are obviously many possibilities for different pole locations with
quantized coefficients.
Ideally, we should select a structure that provides us with a dense set of points in
the regions where the poles lie. Unfortunately, however, there is no simple
and systematic method for determining the filter realization that yields this
desired result.
Given that a higher-order IIR filter should be implemented as a combination of
second-order sections, we still must decide whether to employ a parallel
configuration or a cascade configuration.
direct control of both the poles and the zeros that result from the quantization process.
Direct control on poles only
Undesirable
Cascade is a preferred choice specially with fixed-point implementation