Post on 07-May-2018
Structure/implementation
of discrete-time system
Prasanta Kumar Ghosh
Sep 21, 2017
Consider the class of LTI DT systems characterized by the general
linear constant-coefficient difference equation
Special case
Infinite impulse response; not possible to implement it using discrete
convolution! But
Recursive computation (with initial rest condition)
Basic elements
Adder
Multiplier
Memory
Example
Example
Example
Direct Form I
Implementation
Example (contd.)
Example (contd.)
Example (contd.)
Less delay elements!!
Example (contd.)
Direct Form II
Implementation
Signal Flow Graph
Example
Direct Form II
Implementation
Direct Form I
Implementation
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
Example
Parallel Form
Alternatively,
Parallel Form
Example
Using second-order system Using first-order system
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.
Repeating different forms for FIR case
Direct Form
Transposed Form
Cascade Form
If FIR is linear-phase FIR phase?
M is even M is odd
I
II
III
IV
I
II
III
IV
Type-I linear-phase FIR phase?
Type-III linear-phase FIR phase?
Frequency-sampling structures
Specify desired frequency response at
Frequency-sampling structures
Frequency-sampling structures
With zeros
Parallel bank of
single pole filters
Frequency-sampling structures
For narrowband
filter it results in
efficient
implementation
With symmetry
the
implementation
can be even more
efficient
Lattice structures
Lets begin with
Lattice structures
Direct-form structures of the FIR filter
Lattice structures
Suppose
Single stage-lattice filter
Lattice structures
Lattice structures
Lattice structures
Lattice structures
In general
Lattice structures
In general
Forward
predictor
Backward
predictor
Lattice structures
for IIR systems
Lattice and lattice-ladder structures (all-pole IIR order 1)
Lattice structures
Lattice and lattice-ladder structures (all-pole IIR order 2)
for IIR systems Lattice structures
Lattice and lattice-ladder structures (pole zero system)
Lattice structures
Lattice and lattice-ladder structures (pole zero system)
Lattice structures
Quantization of filter
coefficients
With quantized coefficients
poles
Pole perturbation
Relate perturbation in poles to perturbation to coefficients
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
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
Lets consider a two-pole filter section
For b = 4 there are 169 unique pole positions
Non-uniformity is due to
quantizing r2
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
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