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