Filter Design by pole-zero placements

Design stages for Digital filters

• Performance Specification

• Calculation of filter coefficients

• Realization structure

• finite word-length effects analysis and solutions

• Hardware/Software Implementation and Testing

Types of Frequency Selective filters

• Bandpass

• Lowpass

• Highpass

• Bandstop

ideal FSF have

• completely flat passbands

• completely flat stopbands = 0

• transition band is instantaneous at a single frequency

Practical Frequency Response specification

• The pass band is not flat and has a ripple till the passband edge ripple frequency

• The transition band attenuates gradually

• The stopband ripples past the stopband edge ripple frequency

After performance specifications what comes next

Calculation the filter coefficients b and a of the difference equation or calculation the transfer function

Pole-Zero Placements involve

placing pole & zeros on the z plane so that the resulting filter has a generated response

is Pole-Zero Placement suitable for complex filter designs

NO this is because the filter parameters are needed to be specified for more complex designs.

b₀ is the

gain term selected to normalise the frequency response at a selected frequency |H(Ω₀)| = 1

Geometric Interpretation of the Pole-Zero Placement

The magnitude of the frequency response = the product distance of the zeros to the frequency of choice/ the product distance of the poles to the frequency of choice

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the phase = the sum of phases of the zeros to the frequency of choice − the sum of phases of the poles to the frequency of choice

The presence of a zero close to or on the unit circle results in

attenuation of the magnitude of frequencies close to the zero

The presence of poles close to the unit circle

The magnitude of the frequencies close to the pole will be large

poles at the origin ( 0 point on both axis ) imply

no feedback i.e an FIR Filter

poles at location other than the origin imply

Feedback IIR Filter

The magnitude response is not influenced by The poles and zeros on the origin this is because

There is equal distance from the origin to any point on the unit circle

Do zeros affect the stability of the Filter

no they do not and they can be placed anywhere on the pole-zero plot

Ensuring real valued coefficients.

all complex zeros and poles must occur in complex conjugate pairs

On the z plane the frequency at each quadrant is

1 st is 0 rad or 2π

2nd is π/2 rad

3rd is π

4th is -π/2 or 3π/2

For a first order low pass filter is pole is located on

Ω = π

High frequencies are attenuated

For a first order high pass filter is pole is located on

Ω = 0

Low frequencies are attenuated

In first order FIR filters the poles are located

on the origin

If a second order filters FIR has real zeros they form

Cascade of two

To represent a bandpass filter with a 2nd order filter what is done

The zeros are placed at points -1,1 on the z plane

If the second order zeros are not real they form

complex conjugate pairs

A bandstop filter can be modeled in a 2nd order filter by

• Ensuring the zeros are conjugate pairs

• Setting r = 1

IIR FILTERS are also known as

Resonators

Low pass IIR filter has poles located at

The 0 rads point and it resonates close to 0

A Resonator

A resonator is a device or system that exhibits resonance or resonant behavior. That is, it naturally oscillates with greater amplitude at some frequencies, called resonant frequencies, than at other frequencies.

High pass IIR filter resonates at / close to

π and amplifies the magnitudefrequency’s close as well

For a 2nd order IIR Resonator with complex conjugates it resonates

at /close to θ

z =

rexp(jθ)

What is the effect of r in z for a resonator with no zeros

The resonant frequency can be set to θ, the filter then resonates at z = exp(jθ) → r = 1. The poles can only be placed at points within the circle → r < 1.

The sharpness of resonance in the filter is affected by r as it approaches 1. For lower values of r , the frequency θ resonates but frequencies close to θ also resonate with a lower magnitude. The distance between the point of r is equidistant to all points on the nit circle around θ.

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The opposite of a resonator is

Notch Filter. They decrease the amplitude of the frequency response at the frequency θ.

How to remove an isolated frequency or narrowband interference

The closeness of the pole - zero combination gives a frequency response which is close to unity for all frequencies except those particularly close to zero.

Comb Filters

A filter used to remove harmonically related interference.

Comb filters are based solely on the principle of

factorisation of 1 - r^N - z^N which has roots at rexp(jk2π/N)

The roots of the 1 - r^N -z^N in comb filters must be

evenly spaced around a circle of radius r

In FIR Comb filter, the spacing of the zeros create a

wide transition, resulting in the attenuation of frequencies close to the particular frequency.

In IIR Comb filter, has poles close to the zero, the spacing of the zeros creates a

sharper transition due to the presence of poles

IIR comb filter is a series of

Notch Filters