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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
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.
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
Note
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