7-1 iir filter design by impulse invariance
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7-1 IIR Filter design by Impulse Invariance
Filter
Digital Filter
What are filters…
Filters are a class of LTI systems.
An ideal Frequency-Selective Filter is a system that passes certain frequency components and totally rejects all others.
In a broader context, any system that modifies certain frequencies relative to others is also called a filter.
Basic system for discrete-time filtering of continuous-time signals
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Discrete-time system D/C][ny
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)( jeH
Filtering
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Design of filters…
The design of filters involve the following stages:
1) the of the desired properties of the system;
2) the of the specifications using a causal discrete-time system;
3)
E.g. 1 discrete-time ideal low-pass filter
otherwise
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E.g. 2 Determining specifications for a discrete-time filter
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Discrete-time system D/C][ny
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)(tyc
)( jeH
Filtering
Consider a discrete-time filter that is to be used to lowpass filter a continuous-time signal using the considerations given above. The overall system should have the following properties when the sampling rate is :sT 410
* The gain should be within of unity in the frequency band
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* The gain should be no greater than 0.001 in the frequency band )3000(2
passband transition stopband
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Ideal passband gain: 1 (0dB)
Max passband gain: 1.01 (0.086dB)
Max stopband gain: 0.001 (-60dB)
passband transition stopband
a realizable system
Design of IIR and FIR filters
IIR filter design by Impulse Invariance
Bilinear Transformation
……
FIR filter design using Windows
Frequency-Sampling ……
Design of Discrete-time IIR filters from continuous-time filters
The traditional approach to the design of discrete-time IIR filters involves the transformation of a continuous-time filter into a discrete-time filter meeting the prescribed specifications.
Design of discrete-time IIR filters starts from a set of discrete-timespecifications.
Given the discrete-time specifications, a discrete-time filter is obtained by transforming a prototype continuous-time filter and of course transforming the specifications to the continuous-time domain.
][)( nhzH )()( thsHcc
In such transformations, it is required that the essential properties of the continuous-time frequency response be preserved in the frequency response of the resulting discrete-time filter.
1) the imaginary axis of the s-plane should be mapped onto the unit circle of the z-plane
2) A stable continuous-time filter should be transformed to a stable discrete-time filter.
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inside
Where should the poles be located in the z-plane for a causal and stable LTI system
IIR Filter Design by Impulse Invariance Impulse Invariance
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Discrete-time system D/C][ny
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)(tyc
)( jeH
)( jHc
When is bandlimited, can be obtained by)( jeH)( jHc
),()(T
jHeHc
j
T should be chosen such that
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Impulse Invariance
)(][ nTThnhc
The relationship between the discrete-time system and the continuous-time system can be specified by:
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T
Discrete-time system D/C][ny
T
)(tyc
)( jeH
)( jHc
That is, the impulse response of the discrete-time system is a scaled, sampled version of )(th
c
When the above equation holds for a discrete-time system, it is said to be an impulse-invariant version of the continuous system
Impulse Invariance Proof:
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T
Discrete-time system D/C][ny
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IIR Filter Design by Impulse Invariance In the impulse invariance design procedure for transforming
continuous-time filter is chosen proportional to equally spaced samples of the impulse response of the continuous-time filter, as:
Then the discrete-time filter can result from the sampling of the impulse response of the continuous-time filter.
While as sampling , it is usually carried out by performing transformation on the system function.
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nThTnh
Steps:1) the discrete-time filter specifications to continuous-time filter
:
2) Obtain a continuous-time filter according to the specifications;
3) The system function of the continuous-time filter is transformed to z-plane.
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Assume all poles are single order
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A pole at in the s-plane transforms to a pole at in the z-plane;
the coefficients in the partial fraction expansions of and are equal, except for the scaling multiplier
If the continuous-time filter is stable, corresponding to the real part of being less than zero, then the magnitude of will be less than unity, so that the corresponding pole in the discrete-time filter is inside the unit circle. Therefore the causal discrete-time filter is also stable.
The zeros may not be mapped in the same way.
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What will happen when is not limited to the range of
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Need the continuous-time filter approaching zero at the high frequencies to ignore the aliasing.
Typical frequency-selective continuous-time filters for transformations include:
Butterworth filter
Chebyshev filter
Elliptic filter
Butterworth lowpass filter
Butterworth lowpass filter is defined by the property that the magnitude response is maximally flat in the passband.
And the magnitude response is monotonic both in the passbandand the stopband.
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As N increases, the filter becomes sharper: remain close to unity over more of the passband and become close to zero more rapidly in the stopband.
The magnitude-squared function at the cutoff frequency will always be equal to ½.
c
Chebyshev filtersA more efficient filter distributes the accuracy of the approximation uniformly over the passband or the stopband (or both) -> equiripple behavior rather than a monotonic behavior. (lower order)
The class of Chebyshev filters has the property that the magnitude of the frequency response is either equirippple in the passband and monotonic in the stopband ( ), or monotonic in the passband and equirippple in the stopband (
)
Type I Chebyshev filter)/(1
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Type II Chebyshev filter 122
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Elliptic filtersEquiripple in the passband and the stopband: it is the best that can be achieved for a given filter order N, resulting in a transition band as small as possible.
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E.g. 3 Impulse Invariance with a Butterworth Filter
* Design a lowpass discrete-time filter by applying impulse invariance to an appropriate filter. The specifications for the discrete-time filter are
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eH
What do these inequations mean?
E.g. 4 Impulse Invariance with a Butterworth Filter
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cContinuous-time specifications
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By rounding, the specifications cannot be exactly met.
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The poles of the magnitude-squared function always occur in pairs.
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The poles of the magnitude-squared function always occur in pairs.
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Summary for Filter design based on Impulse Invariance
is to choose an impulse response for the discrete-time filter that is similar in some sense to the impulse response of the continuous-time filter.
If the continuous-time filter is bandlimited, then the discrete-time filter frequency response will closely approximate the continuous-time filter frequency response.
In the impulse invariance design, the relationship between continuous-time and discrete-time frequency is linear. Except for aliasing, the shape of frequency response can be well preserved.
This kind of methods are suitable for bandlimited filters, e.g. low-pass filters, but not so appropriate for highpass or bandstop filters.
Assignment
Preparation for the next lecture:
Filter design by bilinear transformation
Solve the following problems:
7.1 (a)