Impulse invariance: Difference between revisions
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{{More footnotes|date=April 2009}} |
{{More footnotes|date=April 2009}} |
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'''Impulse invariance''' is a |
'''Impulse invariance''' is a technique for designing discrete-time [[infinite-impulse-response]] (IIR) filters from continuous-time filters in which the impulse response of the continuous-time system is sampled to produce the impulse response of the discrete-time system. The frequency response of the discrete-time system will be a sum of shifted copies of the frequency response of the continuous-time system; if the continuous-time system is approximately band-limited to a frequency less than the [[Nyquist frequency]] of the sampling, then the frequency response of the discrete-time system will be approximately equal to it for frequencies below the Nyquist frequency. |
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==Discussion== |
==Discussion== |
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The continuous-time system's impulse response, <math>h_c(t)</math>, is sampled with sampling period <math>T</math> to produce the discrete-time system's impulse response, <math>h[n]</math>. |
The continuous-time system's impulse response, <math>h_c(t)</math>, is sampled with sampling period <math>T</math> to produce the discrete-time system's impulse response, <math>h[n]</math>. |
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Thus, the frequency responses of the two systems are related by |
Thus, the frequency responses of the two systems are related by |
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:<math>H(e^{j\omega}) = \frac{1}{T} \sum_{k=-\infty}^\infty{ |
:<math>H(e^{j\omega}) = \frac{1}{T} \sum_{k=-\infty}^\infty{ TH_c\left(j\frac{\omega}{T} + j\frac{2{\pi}}{T}k\right)}\,</math> |
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If the continuous time filter is approximately band-limited (i.e. <math>H_c(j\Omega) < \delta</math> when <math>|\Omega| \ge \pi/T</math>), then the frequency response of the discrete-time system will be approximately the continuous-time system's frequency response for frequencies below π radians per sample (below the Nyquist frequency 1/(2''T'') Hz): |
If the continuous time filter is approximately band-limited (i.e. <math>H_c(j\Omega) < \delta</math> when <math>|\Omega| \ge \pi/T</math>), then the frequency response of the discrete-time system will be approximately the continuous-time system's frequency response for frequencies below π radians per sample (below the Nyquist frequency 1/(2''T'') Hz): |
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===Comparison to the bilinear transform=== |
===Comparison to the bilinear transform=== |
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Note that aliasing will occur, including aliasing below the Nyquist frequency to the extent that the continuous-time filter's response is nonzero above that frequency. |
Note that aliasing will occur, including aliasing below the Nyquist frequency to the extent that the continuous-time filter's response is nonzero above that frequency. The [[bilinear transform]] is an alternative to impulse invariance that uses a different mapping that maps the continuous-time system's frequency response, out to infinite frequency, into the range of frequencies up to the Nyquist frequency in the discrete-time case, as opposed to mapping frequencies linearly with circular overlap as impulse invariance does. |
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===Effect on poles in system function=== |
===Effect on poles in system function=== |
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If the continuous poles at <math>s = s_k</math>, the system function can be written in partial fraction expansion as |
If the continuous poles at <math>s = s_k</math>, the system function can be written in partial fraction expansion as |
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:<math>H(z) = T \sum_{k=1}^N{\frac{A_k}{1-e^{s_kT}z^{-1}}}\,</math> |
:<math>H(z) = T \sum_{k=1}^N{\frac{A_k}{1-e^{s_kT}z^{-1}}}\,</math> |
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Thus the poles from the continuous-time system function are translated to poles at z = e<sup>s<sub>k</sub>T</sup>. |
Thus the poles from the continuous-time system function are translated to poles at z = e<sup>s<sub>k</sub>T</sup>. The zeros, if any, are not so simply mapped.{{clarify|date=February 2013}} |
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===Poles and zeros=== |
===Poles and zeros=== |
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⚫ | If the system function has zeros as well as poles, they can be mapped the same way, but the result is no longer an impulse invariance result: the discrete-time impulse response is not equal simply to samples of the continuous-time impulse response. This method is known as the [[matched Z-transform method]], or pole–zero mapping. |
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⚫ | If the system function has zeros as well as poles, they can be mapped the same way, but the result is no longer an impulse invariance result: the discrete-time impulse response is not equal simply to samples of the continuous-time impulse response. |
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===Stability and causality=== |
===Stability and causality=== |
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Since poles in the continuous-time system at ''s'' = ''s<sub>k</sub>'' transform to poles in the discrete-time system at z = exp(''s<sub>k</sub>T''), poles in the left half of the ''s''-plane map to inside the unit circle in the ''z''-plane; so if the continuous-time filter is causal and stable, then the discrete-time filter will be causal and stable as well. |
Since poles in the continuous-time system at ''s'' = ''s<sub>k</sub>'' transform to poles in the discrete-time system at z = exp(''s<sub>k</sub>T''), poles in the left half of the ''s''-plane map to inside the unit circle in the ''z''-plane; so if the continuous-time filter is causal and stable, then the discrete-time filter will be causal and stable as well. |
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===Corrected formula=== |
===Corrected formula=== |
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⚫ | When a causal continuous-time impulse response has a discontinuity at <math>t=0</math>, the expressions above are not consistent.<ref>{{Cite journal|title = A correction to impulse invariance|journal = IEEE Signal Processing Letters|date = 2000-10-01|issn = 1070-9908|pages = 273–275|volume = 7|issue = 10|doi = 10.1109/97.870677|first = L.B.|last = Jackson| bibcode=2000ISPL....7..273J }}</ref> |
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⚫ | |||
⚫ | When a causal continuous-time impulse response has a discontinuity at <math>t=0</math>, the expressions above are not consistent.<ref>{{Cite journal|title = A correction to impulse invariance|journal = IEEE Signal Processing Letters|date = 2000-10-01|issn = 1070-9908|pages = 273–275|volume = 7|issue = 10|doi = 10.1109/97.870677|first = L.B.|last = Jackson}}</ref> |
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⚫ | |||
Making this correction gives |
Making this correction gives |
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:<math>h[n] = T \left( h_c(nT) - \frac{1}{2} h_c( |
:<math>h[n] = T \left( h_c(nT) - \frac{1}{2} h_c(0_+)\delta [n] \right) \,</math> |
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:<math>h[n] = T \sum_{k=1}^N{A_ke^{s_knT}} \left( u[n] - \frac{1}{2} \delta[n] \right) \,</math> |
:<math>h[n] = T \sum_{k=1}^N{A_ke^{s_knT}} \left( u[n] - \frac{1}{2} \delta[n] \right) \,</math> |
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:<math>H(z) = T \sum_{k=1}^N{\frac{A_k}{1-e^{s_kT}z^{-1}} - \frac{T}{2} \sum_{k=1}^N A_k}.</math> |
:<math>H(z) = T \sum_{k=1}^N{\frac{A_k}{1-e^{s_kT}z^{-1}} - \frac{T}{2} \sum_{k=1}^N A_k}.</math> |
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The second sum is zero for filters without a discontinuity, which is why ignoring it is often safe. |
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==See also== |
==See also== |
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* Oppenheim, Alan V. and Schafer, Ronald W. with Buck, John R. ''Discrete-Time Signal Processing.'' Second Edition. Upper Saddle River, New Jersey: Prentice-Hall, 1999. |
* Oppenheim, Alan V. and Schafer, Ronald W. with Buck, John R. ''Discrete-Time Signal Processing.'' Second Edition. Upper Saddle River, New Jersey: Prentice-Hall, 1999. |
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* Sahai, Anant. Course Lecture. Electrical Engineering 123: Digital Signal Processing. University of California, Berkeley. 5 April 2007. |
* Sahai, Anant. Course Lecture. Electrical Engineering 123: Digital Signal Processing. University of California, Berkeley. 5 April 2007. |
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* Eitelberg, Ed. "Convolution Invariance and Corrected Impulse Invariance." Signal Processing, Vol. 86, Issue 5, pp. |
* Eitelberg, Ed. "Convolution Invariance and Corrected Impulse Invariance." Signal Processing, Vol. 86, Issue 5, pp. 1116–1120. 2006 |
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==External links== |
==External links== |
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{{DSP}} |
{{DSP}} |
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{{Authority control}} |
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{{DEFAULTSORT:Impulse Invariance}} |
{{DEFAULTSORT:Impulse Invariance}} |
Latest revision as of 06:07, 12 July 2024
This article includes a list of general references, but it lacks sufficient corresponding inline citations. (April 2009) |
Impulse invariance is a technique for designing discrete-time infinite-impulse-response (IIR) filters from continuous-time filters in which the impulse response of the continuous-time system is sampled to produce the impulse response of the discrete-time system. The frequency response of the discrete-time system will be a sum of shifted copies of the frequency response of the continuous-time system; if the continuous-time system is approximately band-limited to a frequency less than the Nyquist frequency of the sampling, then the frequency response of the discrete-time system will be approximately equal to it for frequencies below the Nyquist frequency.
Discussion
[edit]The continuous-time system's impulse response, , is sampled with sampling period to produce the discrete-time system's impulse response, .
Thus, the frequency responses of the two systems are related by
If the continuous time filter is approximately band-limited (i.e. when ), then the frequency response of the discrete-time system will be approximately the continuous-time system's frequency response for frequencies below π radians per sample (below the Nyquist frequency 1/(2T) Hz):
- for
Comparison to the bilinear transform
[edit]Note that aliasing will occur, including aliasing below the Nyquist frequency to the extent that the continuous-time filter's response is nonzero above that frequency. The bilinear transform is an alternative to impulse invariance that uses a different mapping that maps the continuous-time system's frequency response, out to infinite frequency, into the range of frequencies up to the Nyquist frequency in the discrete-time case, as opposed to mapping frequencies linearly with circular overlap as impulse invariance does.
Effect on poles in system function
[edit]If the continuous poles at , the system function can be written in partial fraction expansion as
Thus, using the inverse Laplace transform, the impulse response is
The corresponding discrete-time system's impulse response is then defined as the following
Performing a z-transform on the discrete-time impulse response produces the following discrete-time system function
Thus the poles from the continuous-time system function are translated to poles at z = eskT. The zeros, if any, are not so simply mapped.[clarification needed]
Poles and zeros
[edit]If the system function has zeros as well as poles, they can be mapped the same way, but the result is no longer an impulse invariance result: the discrete-time impulse response is not equal simply to samples of the continuous-time impulse response. This method is known as the matched Z-transform method, or pole–zero mapping.
Stability and causality
[edit]Since poles in the continuous-time system at s = sk transform to poles in the discrete-time system at z = exp(skT), poles in the left half of the s-plane map to inside the unit circle in the z-plane; so if the continuous-time filter is causal and stable, then the discrete-time filter will be causal and stable as well.
Corrected formula
[edit]When a causal continuous-time impulse response has a discontinuity at , the expressions above are not consistent.[1] This is because has different right and left limits, and should really only contribute their average, half its right value , to .
Making this correction gives
Performing a z-transform on the discrete-time impulse response produces the following discrete-time system function
The second sum is zero for filters without a discontinuity, which is why ignoring it is often safe.
See also
[edit]References
[edit]- ^ Jackson, L.B. (1 October 2000). "A correction to impulse invariance". IEEE Signal Processing Letters. 7 (10): 273–275. Bibcode:2000ISPL....7..273J. doi:10.1109/97.870677. ISSN 1070-9908.
Other sources
[edit]- Oppenheim, Alan V. and Schafer, Ronald W. with Buck, John R. Discrete-Time Signal Processing. Second Edition. Upper Saddle River, New Jersey: Prentice-Hall, 1999.
- Sahai, Anant. Course Lecture. Electrical Engineering 123: Digital Signal Processing. University of California, Berkeley. 5 April 2007.
- Eitelberg, Ed. "Convolution Invariance and Corrected Impulse Invariance." Signal Processing, Vol. 86, Issue 5, pp. 1116–1120. 2006
External links
[edit]- Impulse Invariant Transform at CircuitDesign.info Brief explanation, an example, and application to Continuous Time Sigma Delta ADC's.