词条 | Impulse invariance |
释义 |
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. DiscussionThe 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 transformNote 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 functionIf 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.{{clarify|date=February 2013}} Poles and zerosIf 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 causalitySince 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 formulaWhen a causal continuous-time impulse response has a discontinuity at , the expressions above are not consistent.[1] This is because should really only contribute half its value to . Making this correction gives Performing a z-transform on the discrete-time impulse response produces the following discrete-time system function See also
Chebyshev filter Butterworth filter Elliptic filter References1. ^{{Cite journal|title = A correction to impulse invariance|url = http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?tp=&arnumber=870677&isnumber=18861|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}} Other sources{{Refbegin}}
External links
2 : Digital signal processing|Filter theory |
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