Impulse invariance

The continuous-time system's impulse response, $h\_c(t)$, is sampled with sampling period $T$ to produce the discrete-time system's impulse response, $h[n]$.

:$h[n]=Th\_c(nT)\backslash ,$

Thus, the frequency responses of the two systems are related by

:$H(e^\{j\backslash omega\})\; =\; \backslash frac\{1\}\{T\}\; \backslash sum\_\{k=-\backslash infty\}^\backslash infty\{\; TH\_c\backslash left(j\backslash frac\{\backslash omega\}\{T\}\; +\; j\backslash frac\{2\{\backslash pi\}\}\{T\}k\backslash right)\}\backslash ,$

If the continuous time filter is approximately band-limited (i.e. when $|\backslash Omega|\; \backslash ge\; \backslash pi/T$), 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):

:$H(e^\{j\backslash omega\})\; =\; H\_c(j\backslash omega/T)\backslash ,$ for $|\backslash omega|\; \backslash le\; \backslash pi\backslash ,$

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.

If the continuous poles at $s\; =\; s\_k$, the system function can be written in partial fraction expansion as

:$H\_c(s)\; =\; \backslash sum\_\{k=1\}^N\{\backslash frac\{A\_k\}\{s-s\_k\}\}\backslash ,$

Thus, using the inverse Laplace transform, the impulse response is

:$h\_c(t)\; =\; \backslash begin\{cases\}$

\sum_{k=1}^N{A_ke^{s_kt}}, & t \ge 0 \\

0, & \mbox{otherwise}

\end{cases}

The corresponding discrete-time system's impulse response is then defined as the following

:$h[n]\; =\; Th\_c(nT)\backslash ,$

:$h[n]\; =\; T\; \backslash sum\_\{k=1\}^N\{A\_ke^\{s\_knT\}u[n]\}\backslash ,$

Performing a z-transform on the discrete-time impulse response produces the following discrete-time system function

:$H(z)\; =\; T\; \backslash sum\_\{k=1\}^N\{\backslash frac\{A\_k\}\{1-e^\{s\_kT\}z^\{-1\}\}\}\backslash ,$

Thus the poles from the continuous-time system function are translated to poles at z = e^s_kT. The zeros, if any, are not so simply mapped.{{clarify|date=February 2013}}

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.

Since poles in the continuous-time system at s = s_k transform to poles in the discrete-time system at z = exp(s_kT), 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.

When a causal continuous-time impulse response has a discontinuity at $t=0$, the expressions above are not consistent.{{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 }}

This is because $h\_c\; (0)$ has different right and left limits, and should really only contribute their average, half its right value $h\_c\; (0\_+)$, to $h[0]$.

Making this correction gives

:$h[n]\; =\; T\; \backslash left(\; h\_c(nT)\; -\; \backslash frac\{1\}\{2\}\; h\_c(0\_+)\backslash delta\; [n]\; \backslash right)\; \backslash ,$

:$h[n]\; =\; T\; \backslash sum\_\{k=1\}^N\{A\_ke^\{s\_knT\}\}\; \backslash left(\; u[n]\; -\; \backslash frac\{1\}\{2\}\; \backslash delta[n]\; \backslash right)\; \backslash ,$

Performing a z-transform on the discrete-time impulse response produces the following discrete-time system function

:$H(z)\; =\; T\; \backslash sum\_\{k=1\}^N\{\backslash frac\{A\_k\}\{1-e^\{s\_kT\}z^\{-1\}\}\; -\; \backslash frac\{T\}\{2\}\; \backslash sum\_\{k=1\}^N\; A\_k\}.$

The second sum is zero for filters without a discontinuity, which is why ignoring it is often safe.

• Bilinear transform
• Matched Z-transform method

{{Reflist}}

{{Refbegin}}

• 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

• [http://www.circuitdesign.info/blog/2009/02/impulse-invariant-transform/ Impulse Invariant Transform at CircuitDesign.info] Brief explanation, an example, and application to Continuous Time Sigma Delta ADC's.

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Category:Digital signal processing

Category:Filter theory