Impulse invariance: Information from Answers.com

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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. If the continuous-time system is appropriately band-limited, the frequency response of the discrete-time system will be defined as the continuous-time system's frequency response with linearly-scaled frequency.

1 Discussion
2 See also
3 References
4 Further reading

Discussion

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, $h [n]$ .

$h[n]=Th_c(nT)\,$

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

$H(e^{j\omega}) = \sum_{k=-\infty}^\infty{H_c\left(j\frac{\omega}{T} + j\frac{2{\pi}}{T}k\right)}\,$

If the continuous time filter is appropriately band-limited (ie. $H c (j Ω) = 0$ when $|\Omega| \ge \pi/T$ ), then frequency response of the discrete-time system will be defined as the continuous-time system's frequency response with linearly-scaled frequency.

$H(e^{j\omega}) = H_c(j\omega/T)\,$ for $|\omega| \le \pi\,$

Comparison to the Bilinear Transform

Note that if $H_c(j\Omega)\,$ is not band-limited, aliasing will occur. The Bilinear Transform is an alternative to Impulse Invariance that uses a direct unique mapping from the continuous-time frequency axis to the discrete-time frequency axis. Impulse Invariance, however, uses a linear scale between the frequency axes for the continuous-time and discrete-time systems, $\Omega = \omega/T\,$ , which is not true for the Bilinear Transform.

Effect on Poles in System Function

If the continuous-time filter has poles at $s = s k$ , the system function can be written in partial fraction expansion as

$H_c(s) = \sum_{k=1}^N{\frac{A_k}{s-s_k}}\,$

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

$h_c(t) = \begin{cases} \sum_{k=1}^N{A_ke^{s_kt}}, & t \ge 0 \\ 0, & \mbox{otherwise} \end{cases}$

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

$h[n] = Th_c(nT)\,$

$h[n] = T \sum_{k=1}^N{A_ke^{s_knT}u[n]}\,$

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

$H(z) = T \sum_{k=1}^N{\frac{A_k}{1-e^{s_kT}z^{-1}}}\,$

Thus the poles from the continuous-time system function are translated to poles at z = e^s_kT

Stability and Causality

Since poles in the continuous-time system at $s = s_k\,$ transform to poles in the discrete-time system at z = e^s_kT, if the continuous-time filter is causal and stable, then the discrete-time filter will be causal and stable as well.

Corrected Formula

When a causal continuous-time impulse response has a discontinuity at $t = 0$ , the expressions above are not consistent. ^[1] This is because $h c (0)$ should really only contribute half its value to $h [0]$ .

Making this correction gives

$h[n] = T \left( h_c(nT) - \frac{1}{2} h_c(0) \delta [n] \right) \,$

$h[n] = T \sum_{k=1}^N{A_ke^{s_knT} \left( u[n] - \frac{1}{2} \delta[n] \right)}\,$

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

$H(z) = T \sum_{k=1}^N{\frac{A_k}{1-e^{s_kT}z^{-1}} - \frac{T}{2} \sum_{k=1}^N A_k}.$

References

^ [1] L. Jackson, "A correction to impulse invariance", IEEE Signal Processing Letters, Vol. 7, Oct. 2000.

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.
Impulse Invariant Transform at CircuitDesign.info Brief explanation, an example, and application to Continuous Time Sigma Delta ADC's.

v • d • e Digital signal processing

Theory	Discrete frequency \| Nyquist–Shannon sampling theorem \| estimation theory \| detection theory

Sub-fields	audio signal processing \| control engineering \| digital image processing \| speech processing \| statistical signal processing

Techniques	Discrete Fourier transform (DFT) \| Discrete-time Fourier transform (DTFT) \| Impulse invariance \| bilinear transform \| Z-transform, advanced Z-transform

Sampling	oversampling \| undersampling \| downsampling \| upsampling \| aliasing \| anti-aliasing filter \| sampling rate \| Nyquist rate/frequency

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Impulse invariance

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