Impulse invariance

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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. 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, 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 approximately band-limited (i.e. H_c(j\Omega) < \delta when |\Omega| \ge \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\omega}) = H_c(j\omega/T)\, for |\omega| \le \pi\,

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 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 = 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. In the case of all-pole filters, the methods are equivalent.

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 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}.

See also[edit]

References[edit]

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

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.

External links[edit]