Impulse invariance

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.

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

$H(e^\{jomega\})\; =\; sum\_\{k=-infty\}^infty\{H\_cleft(jfrac\{omega\}\{T\}\; +\; jfrac\{2\{pi\{T\}k\; ight)\},$

If the continuous time filter is appropriately band-limited (ie. $H\_c(jOmega)\; =\; 0$ 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.

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

Comparison to the Bilinear Transform

Note that if $H\_c(jOmega),$ 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

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

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

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

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

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

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.

See also

Infinite Impulse Response (IIR)

Bilinear Transform

Continuous Time Filters: Chebyshev Filter, Butterworth Filter, Elliptic Filter

Sources

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. 05 April 2007.