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. 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 })={\frac {1}{T}}\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 $|\Omega |\geq \pi /T$ ), 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 |\leq \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_{k}e^{s_{k}t}},&t\geq 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_{k}e^{s_{k}nT}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_{k}T}z^{-1}}}\,

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.^{[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.

Stability and causality[edit]

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.

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 $h_{c}(0)$ 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\left(h_{c}(nT)-{\frac {1}{2}}h_{c}(0_{+})\delta [n]\right)\,

h[n]=T\sum _{k=1}^{N}{A_{k}e^{s_{k}nT}}\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_{k}T}z^{-1}}}-{\frac {T}{2}}\sum _{k=1}^{N}A_{k}}.

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

References[edit]

^ Jackson, L.B. (1 October 2000). "A correction to impulse invariance". IEEE Signal Processing Letters. 7 (10): 273–275. doi:10.1109/97.870677. ISSN 1070-9908.

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.
Eitelberg, Ed. "Convolution Invariance and Corrected Impulse Invariance." Signal Processing, Vol. 86, Issue 5, pp. 1116–1120. 2006

External links[edit]

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

[1] Jackson, L.B. (1 October 2000). "A correction to impulse invariance". IEEE Signal Processing Letters. 7 (10): 273–275. doi:10.1109/97.870677. ISSN 1070-9908.

[1]

v t e Digital signal processing
Theory	Detection theory Discrete signal Estimation theory Nyquist–Shannon sampling theorem
Sub-fields	Audio signal processing Digital image processing Speech processing Statistical signal processing
Techniques	Z-transform Advanced z-transform Matched Z-transform method Bilinear transform Constant-Q transform Discrete cosine transform (DCT) Discrete Fourier transform (DFT) Discrete-time Fourier transform (DTFT) Impulse invariance Integral transform Laplace transform Post's inversion formula Starred transform Zak transform
Sampling	Aliasing Anti-aliasing filter Downsampling Nyquist rate / frequency Oversampling Quantization Sampling rate Undersampling Upsampling