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

$H(e^{j\omega })=\sum _{k=-\infty }^{\infty }{H_{c}\left(j{\frac {\omega }{T}}+j{\frac {2{\pi }}{T}}k\right)}\,$

If the continuous time filter is appropriately band-limited (ie. $H_{c}(j\Omega )=0$ when $|\Omega |\geq \pi /T$), 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 $|\omega |\leq \pi \,$

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

$h_{c}(t)={\begin{cases}\sum _{k=1}^{N}{A_{k}e^{s_{k}t}},&t\geq 0\\0,&{\mbox{otherwise}}\end{cases}}$

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

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. This is because $h_{c}(0)$ should really only contribute half its value to $h[0]$.

Making this correction gives

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

$h[n]=T\sum _{k=1}^{N}{A_{k}e^{s_{k}nT}\left(u[n]-{\frac {1}{2}}\delta [n]\right)}\,$

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

$H(z)=T\sum _{k=1}^{N}{{\frac {A_{k}}{1-e^{s_{k}T}z^{-1}}}-{\frac {T}{2}}\sum _{k=1}^{N}A_{k}}.$

See also

• Infinite impulse response
• Bilinear transform
• Matched Z-transform method
• Continuous Time Filters:

  Chebyshev filter

  Butterworth filter

  Elliptic filter

Further reading

• 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.