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. ^[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_{k}e^{s_{k}nT}\left(u[n]-{\frac {1}{2}}\delta [n]\right)}\,