Impulse invariance

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.

If the continuous poles at ${\displaystyle s=s_{k}}$, the system function can be written in partial fraction expansion as

${\displaystyle H_{c}(s)=\sum _{k=1}^{N}{\frac {A_{k}}{s-s_{k}}}\,}$

Thus, using the inverse Laplace transform, the impulse response is

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

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

${\displaystyle h[n]=Th_{c}(nT)\,}$

${\displaystyle h[n]=T\sum _{k=1}^{N}{A_{k}e^{s_{k}nT}u[n]}\,}$

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

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

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.

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.

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.

When a causal continuous-time impulse response has a discontinuity at ${\displaystyle t=0}$, the expressions above are not consistent.[1] This is because ${\displaystyle h_{c}(0)}$ has different right and left limits, and should really only contribute their average, half its right value ${\displaystyle h_{c}(0_{+})}$, to ${\displaystyle h[0]}$.

Making this correction gives

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

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

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

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

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