Impulse invariance

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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.

The continuous-time system's impulse response, ${\displaystyle h_{c}(t)}${\displaystyle h_{c}(t)}, is sampled with sampling period ${\displaystyle T}${\displaystyle T} to produce the discrete-time system's impulse response, ${\displaystyle h[n]}${\displaystyle h[n]}.

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

Thus, the frequency responses of the two systems are related by

${\displaystyle H(e^{j\omega })={\frac {1}{T}}\sum _{k=-\infty }^{\infty }{TH_{c}\left(j{\frac {\omega }{T}}+j{\frac {2{\pi }}{T}}k\right)},円}${\displaystyle H(e^{j\omega })={\frac {1}{T}}\sum _{k=-\infty }^{\infty }{TH_{c}\left(j{\frac {\omega }{T}}+j{\frac {2{\pi }}{T}}k\right)},円}

If the continuous time filter is approximately band-limited (i.e. ${\displaystyle H_{c}(j\Omega )<\delta }${\displaystyle H_{c}(j\Omega )<\delta } when ${\displaystyle |\Omega |\geq \pi /T}${\displaystyle |\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):

${\displaystyle H(e^{j\omega })=H_{c}(j\omega /T),円}${\displaystyle H(e^{j\omega })=H_{c}(j\omega /T),円} for ${\displaystyle |\omega |\leq \pi ,円}${\displaystyle |\omega |\leq \pi ,円}

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

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

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

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

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

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

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

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

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

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 ]}

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

Making this correction gives

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

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

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

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

• Bilinear transform
• Matched Z-transform method

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