Editing Infinite impulse response (section)

=== Step Invariance ===
Step invariance is a better design method than impulse invariant. The digital filter has several segments of input with different constants when sampling, which is composed of discrete steps. The step invariant IIR filter is less accurate than the same input step signal to the ADC. However, it is a better approximation for any input than the impulse invariant.<br />
Step invariant solves the problem of the same sample values when T(z) and T(s) are both step inputs. The input to the digital filter is u(n), and the input to the analog filter is u(t). Apply z-transform and Laplace transform on these two inputs to obtain the converted output signal.<br />
Perform z-transform on step input <math>Z[u(n)]=\dfrac{z}{z-1}</math><br />
Converted output after z-transform <math>Y(z)=T(z)U(z)=T(z)\dfrac{z}{z-1}</math><br />
Perform Laplace transform on step input <math>L[u(t)]=\dfrac{1}{s}</math><br />
Converted output after Laplace transform <math>Y(s)=T(s)U(s)=\dfrac{T(s)}{s}</math><br />
The output of the analog filter is y(t), which is the inverse Laplace transform of Y(s). If sampled every T seconds, it is y(n), which is the inverse conversion of Y(z).These signals are used to solve for the digital filter and the analog filter and have the same output at the sampling time.<br />
The following equation points out the solution of T(z), which is the approximate formula for the analog filter. <br />
:<math>T(z)=\dfrac{z-1}{z}Y(z)</math><br />
:<math>T(z)=\dfrac{z-1}{z}Z[y(n)]</math><br />
:<math>T(z)=\dfrac{z-1}{z}Z[Y(s)]</math><br />
:<math>T(z)=\dfrac{z-1}{z}Z[\dfrac{T(s)}{s}]</math><br />