2.1 Prototype analog filter design

Analog filter design

Laplace transform: $$H(s)=\int_{()} \,d t$$∞ ∞ h a t s t Note that the continuous-time Fourier transform is $H(i\lambda )$ (the Laplace transform evaluated on the imaginary axis).

Since the early 1900's, there has been a lot of research on designing analog filters of the form $$H(s)=\frac{b_0+b_{1}s+b_{2}s^2+\mathrm{...}+b_{M}s^M}{1+a_{1}s+a_{2}s^2+\mathrm{...}+a_{M}s^M}$$ A causal IIR filter cannot have linear phase (no possible symmetry point), and design work for analogfilters has concentrated on designing filters with equiriplle ( $()$∞ L ) magnitude responses. These design problems have been solved. We will not concernourselves here with the design of the analog prototype filters, only with how these designs are mapped todiscrete-time while preserving optimality.

An analog filter with real coefficients must have a magnitude response of the form $$\left|H(\lambda )\right|^2=B(\lambda ^2)$$

Recall that an analog filter is stable and causal if all the poles are in the left half-plane, LHP, and is minimum phase if all zeros and poles are in the LHP.

$s=i\lambda$ : $B(\lambda ^2)=B(-s^2)=H(s)H(-s)=Hi\lambda H-(i\lambda )=Hi\lambda \overline{H(i\lambda )}$ we can factor $B(-s^2)$ into $H(s)H(-s)$ , where $H(s)$ has the left half plane poles and zeros, and $H(-s)$ has the RHP poles and zeros.

$\left|H(s)\right|^2=H(s)H(-s)$ for $s=i\lambda$ , so $H(s)$ has the magnitude response $B(\lambda ^2)$ . The trick to analog filter design is to design a good $B(\lambda ^2)$ , then factor this to obtain a filter with that magnitude response.

The traditional analog filter designs all take the form $B(\lambda ^2)=\left|H(\lambda )\right|^2=\frac{1}{1+F(\lambda ^2)}$ , where $F$ is a rational function in $\lambda ^2$ .

Traditional filter designs

Butterworth

$$B(\lambda ^2)=\frac{1}{1+\lambda ^{(2M)}}$$

Chebyshev

$$B(\lambda ^2)=\frac{1}{1+\epsilon ^2{C}_M(\lambda )^2}$$ where $C_M(\lambda )^2$ is an $M^{\mathrm{th}}$ order Chebyshev polynomial. .

Inverse chebyshev

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Elliptic function filter (cauer filter)

$$B(\lambda ^2)=\frac{1}{1+\epsilon ^2{J}_M(\lambda )^2}$$ where $J_M$ is the "Jacobi Elliptic Function." .

The Cauer filter is $()$∞ L optimum in the sense that for a given $M$ , ${\delta }_p$ , ${\delta }_s$ , and ${\lambda }_p$ , the transition bandwidth is smallest.

That is, it is $()$∞ L optimal.