Analog design using digital filters:

When we are needed to calculate an analog filter taking the A/D - H(z) - D/A structure, the specifications consist of the analog frequencies{Ω₁, Ω₂, ..., Ω_N}, the corresponding magnitudes {K₁, K₂, ..., K_N} and the sampling time T. We change to digital specs needing the relation ω_i = Ω_iT.

Example: Design a digital filter H(z) that when needed in an A/D-H(z)-D/A structure provides an equivalent low-pass analog filter with (a) -3.01 dB cut-off frequency of 500Hz, (b) monotonic pass bands and stop, (c) resultant of frequency response down at least 15 dB at 750 Hz, and (d) sample rate of 2000 samples/sec.

Solution

Step 0 First we convert the analog Ω's to digital ω's using ω_i = ω_iT.

Ω₁ = 2pF₁ = 2p 500 = 1000 p rad/sec.,           K₁ = -3.01 dB

Ω₂ = 2pF₂ = 2p 750 = 1500 p rad/sec.,           K₂ = -15 dB

Thus

ω₁ = Ω₁T = 1000Π 1/2000  = 0.5Π rad,           K₁ = -3.01 dB

ω₂ = Ω₂T = 1500Π 1/2000   = 0.75Π rad,         K₂ = -15 dB

Step 1 Pre-warping. Use T = 1. (In Steps 1 and 3 we would have used T = 1/2000 but the two needs of T would cancel out).

Step 2 Design H_a(s), i.e., calculate the low pass Butterworth filter.

Do analog low pass to low pass transformation s→(s/Ωc), i.e., s→(s/2) in order to take the cut- off frequency from 1 to 2 rad/sec. That provides the Ha(s) with pre-warped specs and Ωc = 2 rad/sec.

Step 3 Applying the bilinear transformation  to H (s) with T = 1 will transform the pre-warped analog filter into a digital filter with specific system function H(z) that will satisfy the given needs:

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