C.1 Normalized Butterworth Function

The normalized Butterworth squared magnitude function is given by

(C.1)

where n is the order and ? is the specified ripple on filter passband. The specified ripple in dB is expressed as ? _dB = 20 log ₁₀ dB.

To develop the transfer function P _n( s), we first let s = j ? and then substitute ? ² = ? s ² into Equation (C.1) to obtain

(C.2)

Equation (C.2) has 2 n poles, and P _n( s) has n poles on the left-hand half plane (LHHP) on the s-plane, while P _n( ? s) has n poles on the right-hand half plane (RHHP) on the s-plane. Solving for poles leads to

(C.3)

If n is an odd number, Equation (C.3) becomes

and the corresponding poles are solved as

(C.4)

where k = 0, 1, , 2 n - 1. Thus in the phasor form, we have

(C.5)

When n is an even number, it follows that

(C.6)

where k = 0, 1, , 2 n ? 1. Similarly, the phasor form is given by

(C.7)

When n is an odd number, we can identify the poles on the LHHP as

(C.8)

Using complex conjugate pairs, we have

Notice that

and from a factor from the real pole ( s + r), it follows that

(C.9)

and