Signals and Systems - Electrical Engineering

6.5 Analog Filtering 391

Magnitude Squared Function
The magnitude-squared function of an analog low-pass filter has the general form

|H(j)|^2 =

1

1 +f(^2 )

(6.29)

where for low frequenciesf(^2 )≈0 so that|H(j)|^2 ≈1, and for high frequenciesf(^2 )→∞so
that|H(j)|^2 →0. Accordingly, there are two important issues to consider:

n Selection of the appropriate functionf(.).
n The factorization needed to getH(s)from the magnitude-squared function.

As an example of the above steps, consider theButterworth low-pass analog filter. The Butterworth
magnitude-squared response of orderNis

|HN(j)|^2 =

1

1 +

[



hp

] 2 N (6.30)

wherehpis the half-power frequency of the filter. We then have that for << hp,|HN(j)|≈1,
and for >> hp, then|HN(j)|→0. To findH(s)we need to factorize Equation (6.30). LettingS
be a normalized variableS=s/hp, the magnitude-squared function (Eq. 6.30) can be expressed in
terms of theSvariable by lettingS/j=/hpto obtain

H(S)H(−S)=

1

1 +(−S^2 )N

since|H(j′)|^2 =H(j′)H∗(j′)=H(j′)H(−j′). As we will see, the poles ofH(S)H(−S)are sym-
metrically clustered in thes-plane with none on thejaxis. The factorization then consists of
assigning poles in the open left-hands-plane toH(S), and the rest toH(−S). We thus obtain

H(S)H(−S)=

1

D(S)

1

D(−S)

so that the final form of the filter is

H(S)=

1

D(S)

whereD(S)has roots on the left-hand s-plane. A final step is the replacement of S by the
unnormalized variables, to obtain the final form of the filter transfer function:

Butterworth low-pass filter: H(s)=H(S)|S=s/hp (6.31)

Filter Specifications
Although an ideal low-pass filter is not realizable (recall the Paley-Wiener condition in Chapter 5)
its magnitude response can be used as prototype for specifying low-pass filters. Thus, the desired
magnitude is specified as

1 −δ 2 ≤|H(j)|≤ 1 0 ≤≤p (passband)

0 ≤|H(j)|≤δ 1 ≥s (stopband) (6.32)