Signals and Systems - Electrical Engineering

392 C H A P T E R 6: Application to Control and Communications

FIGURE 6.22

Magnitude specifications for

a low-pass filter.

Ω Ω

1

0

|H(jΩ)|

1 −δ 2

δ 1

Ωp Ωp

α(Ω)

αmax

αmin

Ωs Ωs

for some small valuesδ 1 andδ 2. There is no specification in the transition regionp<  < s. Also

the phase is not specified, although we wish it to be linear at least in the passband. See Figure 6.22.

To simplify the computation of the filter parameters, and to provide a scale that has more resolution

and physiological significance than the specifications given above, the magnitude specifications are

typically expressed in a logarithmic scale. Defining the loss function (in decibels, or dBs) as

α()=−10 log 10 |H(j)|^2

=−20 log 10 |H(j)| dBs (6.33)

an equivalent set of specifications to those in Equation (6.32) is

0 ≤α()≤αmax 0 ≤≤p (passband)

α()≥αmin ≥s (stopband) (6.34)

whereαmax=−20 log 10 ( 1 −δ 2 )andαmin=−20 log 10 (δ 1 ).

In the above specifications, the dc loss was 0 dB corresponding to a normalized dc gain of 1. In

more general cases,α( 0 )6=0 and the loss specifications are given asα( 0 )=α 1 ,α 2 in the passband

andα 3 in the stopband. To normalize these specifications we need to subtractα 1 , so that the loss

specifications are

α( 0 )=α 1 (dc loss)

αmax=α 2 −α 1 (maximum attenuation in passband)

αmin=α 3 −α 1 (minimum attenuation in stopband)

Using{αmax,p,αmin,s}we proceed to design a magnitude-normalized filter, and then useα 1 to

achieve the desired dc gain.

The design problem is then: Given the magnitude specifications in the passband (α( 0 ),αmax, andp)

and in the stopband (αminands) we then

1. Choose the rational approximation method (e.g., Butterworth).


2. Solve for the parameters of the filter to obtain a magnitude-squared function that satisfies the
   given specifications.