Signals and Systems - Electrical Engineering

650 C H A P T E R 11: Introduction to the Design of Discrete Filters

Solution

The loss specifications are then

0 ≤α(ejω)≤0.92 0 ≤ω≤π/ 2

α(ejω)≥ 20 3 π/ 4 ≤ω≤π

whereαmax=0.92 dB andαmin=20 dB. These specifications indicate that in the passband the loss

is small, or that the magnitude would change from 1 to

10 −αmax/^20 = 10 −0.92/^20 =0.9

while in the stopband we would like a large attenuation, at leastαmin, or that the magnitude would

have values smaller than

10 −αmin/^20 =0.1 n

Remarks

n The dB scale is an indicator of attenuation: If we have a unit magnitude the corresponding loss is 0 dB,

and for every 20 dB in loss this magnitude is attenuated by 10 −^1 , so that when the loss is 100 dB the unit

magnitude would be attenuated to 10 −^5. The dB scale also has the physiological significance of being a

measure of how humans detect levels of sound.

n Besides the physiological significance, the loss specifications have intuitive appeal. They indicate that

in the passband, where minimal attenuation of the input signal is desired, the “loss” is minimal as it is

constrained to be below a maximum loss ofαmaxdB. Likewise, in the stopband where maximal attenuation

of the input signal is needed, the “loss” is set to be larger thanαmindB.

n When specifying a high-quality filter theαmaxvalue should be small, theαminvalue should be large,

and the transition band should be as narrow as possible—that is, approximating as much as possible the

frequency response of an ideal low-pass filter. The cost of this is a large order for the resulting filter, making

the implementation expensive computationally and requiring large memory space.

Magnitude Normalization

The specifications of the low-pass filter in Figure 11.6 are normalized in magnitude: The dc gain

is assumed to be unity (or the dc loss is 0 dB), but there are many cases where that is not so. See

Figure 11.8.

Not-normalized magnitude specifications: In general, thedcloss is different from0 dB, so that the loss

specifications are

α 1 ≤ˆα(ejω)≤α 2 0 ≤ω≤ωp

α 3 ≤ˆα(ejω) ωst≤ω≤π

Writing the above loss as

α(ˆejω)=α 1 +α(ejω) (11.14)