Signals and Systems - Electrical Engineering

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

Then the dc loss is 10 dB andαmax=1 andαmin=40 dB. Suppose that we design a filterH(z)that

satisfies the normalized filter specifications. If we letHˆ(z)=KH(z)be the filter that satisfies the

given loss specifications, at the dc frequency we must have that

−20 log 10 |Hˆ(ej^0 )|=−20 log 10 K−20 log 10 |H(ej^0 )|

10 =−20 log 10 K+ 0

so thatK= 10 −0.5= 1 /

√

10. n

Frequency Scales

Given that discrete filters can be used to process continuous-time as well as discrete-time signals,

there are different equivalent ways in which the frequency of a discrete filter can be expressed (see

Figure 11.9).

In the discrete processing of continuous-time signals the sampling frequency (fsin hertz orsin rad/sec) is

known, and so we have the following possible scales:

n Thef(Hz)scale from 0 tofs/ 2 , the foldover or Nyquist frequency, that comes from the sampling theory.

n The scale= 2 πf(rad/sec)wherefis the previous scale, the frequency range is then from 0 tos/ 2.

n The discrete frequency scaleω=Ts(rad)ranging from 0 toπ.

n A normalized discrete-frequency scaleω/π(no units) ranging from 0 to 1.

If the specifications are in the discrete domain, the scale is theω(rad)or the normalizedω/π.

RemarksOther scales are possible, but less used. One of these consists in dividing by the sampling frequency

either in hertz or in rad/sec: The f/fs(no units) scale goes from 0 to 1 / 2 , and so does the/s(no units)

scale. It is clear that when the specifications are given in any scale, it can be easily transformed into any

other desired scale. If the filter is designed for use in the discrete domain only the scales in radians and the

normalizedω/πare meaningful.

11.3.2 Time-Domain Specifications

Time-domain specifications consist in giving a desired impulse responsehd[n]. For instance, when

designing a low-pass filter with cut-off frequencyωcand linear phaseφ(ω)=−Nω, the desired

FIGURE 11.9

Frequency scales used in discrete filter design.

0 ω(rad)

0 Ω(rad /sec)

0 f(Hz)

π

0.5 Ωs

0.5 fs

0 1

ωp

Ωp

fp

ωp

π

ωst

Ωst

fst

π

ωst ω

π