IDEAL AND PRACTICAL FILTERS

7.5 IDEAL AND PRACTICAL FILTERS

Ideal filters allow distortionless transmission of a certain band of frequencies and completely suppress the remaining frequencies. The ideal lowpass filter ( Fig. 7.31) , for example, allows all components below ω = W rad/s to pass without distortion and suppresses all components above ω = W. Figure 7.32 illustrates ideal highpass and bandpass filter characteristics.

Figure 7.31: Ideal lowpass filter: (a) frequency response and (b) impulse response.

Figure 7.32: Ideal (a) highpass and (b) bandpass filter frequency responses. The ideal lowpass filter in Fig. 7.31a has a linear phase of slope −t d , which results in a time delay of t d seconds for all its input

components of the frequencies below W rad/s. Therefore, if the input is a signal x(t) bandlimited to W rad/s, the output y(t) is x(t) delayed by t d : that is,

The signal x(t) is transmitted by this system without distortion, but with time delay t d . For this filter |H( ω)| = rect (ω/2W) and ∠H(ω) =

e −j ωt d , so that

The unit impulse response h(t) of this filter is obtained from pair 18 ( Table 7.1) and the time-shifting property

Recall that h(t) is the system response to impulse input δ(t), which is applied at t = 0. Figure 7.31b shows a curious fact: the response h(t) begins even before the input is applied (at t = 0). Clearly, the filter is noncausal and therefore physically unrealizable. Similarly, one can show that other ideal filters (such as the ideal highpass or ideal bandpass filters depicted in Fig. 7.32) are also physically unrealizable.

For a physically realizable system, h(t) must be causal: that is, In the frequency domain, this condition is equivalent to the well-known Paley-Wiener criterion, which states that the necessary and

sufficient condition for the amplitude response |H( ω)| to be realizable is [ † ]

If H( ω) does not satisfy this condition, it is unrealizable. Note that if |H(ω)| = 0 over any finite band, | In |H(ω)‖ = ∞ over that band, and condition (7.61) is violated. If, however, H( ω) = 0 at a single frequency (or a set of discrete frequencies), the integral in Eq. (7.61) may still be finite even though the integrand is infinite at those discrete frequencies. Therefore, for a physically realizable system, H( ω) may be zero at some discrete frequencies. Therefore, for a physically realizable system, H( ω) may be zero at some discrete frequencies, but it cannot be zero over any finite band. In addition, if |H( ω)| decays exponentially (or at a higher rate) with ω, the integral in (7.61) goes to infinity, and |H( ω)| cannot be realized. Clearly, |H(ω)| cannot decay too fast with ω. According to this criterion, ideal filter characteristics ( Figs. 7.31 and 7.32) are unrealizable.

The impulse response h(t) in Fig. 7.31 is not realizable. One practical approach to filter design is to cut off the tail of h(t) for t < 0. The resulting causal impulse response

, given by

will be a close approximation of h(t), and the resulting filter

is physically realizable because it is causal ( Fig. 7.33) . If t d is sufficiently large,

will be a good approximation of an ideal filter. This close realization of the ideal filter is achieved because of the increased value of time delay t d . This observation means that the price of close realization is higher delay in the output; this situation is common in noncausal systems. Of course, theoretically, a delay t d = ∞ is needed to realize the ideal characteristics. But a glance at

Fig. 7.31b shows that a delay t d of three or four times will make ĥ(t) a reasonable close version of h(t −t d ). For instance, an audio filter is required to handle frequencies of up to 20 kHz (W = 40,000 π). In this case, a t d of about 10 −4 (0.1 ms) would be a reasonable choice. The truncation operation [cutting the tail of h(t) to make it causal], however, creates some unsuspected problems. We discuss

these problems and their cure in Section 7.8 .

Figure 7.33: Approximate realization of an ideal low-pass filter by truncation of its impulse response. In practice, we can realize a variety of filter characteristics that approach the ideal. Practical (realizable) filter characteristics are gradual,

without jump discontinuities in amplitude response.