[7.5 IDEAL AND](#page-13-0) PRACTICAL FILTERS

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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. 7.32), for example, allows all components below ω = W rad/s to pass without distortion and suppresses all components above ω = W. Figure 7.33 illustrates ideal highpass and bandpass filter characteristics.

The ideal lowpass filter in Fig. 7.32a has a linear phase of slope −td, which results in a time delay of td seconds for all its input components of 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 td: that is,

$$y(t) = x(t - t_d)$$

The signal x(t) is transmitted by this system without distortion, but with time delay td. For this filter, |H(ω)| = rect(ω/2W) and H(ω) = e−jωtd so that

Figure 7.32 Ideal lowpass filter: (a) frequency response and (b) impulse response.

Figure 7.33 Ideal (a) highpass and (b) bandpass filter frequency responses.

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

$$h(t) = \mathcal{F}^{-1} \left[ \text{rect}\left(\frac{\omega}{2W}\right) e^{-j\omega t_d} \right] = \frac{W}{\pi} \operatorname{sinc}[W(t - t_d)]$$

Recall that h(t) is the system response to impulse input δ(t), which is applied at t = 0. Figure 7.32b 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. 7.33) are also physically unrealizable.

For a physically realizable system, h(t) must be causal; that is,

$$h(t) = 0 \qquad \text{for } t < 0$$

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†

$$\int_{-\infty}^{\infty} \frac{|\ln|H(\omega)|}{1 + \omega^2} d\omega < \infty$$

\n(7.43)

If H(ω) does not satisfy this condition, it is unrealizable. Note that if |H(ω)| = 0 over any finite band, |ln|H(ω)|| = ∞ over that band, and Eq. (7.43) is violated. If, however, H(ω) = 0 at a single frequency (or a set of discrete frequencies), the integral in Eq. (7.43) 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, 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 Eq. (7.43) 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. 7.32 and 7.33) are unrealizable.

The impulse response h(t) in Fig. 7.32 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:h(t), given by

$$h(t) = h(t)u(t)$$

is physically realizable because it is causal (Fig. 7.34). If td is sufficiently large, :h(t) will be a close approximation of h(t), and the resulting filter H :(ω) 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 td. 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 td = ∞ is needed to realize the ideal characteristics. But a glance at Fig. 7.32b shows that a delay td of three or four times ^{π W} will make :h(t) a reasonably close version of h(t − td). For instance, an audio filter is required to handle frequencies of up to 20 kHz (W = 40,000π). In this case, a td of about 10−⁴

$$\int_{-\infty}^{\infty} |H(\omega)|^2 d\omega < \infty$$

Note that the Paley–Wiener criterion is a criterion for the realizability of the amplitude response |H(ω)|.

^† We are assuming that ^|H(ω)^| is square integrable, that is,

Figure 7.34 Approximate realization of an ideal lowpass filter by truncation of its impulse response.

(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 Sec. 7.8.

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.

DR ILL 7.11 The Unrealizable Gaussian Response

Show that a filter with Gaussian frequency response ^H(ω) ^{= e}−αω² is unrealizable. Demonstrate this fact in two ways: first by showing that its impulse response is noncausal, and then by showing that |H(ω)| violates the Paley–Wiener criterion. [Hint: Use pair 22 in Table 7.1.]

THINKING IN THE TIME AND FREQUENCY DOMAINS: A TWO-DIMENSIONAL VIEW OF SIGNALS AND SYSTEMS

Both signals and systems have dual personalities, the time domain and the frequency domain. For a deeper perspective, we should examine and understand both these identities because they offer complementary insights. An exponential signal, for instance, can be specified by its time-domain description such as e−2*^t u*(t) or by its Fourier transform (its frequency-domain description) 1/(jω +2). The time-domain description depicts the waveform of a signal. The frequency-domain description portrays its spectral composition [relative amplitudes of its sinusoidal (or exponential) components and their phases]. For the signal e−2*^t* , for instance, the time-domain description portrays the exponentially decaying signal with a time constant 0.5. The frequency-domain description characterizes it as a lowpass signal, which can be synthesized by sinusoids with amplitudes decaying with frequency roughly as 1/ω.

An LTIC system can also be described or specified in the time domain by its impulse response h(t) or in the frequency domain by its frequency response H(ω). In Sec. 2.6, we studied intuitive insights in the system behavior offered by the impulse response, which consists of characteristic modes of the system. By purely qualitative reasoning, we saw that the system responds well to signals that are similar to the characteristic modes and responds poorly to signals that are very different from those modes. We also saw that the shape of the impulse response h(t) determines the system time constant (speed of response), and pulse dispersion (spreading), which, in turn, determines the rate of pulse transmission.

The frequency response H(ω) specifies the system response to exponential or sinusoidal input of various frequencies. This is precisely the filtering characteristic of the system.

Experienced electrical engineers instinctively think in both domains (time and frequency) whenever possible. When they look at a signal, they consider its waveform, the signal width (duration), and the rate at which the waveform decays. This is basically a time-domain perspective. They also think of the signal in terms of its frequency spectrum, that is, in terms of its sinusoidal components and their relative amplitudes and phases, whether the spectrum is lowpass, bandpass, highpass, and so on. This is a frequency-domain perspective. Experienced electrical engineers think of a system in terms of its impulse response h(t). The width of h(t) indicates the time constant (response time): that is, how quickly the system is capable of responding to an input, and how much dispersion (spreading) it will cause. This is a time-domain perspective. From the frequency-domain perspective, these engineers view a system as a filter, which selectively transmits certain frequency components and suppresses the others [frequency response H(ω)]. Knowing the input signal spectrum and the frequency response of the system, they create a mental image of the output signal spectrum. This concept is precisely expressed by Y(ω) = X(ω)H(ω).

We can analyze LTI systems by time-domain techniques or by frequency-domain techniques. Then why learn both? The reason is that the two domains offer complementary insights into system behavior. Some aspects are easily grasped in one domain; other aspects may be easier to see in the other domain. Both time-domain and frequency-domain methods are as essential for the study of signals and systems as two eyes are essential to a human being for correct visual perception of reality. A person can see with either eye, but for proper perception of three-dimensional reality, both eyes are essential.

It is important to keep the two domains separate, and not to mix the entities in the two domains. If we are using the frequency domain to determine the system response, we must deal with all signals in terms of their spectra (Fourier transforms) and all systems in terms of their frequency responses. For example, to determine the system response y(t) to an input x(t), we must first convert the input signal into its frequency-domain description X(ω). The system description also must be in the frequency domain, that is, the frequency response H(ω). The output signal spectrum Y(ω) = X(ω)H(ω). Thus, the result (output) is also in the frequency domain. To determine the final answer y(t), we must take the inverse transform of Y(ω).