B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

96 ANALYSIS AND TRANSMISSION OF SIGNALS
Figure 3.27
Ideal high-pass
and bandpass
filter frequency
responses.
IHCn l
0 · - . .
B
f-+
(a)
/0/f)
. )I
IHUl l
0
J-+
(b)
The unit impulse response h(t) of this filter is found from pair 18 in Table 3.1 and the timeshifting
property:
h(t) = y - 1 [ TI ({B) e -j 2 n:ftd ]
= 2B sine [2rr B(t - td) I (3.59b)
Recall that h(t) is the system response to impulse input 8 (t ), which is applied at t = 0. Figure
3.26b 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 unrealizable; that is, such a system is physically
impossible, since no sensible system can respond to an input before it is applied to the system.
Similarly, one can show that other ideal filters (such as the ideal high-pass or the ideal bandpass
filters shown in Fig. 3.27) are also physically unrealizable.
For a physically realizable system, h(t) must be causal; that is,
h(t) = 0 for t < 0
In the frequency domain, this condition is equivalent to the Paley-Wiener criterion, which
states that the necessary and sufficient condition for IH (f ) I to be the amplitude response of a
realizable ( or causal) system is*
00
1 !In IH (fll I
--- 2
df < oo
-oo 1 + (2rr.f)
(3.60)
If H (f) does not satisfy this condition, it is unrealizable. Note that if IH (f ) I = 0 over any
finite band, I lnlH (f) 11 = oo over that band, and the condition (3.60) is violated. If, however,
H(f) = 0 at a single frequency (or a set of discrete frequencies), the integral in Eq. (3.60)
may still be finite even though the integrand is infinite. Therefore, for a physically realizable
system, H (f) may be zero at some discrete frequencies, but it cannot be zero over any finite
band. According to this criterion, ideal filter characteristics (Figs. 3.26 and 3.27) are clearly
unrealizable.
* IH (f) I is assumed to be square integrable. That is,
1-: IH(f) l 2 elf
is assumed to be finite.