5.7 Convolution and Filtering 333FIGURE 5.8
Ideal filters: (a) low pass, (b) band pass, (c) band
eliminating, and (d) high pass.
|Hlp(jΩ)|Ω 1 Ω
(a)|Hbp(jΩ)|Ω 1 Ω 2 Ω
(b)
|Hbe(jΩ)|Ω 1 Ω 2 Ω
(c)|Hhp(jΩ)|Ω 3 Ω
(d)for all frequencies, since in this case we chose the frequencies 1 and 2 so that the response of these
filters add to unity. An ideal multi-band filter can be obtained as a combination of the low-, band-,
and high-pass filters. Figure 5.8 displays the frequency responses of the ideal filters discussed here.
Remarks
n If hlp(t)is the impulse response of a low-pass filter, applying the modulation property we get that
2 hlp(t)cos( 0 t)(where 0 >> 1 and 1 is the cut-off frequency of the low-pass filter) corresponds to
the impulse response of a band-pass filter centered around 0. Indeed, its Fourier transform is given by
F[2hlp(t)cos( 0 t)]=Hlp(j(− 0 ))+Hlp(j(+ 0 ))which is the frequency response of the low-pass filter shifted to new center frequencies 0 and− 0 ,
making it a band-pass filter.
n A zero-phase ideal low-pass filter Hlp(j)=u(+ 1 )−u(− 1 )has as impulse response a sinc
function with a support from−∞to∞. This ideal low-pass filter is clearly noncausal as its impulse
response is not zero for negative values of time t. To make it causal we could approximate its impulse
response by a function h 1 (t)=hlp(t)w(t)where w(t)=u(t+τ)−u(t−τ)is a rectangular window
where the value ofτis chosen so that outside the window the values of the impulse response hlp(t)are very
close to zero. Although the Fourier transform of h 1 (t)is a very good approximation of the desired frequency
response, the frequency response of h 1 (t)displays ringing around the cut-off frequency 1 because of
the rectangular window. Finally, we delay h 1 (t)byτto get a causal filter with linear phase. That is,
h 1 (t−τ)has as its magnitude response|H 1 (j)|≈|Hlp(j)|and its phase response is∠H 1 (j)=
−τ. Although the above procedure is a valid way to obtain approximate low-pass filters with linear
phase, they are not guaranteed to be rational and would be difficult to implement. Thus, other methods
are used to design filters.
n Since ideal filters are not causal they cannot be used in real-time applications—that is when the input
signal needs to be processed as it comes to the filter. Imposing causality on the filter restricts the frequency
response of the filter in significant ways. According to thePaley-Wiener integral condition, a causal