Page 344 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 61 FOURIER ANALYSIS OF DISCRETE-TIME SIGNALS AND SYSTEMS

Since - 1 = el", we can write
h[n] = ( - l)"hLPF[n] eJ""hLPF[n] ( 6.152)
=
Taking the Fourier transform of Eq. (6.152) and using the frequency-shifting property (6.44),
we obtain
H(R) =HLPF(R-~)
which represents the frequency response of a high-pass filter. This is illustrated in Fig. 6-22.

-n -a, 0 R, 7r R -n -7r + -a, 0 n-R, n R
Fig. 6-22 Transformation of a low-pass filter to a high-pass filter.

6.37. Show that if a discrete-time low-pass filter is described by the difference equation

then the discrete-time filter described by

is a high-pass filter.
Taking the Fourier transform of Eq. (6.153), we obtain the frequency response HLpF(R) of
the low-pass filter as
M

If we replace R by (R - a) in Eq. (6.155), then we have

which corresponds to the difference equation