Hilbert-Transform Impulse Response

Let denote the convolution kernel of the continuous-time Hilbert transform from (4.17) above:

Convolving a real signal with this kernel produces the imaginary part of the corresponding analytic signal. The way the ``window method'' for digital filter design is classically done is to simply

Note that we cannot allow a time-domain sample at time **0**
in
(4.22) because it would be infinity. Instead, time **0**
should be taken to lie between two samples, thereby introducing a
small non-integer advance or delay. We'll choose a half-sample delay.
As a result, we'll need to delay the real-part filter by half a sample
as well when we make a complete single-sideband filter.

The matlab below illustrates the design of an FIR Hilbert-transform
filter by the window method using a *Kaiser window*. For a more
practical illustration, the sampling-rate assumed is set to
Hz instead of being normalized to 1 as usual. The
Kaiser-window
parameter is set to
, which normally gives
``pretty good'' audio performance (*cf.* Fig.3.28). From
Fig.3.28, we see that we can expect a stop-band attenuation
better than
dB. The choice of
, in setting the
time-bandwidth product of the Kaiser window, determines both the
stop-band rejection and the transition bandwidths required by our FIR
frequency response.

M = 257; % window length = FIR filter length (Window Method) fs = 22050; % sampling rate assumed (Hz) f1 = 530; % lower pass-band limit = transition bandwidth (Hz) beta = 8; % beta for Kaiser window for decent side-lobe rejectionRecall that, for a rectangular window, our minimum transition bandwidth would be Hz, and for a Hamming window, Hz. In this example, using a Kaiser window with ( ), the main-lobe width is on the order of Hz, so we expect transition bandwidths of this width. The choice above should therefore be sufficient, but not ``tight''.

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