Hilbert Transform Design Example

We will now use the window method to design a *complex*
bandpass filter which passes positive frequencies and rejects
negative frequencies.

Since every real signal
possesses a *Hermitian spectrum*
, *i.e.*,
, it follows that, if
we filter out the negative frequencies, we will destroy this spectral
symmetry, and the output signal will be complex for every nonzero real
input signal (excluding dc and half the sampling rate). In other
terms, we want a filter which produces a ``single sideband'' (SSB)
output signal in response to any real input signal. The Hermitian
spectrum of a real signal can be viewed as *two* sidebands about
dc (with one sideband being the ``conjugate flip'' of the other). See
§2.3.3 for a review of Fourier symmetry-properties for
real signals.

An ``*analytic signal*'' in signal processing is defined as any
signal
having only positive or only negative frequencies, but
not both (typically only positive frequencies). In principle, the
imaginary part of an analytic signal is computed from its real part by
the *Hilbert transform* (defined and discussed below). In other
words, one can ``filter out'' the negative-frequency components of a
signal
by taking its Hilbert transform
and forming the analytic signal
. Thus, an
alternative problem specification is to ask for a (real) filter which
approximates the Hilbert transform as closely as possible for a given
filter order.

- Primer on Hilbert Transform Theory

- Preparing the Desired Impulse Response

- More General FIR Filter Design
- Comparison to Optimal Chebyshev FIR Filter
- Conclusions

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