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. In other terms, we want a filter which produces a ``single sideband'' (SSB) output signal in response to a real input signal (which has two sidebands about dc, with one sideband being the ``conjugate flip'' of the other).

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 (see §B.9). In other words, one can ``filter out negative-frequencies'' in 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.

The price we pay for choosing a small filter order is greater attenuation at low frequencies. In general, the analytic signal concept is most useful for narrowband signals which look like sinusoids with slow amplitude and/or frequency modulations, such as the output of a single channel of the phase vocoder. Near DC, they tend to break down since, for example, the delay of the imaginary part relative to the real part approaches infinity as frequency goes to zero.

M = 257;      % window length = FIR filter length (Window Method)
fs = 22050;   % sampling rate (Hz) - low by 1/2 for extra LF octave
T = 1/fs;     % sampling period (sec)
f1 = 530;     % lower passband limit (Hz) - LF not available at M=257
              %   and sideband level sought.  Also, f1/fs = 0.024 
              %   as used in paper by Reilly et al. (cited below)
beta = 8;     % beta for Kaiser window for decent sidelobe rejection
saveplots = 1;% set nonzero to write out figures
doremez = 1;  % set to 0 for faster replay (\eg, to touch up plots)
dopause = 0;  % set to 1 to pause on each plot
fn = fs/2;    % Nyquist limit
f2 = fn - f1; % upper passband limit. See Rabiner and Gold 1975, 
              %   p. 172 for why this constraint

N = 2^(nextpow2(8*fs/f1))   % large fft size (for ``ideal'' IR)
if N<8*M, N = 8*(M-1); end; % for nice displays
k1 = N*f1/fs;           % bandlimit in bins
k1 = round(k1);
if k1<2, k1=2; end;     % cannot have DC or Nyquist response
kn = N/2 + 1;           % bin index (1 based) at "Nyquist limit"
k1, k2 = kn-k1+1        % print k1, k2 = HF transition bandlimit

f1 = k1*fs/N            % actual band limits
f2 = k2*fs/N
clf;                    % clear graphics

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Subsections

• Choice of Kaiser Window
• Windowing an ``Ideal'' Impulse Response
• Hilbert Transformer by the Window Method
• Comparison to Optimal Chebyshev FIR Filter
  • Conclusions
  
  
• Comparison to use of the hilbert function

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Order a Hardcopy of Spectral Audio Signal Processing

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.