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Complex Sine-Wave Analysis

To illustrate the use of complex numbers in matlab, we repeat the previous sine-wave analysis of the simplest lowpass filter using complex sinusoids instead of real sinusoids.

Only the sine-wave analysis function needs to be rewritten, and it appears in Fig.2.10. The initial change is to replace the line

  s = ampin * cos(2*pi*f(k)*t + phasein); % real sinusoid
with the line
  s = ampin * e .^ (j*2*pi*f(k)*t + phasein); % complex
Another change in Fig.2.10 is that the plotsignals option is omitted, since a complex signal plot requires two real plots. This option is straightforward to restore if desired.

In the complex-sinusoid case, we find that measuring the amplitude and phase of the output signal is greatly facilitated. While we could use the previous method on either the real part or imaginary part of the complex output sinusoid, it is much better to measure its instananeous amplitude and instananeous phase by means of the formulas

\begin{eqnarray*}
A &=& \left\vert A e^{j\theta}\right\vert\\
\theta &=& \angle{A e^{j\theta}}.
\end{eqnarray*}

Furthermore, since we should obtain the same answer for each sample, we can average the results to minimize noise due to round-off error:

  ampout = mean(abs(yss)); % avg instantaneous amplitude
  gains(k) = ampout/ampin; % amplitude response sample
  sss = s(ntransient+1:length(y)); % chop input like output
  phases(k) = mean(mod2pi(angle(yss.*conj(sss))));
The expression angle(yss.*conj(sss)) in the last line above produces a vector of estimated filter phases which are the same for each sample (to within accumulated round-off errors), because the term $ e^{j2\pi f(k) nT+\phi}$ in yss is canceled by the conjugate of that term in conj(sss). We must be certain that the filter phase-shift is well within the interval $ [-\pi ,\pi )$ ; otherwise a call to mod2pi would be necessary for each element of angle(yss.*conj(sss)).

The final measured frequency response is plotted in Fig.2.9. The test conditions were as in Fig.2.7, i.e., the highest test frequency was fmax = $ f_s/2$ , and the number of samples in each test sinusoid was tmax $ = 10$ . Unlike the real sine-wave analysis results in Fig.2.7, there is no visible error associated with complex sine-wave analysis. Because instantaneous amplitude and phase are available from every sample of a complex sinusoid, there is no need for signal interpolation of any kind. The only source of error is now round-off error, and even that can be ``averaged out'' to any desired degree by enlarging the number of samples in the complex sinusoids used to probe the system.

Figure 2.9: Overlay of measured and theoretical frequency responses of the simplest lowpass filter, obtained using complex sine-wave analysis.
\includegraphics[width=\twidth ]{eps/swanalcmain}

This example illustrates some of the advantages of complex sinusoids over real sinusoids. Note also the ease with which complex vector quantities are manipulated in the matlab language.

Figure 2.10: Matlab function for performing complex sine-wave analysis.

 
function [gains,phases] = swanalc(t,f,B,A)
% SWANALC - Perform COMPLEX sine-wave analysis on the 
%           digital filter having transfer function 
%           H(z) = B(z)/A(z)
  
ampin = 1;      % input signal amplitude
phasein = 0;    % input signal phase

N = length(f);       % number of test frequencies
gains = zeros(1,N);  % pre-allocate amp-response array
phases = zeros(1,N); % pre-allocate phase-response array

if length(A)==1, ntransient=length(B)-1, else
  error('Need to set transient response duration here');
end

for k=1:length(f)    % loop over analysis frequencies
  s = ampin*e.^(j*2*pi*f(k)*t+phasein); % test sinusoid
  y = filter(B,A,s); % run it through the filter
  yss = y(ntransient+1:length(y)); % chop off transient
  ampout = mean(abs(yss)); % avg instantaneous amplitude
  gains(k) = ampout/ampin; % amplitude response sample
  sss = s(ntransient+1:length(y)); % align with yss
  phases(k) = mean(mod2pi(angle(yss.*conj(sss))));
end


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``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition).
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA