Example 4: Phase Unwrapping, Forced Odd Phase Symmetry

To solve the problem with Example 3, we can simply unwrap the positive-frequency portion of the desired frequency response, and construct the negative-frequency frequency response as the conjugate-flip of the positive-frequency response.

The result of this approach is shown in Fig.B.25d on page , and the matlab code is given in Fig.B.29. To illustrate separate processing of phase and magnitude data, this example works in the linear domain instead of the log domain, but either method works just as well.

As a fine point, note that the phase-response sample at exactly half the sampling rate must be zeroed in order that the phase response be an odd sequence. Forgetting to do this will result in a prominent imaginary alternating sequence in the final impulse response. Zeroing the imaginary part at is often needed when performing nonlinear spectral modifications.

% Example 4: Work on magnitude and phase separately,
%           and enforce spectral conjugate symmetry:

No2 = N/2;
Nspec = No2+1;   % Index of fs/2 in all spectrum arrays
Sh = S(1:Nspec); % Concentrate on (+) frequencies first
Sm = abs(Sh);
Sp = angle(Sh);
Spu = unwrap(Sp);
if Spu(1) ~= 0, error('DC phase should be zero'); end

% Apply spectral transformation separately 
% to magnitude and phase:
Stm = Sm .^ power;
Stpu = Spu * power;
Stu = Stm .* exp(j*Stpu); % Reassemble complex spectrum

% Append negative-frequency spectrum:
Stu = [Stu,conj(Stu(No2:-1:2))]; % append conjugate flip
Stu(No2+1)=real(Stu(No2+1)); % enforce odd imag part
Stpu = angle(Stu);

plottitle = sprintf('angle(abs(S)^{%0.1f}',...
	'*exp(-j*%0.1f*unwrap(angle(S))), ',...
	'oddified',power,power);
phasesubplot(Stpu,plottitle,4,4);

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