# Phase Offtset Bandlimited Saw Wave

I'm able to make a standard band limited saw wave, but at some point in the past, I remember reading that if you phase offset the sinusoids, you get a different looking wave form that sounds the same.

Trying to implement this by offsetting the phase 270 degrees, I'm seeing a different looking wave form but it doesn't quite sound the same. It has less volume and sounds a little less bright.

Am I remembering correctly / implementing correctly? Here's the saw and "other" wave below:

Thanks!

Have you normalized the waveforms differently? The lowest point of the phase-shifted saw should be at amplitude -2 ln(2) / pi = -0.44 if the amplitude of the saw is normalized to -1 .. 1. The positive peak of the phase-shifted saw approaches infinity as you increase the bandwidth. So you could also have a clipping problem. Nonlinearities of the physical world affect the phase-shifted saw more strongly. The two waveforms have the same power spectrum and "complex envelope", but human hearing is more complex and may be able to detect some difference.

• You are right... I was normalizing each to have a max peak of -3db and thus the overall volume changed. If I leave the volume alone, the 2nd form does indeed clip and that changes the sound quite a bit. Thanks olli! – Alan Wolfe Apr 9 '15 at 14:36

if you have MATLAB, check this out and see if you can hear a difference:

%
%   sawtooth_phase.m
%
%   a test to see if we can really hear phase changes
%   in the harmonics of a Nyquist limited sawtooth wave.
%
%   (c) 2004 rbj@audioimagination.com
%

if ~exist('Fs')
Fs = 44100                      % sample rate, Hz
end

if ~exist('f0')
f0 = 110.25                     % fundamental freq, Hz
end

if ~exist('tone_duration')
tone_duration = 2.0             % seconds
end

if ~exist('change_rate')
change_rate = 1.0               % Hz
end

if ~exist('max_harmonic')
max_harmonic = floor((Fs/2)/f0) - 1
end

if ~exist('amplitude_factor')
amplitude_factor = 0.25         % this just keeps things from clipping
end

if ~exist('outFile')
outFile = 'sawtooth_phase.wav'
end

% make sure we don't uber-Nyquist anything
max_harmonic = min(max_harmonic, floor((Fs/2)/f0)-1);

t = linspace(0.0, tone_duration, Fs*tone_duration+1);

detune = change_rate;

n = 2;                               % continue with 2nd harmonic
while (n <= max_harmonic)
if ((n-1) == 2*floor((n-1)/2))  % lessee if it's an "even" or "odd" term
x = x + (1/n)*sin(2*pi*n*f0*t);
else
x = x - (1/n)*sin(2*pi*(n*f0+detune)*t);
detune = -detune;       % comment this line in an see some
end                         % funky intermediate waveforms
n = n + 1;                      % continue with next harmonic
end

x = amplitude_factor*x;

% x = sin((pi/2)*x);               % toss in a little soft clipping

plot(t, x);                      % see
sound(x, Fs);                    % hear
wavwrite(x, Fs, outFile);        % remember


and try reducing the max_harmonic to something like 10 or 20.