How to represent waveform (sum of sinusoids) in complex notation

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Nathan Kennedy
Nathan Kennedy 2018-1-9
评论： Jan 2018-1-9
Hi,
I have a sum sinusoids to make a waveform, each sinusoid has a different phase attached to it.
clear
f = (20.2 : 0.01 : 21.2)*10^9;
Fs = 5*max(f);
Ts = 1/Fs;
end_t = 0.2*10^(-6);
dt = 0 : Ts : end_t-Ts;
for a = 1:length(f)-1
random_phase = 2*pi*rand(1,1);
%y(a,:) = 2 * sin(2*pi .* f(a) .* dt + random_phase);
end
waveform = sum(abs(y))
plot(dt,waveform)
I need to amplify this waveform, but the amplification being applied is in the form of a vector, it has an increase amplitude (gain) component and also a phase change component.
So the waveform is amplified but also subjected to phase change at instances in time.
To do this, my original waveform must be complex so I can multiply two complex numbers together to get the amplified waveform.
How do I do get my original code in a complex form?

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Jan
Jan 2018-1-9
编辑：Jan 2018-1-9
What about:
y(f,:) = cos(2*pi .* f .* dt + random_phase) + ...
1i * sin(2*pi .* f .* dt + random_phase);
Or equivalently:
y(f,:) = exp(1i * (2*pi .* f .* dt + random_phase));
  2 个评论
Nathan Kennedy
Nathan Kennedy 2018-1-9
编辑：Nathan Kennedy 2018-1-9
I had to edit my original post because I pasted in test code that wasn't working. My original post now has working code.
I tried your idea and it works, but why is the imaginary component (sin) exactly the same as the real part (cos) in your code. Can this be trimmed down?
Jan
Jan 2018-1-9
Trimmed down to what? A complex sin wave can be expressed as
y = A * exp(k * t + a)
to define amplitude, frequency and phase shift. Together with the Euler formula: exp(ix) = cos(x) + i * sin(x) you get the shown code.

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