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

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?

Answer 1

[Jan]

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));

[Nathan Kennedy]

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]

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.