Dot product of a sinusoid with a complex tone in Octave

Question

I am trying to figure out how to solve this problem;

Now make a new sinusoid with amplitude 1 and frequency 1000Hz. Calculate the dot product of this sinusoid with your complex tone using vector multiplication between the heterodyne and the transposed complex tone signal.

So far no good. I have managed to represent a complex tone by:

octave:65> t = [0 : 441];
octave:68> A = 0.25;
octave:69> x = A*sin(2*pi*t*441) + A*sin(2*pi*t*882) + A*sin(2*pi*t*1323) + A*sin(1764*pi*t*2);

And the new sinusoid

 octave:66> y = sin(pi*2*t*1000);

Now I find myself trapped with the heterodyne concept and its application. I can tell that the vector multiplication would be like;

octave:75> abs(y*x')

However, parameter 'y' which needs to be the the heterodyne of 'y'(I guess), is driving me nuts...

Any suggestion would be appreciated

Answer 1

I don't fully understand the assignment's questions/tasks, so hopefully the below general information will be helpful in some way.

Heterodyning refers to mixing a signal with a tone in order to shift the signal's spectrum. Usually (but not always) heterodyning is used to baseband/carry a signal to a lower/higher frequency for efficient transmission. For instance, some frequencies will propagate through the atmosphere with little attenuation while others will not. If we want to transmit a signal $x(t)$ from point A to point B, then we want to carry $x(t)$ on a frequency that will not attenuate drastically during propagation so that we can use less power. Suppose that $f_0$ is such a frequency. Then we can shift the spectrum of $x(t)$ to be centered at $f_0$ by mixing $x(t)$ with a complex sinusoid at that frequency:

$$ x_0(t) = x(t) \circ e^{j2\pi f_0 t} $$

You can see the shift affected by the above by looking at the Fourier transform:

$$ X_0(f) = X(f) \ast \delta( f - f_0 ) = X(f - f_0) $$

To make a complex tone in Matlab/Octave, use the exp function and $\sqrt{-1}$, which is 1j or 1i:

fs = 5e3;                         % Sampling frequency (Hz)
f0 = 1e3;                         % Tone frequency (Hz)
T  = 10;                          % Tone duration (s)
t  = 0 : 1/fs : T;                % Sample times (s)
y  = exp( 1j * 2*pi * f0 * t );   % Complex tone.