CWatters said:
Building and simulating may not be the best way to learn about this. Google
for articles on the Superhet principle.
Figure out why this equation is important..
sin(x) sin(y) = 0.5 cos(x ? y) ? 0.5 cos(x + y),.
Better yet, go for broke. Pretend that the whole world is made of
nothing but complex numbers. In other words, pretend that complex
numbers are not a special case of regular numbers that we learned in
grade school, but the end-all in general of quantities, and that it is
we, the humans, who have been operating in a mode of deficiency since
the very first time we learned to count.
Then you can assert that all functions are complex, where every part
that make them up is potentially complex. Then, for a wide variety of
functions, it is true that those functions can be represented as sums
of complex exponentials on t:
x(t) =Sum {-infinity, +infinity} (complex coefficient)*e-to-j-omega-t.
It would do you great benefit to take random "grade-school" functional
patterns of t that you make up yourself (sines, cosines, ramps, boxes),
and see if you an represent the functions as a sum of clumps where each
clump is a complex co-factor applied to e raised to j omega t. Keep
figeting with the per-omega clumps to get the signal to look right in
the time domain. This is most likely what Fourier did before he
arrived at his convictions.
If you view the world this way, as if all numbers were complex,
including the number of pieces of fruit that you last bought at the
supermarket, you will feel a lot better about all of this, because
there will be no more special cases, as everything will be complex, and
the vast majority of quantities that we experience each day, the
complex part just happens to be zero.
Then take the two pure sinusoids that you plan to mix, use Euler's
Theorem to treat them as two complex functions as above.
Multiplying them together (heterodyning) will quickly reveal, by
definition of multiplication of *any* two exponential functions (add
the exponents), that the frequencies will add in the resulting signal.
Then if you take a x1 to be sum of two sinusoids, and x2 to also be sum
of two sinusoids, and multiply them, you can see the blobs that they
make in the frequency domain (again by adding).
If you keep adding sinusoids to x1 and x2 so that they become "rich" in
time (and therefore spectral pattern becomes less spike-like), you will
see that the multiplication in time domain results in convolution in
frequency domain.
-Le Chaud Lapin-