I have a problem to solve in my signal processing course. The chapter of this question is "Sampling and aliasing".
Let's call $g_f(t) = \sin(2\pi.f.t) + \sin(2\pi.(f+\Delta_f).t) + \sin(2\pi.(f+2\Delta_f).t) + 2.\sin(2\pi.(f+3\Delta_f).t)$.
The first question was to visualize the frequency representation of $g_{1000}$. The result was coherent : 4 negative Diracs in $f$, $f+\Delta_f$, $f+2\Delta_f$, $f+3\Delta_f$ and 4 positive Diracs in $-f$, $-f-\Delta_f$, $-f-2\Delta_f$, $-f-3\Delta_f$. Then I had to visualize the frequency representaion of $g_{2250}$. This time, the result was not according to theory because of aliasing ($2250 > \frac{f_e}{2}$, with $f_e$ the sempling frequency). Here is the next question :
Find all the $f$ so $g_f(t)$ is equal to a unique sinus of another specific frequency.
Any idea about how to begin that ?
Let $f = -\Delta_f$. We'll have, $$g_f(t) = \sin(2\pi.-\Delta_f.t) + \sin(2\pi.(-\Delta_f+\Delta_f).t) + \sin(2\pi.(-\Delta_f+2\Delta_f).t) + 2.\sin(2\pi.(-\Delta_f+3\Delta_f).t)$$ So, $$g_f(t) = \sin(2\pi.-\Delta_f.t) + \sin(2\pi.\Delta_f.t) + 2.\sin(2\pi.(2\Delta_f).t)$$
If we think in terms of frequency, the first sine has one negative Dirac peak in $-\Delta_f$ and one positive in $\Delta_f$. The second one has one positive in $-\Delta_f$ and one negative in $\Delta_f$. They cancel each other.
Finally it will only result 2 peaks, one positive in $-2\Delta_f$ and one negative in $2\Delta_f$ corresponding to the last sine.
So $g_{-\Delta_f.t}$ is equivalent to a unique sine of frequency $2\Delta_f$.