Minimum number of data points needed for a DFT to avoid spectrum leakage?

Question

Let $x(t) = 0.4 + 0.5 \cos (2 \pi f_1 t) + 2 \cos (2\pi f_2 t) + \sin(2 \pi f_2 t) + 1.5 \cos(2\pi f_3 t)$ where $f_1 = 3 kHz, f_2 = 5 kHz, f_3 = 8 kHz$. If $x[n]$ is obtained by sampling $x(t)$ with a sampling frequency of $f_s = 16 kHz$ what is the minimum number of data points needed for a DFT in order to resolve all the frequencies without spectrum leakage?

Can someone point me in the right direction on this?

UPDATE work:

With $f_s = 16000, T_s = \frac{1}{16000}$, we have:

\begin{align*} x(t) &= 0.4 + 0.5 \cos (2 \pi f_1 t) + 2 \cos (2\pi f_2 t) + \sin(2 \pi f_2 t) + 1.5 \cos(2\pi f_3 t) \\ x[n] &= 0.4 + 0.5 \cos (2 \pi \frac{3}{16} n) + 2 \cos (2\pi \frac{5}{16} n) + \sin(2 \pi \frac{5}{16} n) + 1.5 \cos(2\pi \frac{1}{2} n) \\ \end{align*}

The discrete periods would be $\frac{16}{3}, \frac{16}{5}, 2$, right?

Answer 1

What you're actually asked to do is find the periods of the discrete-time sinusoids, and choose a DFT-length, such that all sinusoids occur an integer number of periods inside that window, hence no spectral leakage will occur.

I leave it as an exercise for you to determine the periods of the sampled sinusoids, but the end result of the DFT looks like this:

You can see that all frequencies are clearly resolved.

The problem (mentioned in the comments) of one sinusoid being located at exactly the Nyquist frequency (half the sampling rate) is not relevant for this toy example, even though in practice it is. What it means is that that sinusoid cannot generally be reconstructed uniquely. However, in your DFT you can see that there's a component at that frequency, and that's all they asked for.