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

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?

• Since f3 isn't below half the sampling rate, even an infinite number of samples might not be enough. Trick question perhaps? – hotpaw2 Oct 29 '15 at 7:38
• @MattL. Let's call the whole thing off! :-) – Peter K. Oct 29 '15 at 9:36