# Signal's Fundamental Frequency is different from Plotted Signal

I've been attempting to plot the following function using MATLAB: $$x(k)=\sum_{n=11}^{50} \sqrt{n} \sin (2n\pi k) +\sum_{n=1}^{40}\sqrt[3]{n} \sin (3n\pi k)$$

Note that $k$ is a continuous variable.

The problem I am facing with MATLAB is when I plot it, I get a signal which has a fundamental period of 2 seconds, and not 1.5 seconds.

Have a look below:

I've tried a higher sampling rate:

So, what is going on wrong? Why isn't the fundamental period 1.5Hz?

• Assuming your horizontal axis is in seconds, you have 3 full cycles in 2 seconds, which gives a fundamental period of 3/2 Hz = 1.5 Hz, as you were expecting. Mar 12 '16 at 23:34
• @MattL. Thanks, Matt, but $x(k)$ is defined as a continuous periodic function. You could replace k by t. Mar 13 '16 at 6:37
• @ Jazzmaniac Thanks for pointing that out, but... how do you identify the three cycles visually? I can't see them, or see a form of $x(n+N)$ which implies periodicity. Mar 13 '16 at 6:38

With $k$ a continuous variable, the frequencies of the sinusoids in the first sum are $f_n=n$, i.e. they are integers. The frequencies in the second sum are integer multiples of $\frac32$. The highest fundamental frequency of which all those frequencies are integer multiples is $f_0=\frac12$. The frequencies in the first sum are given by $2mf_0$ (with integer $m$), and the frequencies in the second sum are $3mf_0$. With a fundamental frequency of $f_0=\frac12$, the fundamental period is $T=2$, as observed by you.