calculate the normalized angular frequencies

Question

I have two discrete time signals $x_1[n] = 2\sin\left(\frac{2\pi 3.5n}{64}\right)$ and $x_2[n] = \cos\left(\frac{9n}{64}\right)$

I have to calculate the normalized angular frequencies.

What I have are:

normalized angular freq. of $x_1[n]$ is $\frac{2\pi 3.5n}{64}$ with sampling frequency being 64 and $f_0 = 3.5$ and the normalized angular frequency is defined to be $\frac{2\pi F}{Fs}$. $F$ here is equal to $f_o$.

Use the same argument the normalized angular freq. for $x_2[n]$ is $\frac{2\pi 6 }{64}$

EDIT extention question: $X_1[n]$ is periodic and $x_2[n]$ is not periodic

Is this correct?

Answer 1

The normalized angular frequency for $x_1[n]$ is $2\pi3.5/64$. $n$ is the sampling index so would not be included in the angular frequency. The units of normalized angular frequency are radians/sample, so we see for each sampling index $n$ the sinusoid will have indexed $2\pi3.5/64$ radians.

By the same consideration, the angular frequency for $x_2[n]$ is simply $9/64$ radians per sample.

Both $x_1[n]$ and $x_2[n]$ are samples of periodic sinusoidal functions, but we note in that case that it is the sinusoidal functions sampled that are periodic but not necessarily the sampled results themselves. For the case of $x_2[n]$, the resulting samples will never actually repeat again since $\pi$ is not a rational number and the sine and cosine functions repeat every $2\pi$ radians, so strictly speaking $x_2[n]$ is not periodic. Since we do have an integer ratio multiplied by $2\pi$ for the angle in $x_1[n]$, this will have results that will repeat exactly and is itself a periodic function, but that is not the case for $x_2[n]$!