How can I sample a signal at 4 samples per cycle?

Question

I am in Chapter 5 of "Understanding Digital signal processing" 3rd edition by Richard G. Lyons. I am stuck with Section 5.4.

In order to get the bandpass filter centred at $F_s/4 \mathrm{Hz}$, I need to multiply the low pass filter coefficients by a sinusoid of $F_s/4 \mathrm{Hz}$ samples at 4 samples per cycle. I have tried to do this in MATLAB but cannot figure out how to sample at four samples per cycle. Since samplerate was not given, I created a sinusoid: $$ n=0:30;\\ x=\sin(n\pi/2)% $$ Is this correct for $F_s/4$ signal?

How do I sample it at four samples per cycle in MATLAB?

I feel like I missing something very fundamental.

Answer 1

You need to understand that $f=f_s/4$ and "$4$ samples per cycle" are two ways of saying the same thing:

$$\text{ # samples per cycle}=\frac{f_s\text{ samples per second }}{f \text{ cycles per second}}$$

The number of cycles per second is equal to the ratio $f_s/f$, where $f_s$ is the sampling frequency, and $f$ is the signal's frequency.

So you just need a sampled sinusoid with a frequency $f=f_s/4$ (or, equivalently, $4$ samples per cycle). The value of $f_s$ is irrelevant.

$$x[n]=\sin(2\pi f\cdot n/f_s)=\sin(n\pi/2)$$

which is exactly what you have already come up with. Multiply this signal with your low pass impulse response, and you'll get a band pass filter centered at $f_s/4$.