# Practical Signal Processing by Mark Owen Exercise 2.1. and 2.2

The following exercises don't have answers in Practical Signal Processing by Mark Owen. What is the author looking for here? A mathematical proof?

2.1 A cosine wave of frequency f is sampled at $$t = n\times 1 / f_{s}$$ where $$n$$ is the sample number (an integer) and $$f_{s}$$ is the sample rate. Verify that $$\cos2\pi ft = \cos2\pi(kf_{s} - f)t = \cos2\pi(kf_{s} + f)t$$ for all integers k, positive or negative.

2.2 Verify that $$±(kf_{s} +f)$$ for all integers $$k$$, positive or negative, is precisely the same set of values as $$±(kfs± f)$$ for all integers $$k$$, positive or negative, justifying the conclusions in Section 2.4.

Some background: I'm new here. No, I'm not a student. This book was recommended by Michael Ossmann for SDR tinkerers.

• "Proof" is too strong a word in this case. When asked to "verify" something, I interpret that to mean "show the statement is true by simple mathematical simplification", or "do a few numerical calculations to corroborate the equations are correct". Try to come up with an argument that would convince a reasonable engineer, not an unreasonable rigorous mathematician. – MBaz Apr 26 at 0:48