Discussion
Discrete time sinusoidal signals are identical when their frequencies are separated by the integer multiple of ______.
Answer: A
A discrete time sinusoid \(x(n) = \cos(\omega n)\) satisfies
\[ \begin{aligned} \cos((\omega + 2\pi k)n) \\ &= \cos(\omega n + 2\pi k n) = \cos(\omega n) \end{aligned} \]
because \(kn\) is an integer. So two sinusoids whose frequencies differ by an integer multiple of \(2\pi\) are the same signal, which is why the frequency of a discrete time signal is only meaningful in an interval of length \(2\pi\). A difference of \(\pi\), \(\pi/2\) or \(\pi/3\) gives different sequences (for example, a difference of \(\pi\) changes the sign of every other sample). The answer is \(2\pi\), option A.
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