# periodicity of constant discrete time signals

are constant discrete time signals periodic?
example $$$$e^{i10\pi n}$$$$ my proffesor says that this signal is aperiodic, in the discrete sense. but it seems wrong, because unlike in the continuous case, i can calculate the smallest time period , which is 1.

• Welcome to SE.SP! As you say, $e^{\imath 10 \pi n}$ at integer $n$ is a constant. Are constants periodic? See this question and answer on SE.math. – Peter K. Jan 31 at 19:28
• @PeterK. Since this is a discrete signal, I'd say it is periodic with period 1, as abhishek suspects. fourier.eng.hmc.edu/e101/lectures/Fundamental_Frequency/… – MBaz Jan 31 at 19:47
• @MBaz Yes, it looks like you're correct, it just doesn't have a fundamental or minimal period. – Peter K. Jan 31 at 20:00
• @PeterK. the document you gave a reference to says that a constant continuous time signal has no fundamental or minimal period. The document says nothing about a constant discrete time signal. – abhishek Feb 1 at 16:15
• @abhishek Surely that’s true in discrete time too? If the fundamental period must be greater than zero? – Peter K. Feb 1 at 20:56

normally, "$$n$$" is the symbol we use here for discrete-time. if your professor said that:
\begin{align} x[n] &= e^{i10 \pi n} \\ &= e^{i 2 \pi (5n)} \\ \end{align}
is not periodic with a period of $$1$$ (assuming $$n \in \mathbb{Z}$$) or a period of $$\frac15$$ (assuming $$n \in \mathbb{R}$$), then your professor is mistaken.
• More generally, if the frequency is a rational multiple of $pi$, then the sequence is periodic. $10pi$ is a rational multiple of $pi$. – Juancho Feb 1 at 14:46