# How do I know if the given signal is discrete?

If I am given an expression of a signal then how do I know if it is a discrete signal or not?

If this signal is given how do I know if it's a discrete or continuous?

$$x(n) = e^{j (n \pi/2 + \pi/8) }$$

Fortunately, one typically does not receive signals from strangers in real life. One knows because one is working on an application.

A discrete signal can be equivalent to a continuous signal so you need a set of assumptions tying the two together, like a Nyquist sampling criteria and then you need to test for equivalence under those assumptions.

A function that only exists at discrete points( like time points) and there is no meaningful notion of intermediate values between those points is discrete. Some financial time series are like that. Arrival times of a Poisson process is another discrete function. If intermediate values on a continuous range are meaningful, a continuous function is likely.

In general you cannot know, but in DSP (and in other fields too) there's a convention that $n$ denotes a natural number, and not a real number. So the given signal $x(n)$ almost surely denotes a discrete-time (-space, etc.) signal. Some authors, but by far not all, use the additional convention that brackets denote sequences, and parentheses denote functions (of continuous variables). So those authors would have written $x[n]=\ldots$, but usually the variable $n$ is sufficient to make clear that we're talking about a sequence defined for integer arguments, and not about a function of a continuous variable.

Two sides could be checked: sampling and quantization, two sides of discretization. Discrete somehow refers to a grouping, a set of things that are distinct. Continuous refer to very close things. This does not seen very precise mathematically, let us be more specific.

A signal $s(t)$ can be discrete/continuous either in the ordinal variable $t$ (a variable that exists outside $s$, like the time, or a space measure), or in the different values $s$ can take across all $t$. To me, a truly discrete signal should be discrete in both aspects; a regularly-spaced signal $s[nT]$, $T>0$ where the values are coded on 8-bits (taking at most 256 different values) is fully discrete. On the other side, the signal $t\to t\sin t$, $t\in \mathbb{R}$ is continuous, and can take any given value in $\mathbb{R}$.

How to draw a line between them is complicated. In many textbook cases, discrete refers only to the ordinal variable. Then, you know than one can take frequency as the ordinal variable, and talk about a discrete spectrum. In that case, I'd consider a data discrete if for any two different ordinal locations $t_k$ and $t_l$ where the signal is defined, the difference $|t_k-t_l|$ is always above some predefined threshold. A similar version can be used on the amplitudes, and looking at closed finite intervals.

In your case, where $n$ is likely to be an integer, the difference $|(n_kπ/2+π/8)- (n_lπ/2+π/8)|$ is always greater than a non-zero positive constant. Now, what happens to the values: if $n$ is an integer, then:

$$x(n) = e^{j (n \pi/2)} e^{j\pi/8) } = (e^{j \pi/2})^n e^{j\pi/8} = j^n e^{j\pi/8}$$

so it can take only four different values (or 2 bits): $0$, $\pm j e^{j\pi/8}$ or $e^{j\pi/8}$, so it is discrete in both aspects.