Discrete Time Periodicity

Mar 2020

We say that a function is periodic if, for some p, f(x) = f(x + p) for all x. This is all pretty familiar. If you've ever taken a geometry class before, you've probably encountered quite a few trig functions who are periodic. Here, for example, is our dear friend sin(x) plotted in continuous time from 0 to 6π.

sin of x from zero to 6π

Both a visual and mathematical examination of this function will reveal that it is periodic. We learn this pretty early along in our math careers and a simple google search will tell you that its period is .

More generally, the condition for this function to be periodic is that there exists a positive p for which sin(wx) = sin(w(x+p)) = sin(wx + wp). Because we know that the underlying sin function here is periodic, this requirement holds if wp = 2πk for some k.

In our continuous time case, we know exactly how to solve this one. Dividing by w, we find the period to be 2πk/w.

What about the discrete time case though? Much to my surprise recently, it actually turns out that sin(wx) is only periodic under rather draconian conditions in discrete time. Consider our signal above, but this time its discretized version.

sin of x from zero to 6π

Things still look fairly periodic, but something isn't quite right. The wave looks like its repeating, but our samples on each hump aren't quite the same. Is our wave still periodic?

Well, lets return to our earlier definition of periodicity. We know that this wave will be periodic with period p if sin(wn) = sin(w(n+p)) = sin(wn + wp) (I've switched to using n here as our variable as we're now in discrete time). Working this out, we arrive at a familiar formula: p = 2πk/w, but in our sin(n) function w = 1!

Something is different, our period is now an irrational number. In continuous time this wasn't a problem because we can represent irrational numbers the same way that we would anything else, but in discrete time where x can only take on integer values there is no way to represent an irrational x value. This means that in discrete time you can only have integer periods.

Another way to think about this is that even if the signal is periodic, if this period never lands on one of our discrete time samples, then it is no longer periodic in discrete time. In our example the sin(wx) is periodic with period 2πk/w in continuous time but because there is no integer k such that2πk is also an integer, in discrete time this function is no longer periodic.

I thought this was quite interesting.