How can a signal be both periodic and random?

Question

Do any examples of such signals exist where the signal is both periodic and random? Because as I see it, if a signal is periodic then the randomness kinda goes away because you know what the signal will look like after some time.

Answer 1

If you are talking about a given signal as "a deterministic realization of a phenomenon", it can be periodic, but not really random.

However, some physical systems are prone to produce randomness and periodicity, like rotating machines, gears, cyclic engines, that produce signals similar to:

Naturally rotating bodies (stars, planets) also produce random periodic events (sunspots, daily temperatures).

A whole branch of digital processing is devoted to those. In terms of processes (that may generate realizations in terms of signals), then there exists notions that combine stochasticity and periodicity, like:

• random periodic processes,
• cyclostationnarity, or periodically processes.

A stochastic process $x(t)$ with expectation $\operatorname{E}[x(t)]$ and autocorrelation $R_x(t,\tau) = \operatorname{E} \{ x(t + \tau) x^*(t) \},$ is said wide-sense cyclostationary for $T_0> 0$ if

• $\operatorname{E}[x(t)] = \operatorname{E}[x(t+T_0)]$,
• $R_x(t,\tau) = R_x(t+T_0; \tau)$.

Answer 2

An example you run typically across in a text book (Papoulis as an example) is the sine with random phase $$ x(t)=\sin(2\pi f + \phi) $$ where $\phi$ is a random variable, distributed uniformly, over $0$ to $2\pi$.

Any realization will have $\phi$ take on a particular value, but it’s random, just like a 6 on a dice after a throw. You could not predict it would be 6 before the throw or in this case you could not predict $\phi$ before you received the signal.

At the other extreme, you can construct a very “random” looking waveform using something like randn() in Matlab. It will pass several statistical tests for randomness but if you use the same seed, the waveform is perfectly repeatable (deterministic).

Every waveform has a story ( a context ) and the story is how you judge if something is random or not.

Answer 3

Most realistic signals are both random and periodic.

For example, you can modulate a harmonic oscillator with a slow enough random signal that moves its frequency around a $\mu_{f}, \sigma_f$. This looks like:

$$y= \sin \left( \frac{2 \pi \mathcal{N_s}(\mu_f, \sigma_f) n}{Fs} \right )$$

Where $\mathcal{N_s}(\cdot)$, denotes a normally distributed random variable that evolves much slower than once every sample.

For short segments, this signal is approximately periodic but because of the random phase, it never quite is periodic.

In fact, the signal does not even have to have any resemblance to trigonometric functions.

The Electrocardiogram (ECG) for example is approximately periodic for an average human being at rest. However, the heart modulates its function at every beat. Therefore, in the long term, while there is some structure in this signal, its period is random.

And finally, taken to the extreme, the signal does not even have to have any resemblance to "periodicity" and yet still be periodic. These are chaotic signals. These can appear to look like noise, like the output of a laser for example but careful examination of their state space reveals some form of regularity.

...the randomness kinda goes away because you know what the signal will look like after some time.

The key phrase here is "...after some time" and the question is "How much time"?

Consider for example a binary sequence of length 3. It has 8 distinct states ($000, 001, 010, 011, ...$). This means that it will "run out of combinations" very quickly and its patterns would have to start re-appearing. But in reality, the phenomena that give rise to signals do not have to occur in "small finite batches". If you were to start looking for subsequences of length 3 you would find that even they tend to appear in more complex batches of pairs of sequences of threes...And then what about pairs of pairs of subsequences and so on.

Therefore, yes, periodicity, but at what timescale?

Hope this helps.