Once More

Let’s take a signal which everybody agrees is broadband. It’s called the sinc function, and I’ll choose the form

.

Its Fourier transform is a rectangle function, equal to 1 for frequency between -0.5 and +0.5, equal to zero otherwise.

Let’s sample this function at equal time intervals (even sampling), starting at t=50, with a sampling rate of only 0.33, which is well below the limit we need to reconstruct the signal by traditional theory. Then let’s apply up-to-date Fourier methods such as used, say, in astronomy. We can easily model the observed data at the times of observation. Here’s the data as black dots and the model function as red circles:

Clearly the model got it right for the observation times. But that’s not much help — we already know what the value is when we observed it!

So let’s take our model function and use it to estimate the signal at times we did not observe. I’ll plot it over a small enough time span (much less than covered by the data) so we can see the details:

The solid black line shows the true function (which we know because we chose it beforehand). The thin red line shows the reconstructed signal. It ain’t right. Not even close. Of course at the times of observation (shown by black dots) it’s right. But that’s not much of an accomplishment, since we know the values at those times by direct observation.

OK let’s do the same thing, but using uneven time sampling. Same overall time span, same mean sampling rate (much lower than Nyquist), same methodology. Here’s the data and the model at the observation times:

We got it again, but it’s still no big whoop because we already know the signal value at the observed times. Can we reconstruct the signal at times we did not observe? Here’s the true signal in black, the reconstructed signal in red, and observed data as dots:

If you’re having trouble telling the difference between the true and reconstructed signals, that’s because the reconstruction is so good.

This reconstruction — which is dead on — did not require any a priori knowledge that the signal was band-limited. Or what the bandwidth was, or where the band was located. Or sampling the data at a high enough frequency to satisfy customary limits.

If all we knew about this signal was the observed data, then we most certainly could not make blanket statements about its character or its spectrum. We couldn’t say for sure that it’s a sinc function, even though modelling it with a sinc function by nonlinear least squares gives a perfect fit.

Therefore some might object that I haven’t “really” fit a broadband signal. After all, the discrete Fourier transform of the data looks like this:

Which doesn’t much resemble a rectangle function and doesn’t seem to be very “broadband.” So, the argument might go, it’s still not possible to model a broadband signal sampled less often than traditional ideas require.

But the “signal” — the underlying signal from which we sampled data — is broadband. We know because we chose it ahead of time. And the sampling rate was way too low to reconstruct the signal by traditional theory.

If you want to say that the observed data are not a broadband signal, I’ll say fine. After all, I can reconstruct the data exactly — exactly — with a finite number of discrete Fourier components. I can do so in an infinite number of ways. Each of those would give different values at the times we did not observe, but the same values at times we did observe.

That’s just a reflection of the fact that we don’t know what the signal does when we don’t observe it. And we certainly don’t know what it did before we started looking, or what it will do in the future.

Given any finite set of data and I can express it as a finite Fourier series. If you then want to claim that all the nonzero values of the Fourier transform have bandwidth zero, then I’ll retort that you’re just hand-waving because it’s not possible to compute the continuous Fourier transform from a finite amount of data.

What we can do is choose from among the infinite number of possibilities that which is most plausible. If, for instance, I gave you data which followed a sine curve perfectly, then you’d be justified in suggesting that the signal was a sinusoid. I could easily construct different signals, with broadband spectra or multiple discrete Fourier components or some combination, which gave exactly the same data. There’s no way to prove which is the actual signal just using the available data. But if we had to bet on what future observed values would be, I’d go with the sinusoid.

And the truth of the matter is, that uneven time sampling really does enable us to discriminate among aliases without a priori knowledge of where the “band” is. And it really does enable us to reconstruct broadband signals when the mean sampling rate doesn’t measure up to traditional requirements. It doesn’t work all the time, in fact it’s easy to construct examples which will confound even the most state-of-the-art methods. But in the real world? It works.