RLS adaptive filter intuitive explanation of the so-called desired signal

Question

This wikipedia page https://en.wikipedia.org/wiki/Recursive_least_squares_filter (and in fact other sources) do not explain the apparent paradox of the cost function that computes the MSE of the output and the "desired" (or "ideal") signal.

For someone who tries, like me, to understand this RLS algorithm, it seems a priori absurd to compute a cost function w.r.t. a signal that we do not know, and that we are looking for : this "desired" signal !

I have trained neural nets in a supervised learning manner, i.e. with a "ground truth" so this is the only scenario that I can imagine where we have access to a "desired" signal... but for the adaptive filter, there is no "training" mentioned and as I understand it should continuously adapt its parameters in real time as the input signal (to be filtered) arrives (so training would not really make any sense anyway ?)... So can anyone please explain clearly what this desired signal means ? Especially in the context of filtering a signal corrupted by e.g. noise or other perturbations, for example the wiki example of the ECG corrupted by AC noise. They never explain what this "ideal signal" is... it seems absurd because we DO NOT KNOW the clean ECG signal, this IS what we try to obtain with the filter ... how could we compute any cost function then?

Also, this wiki page should clearly be modified to explain this apparent absurdity. I really don't see how someone who tries to learn about these filters could understand anything about this without this basic explanation.

Answer 1

A good example of an adaptive filter is an Acoustic Echo Canceller as shown in this picture (source)

The problem here is that the sound emitted by the loudspeaker will be picked up from the microphone and needs to be removed to get the "clean" speech signal. Ignoring the noise ($w$ in the picture) the signal at the microphone, $m(t)$ is

$$m(t) = v(t) + x(t)*h(t) \tag{1}$$

where $v(t)$ is the speech signal, $x(t)$ is the input signal to the loudspeaker, $h(t)$ is the room impulse response (RIR), and $*$ the convolution operator. We DO know the input to the loudspeaker but we don't know the RIR and often it's time variant as speaker and/or microphone move around.

We want the adaptive filter to estimate the RIR: we give it $x(t)$ as the input and $m(t)$ as the desired signal. An adaptive filter minimizes the error, $E$, between the input and the desired signal which we can write as:

$$E = <(x(t)*\hat{h}(t) - m(t))^2> \tag{2}$$

The error is clearly minimized if the response, $\hat{h}(t)$, of the adaptive filter is identical to the RIR, i.e. $\hat{h}(t) = h(t)$ and that's what the adaptive filter will try to converge to.

In other words: we use the adaptive filter to estimate the RIR so we can remove as much of the filtered loudspeaker signal as possible. The best estimate of the speech signal is simply the residual error:

$$v(t) \approx m(t) - x(t)*\hat{h}(t) \tag{3}$$

signal that we do not know, and that we are looking for : this "desired" signal !

You misunderstand "desired" here. It doesn't mean "it's the thing we want to calculate". It just means "it's the target for the adaptive filter". In most cases we use adaptive filters to identify impulse responses we don't know using signals we do know, which includes the "desired" signal. You can argue the naming convention, but it's the established name for it.

Also, this wiki page should clearly be modified to explain this apparent absurdity

There is no need for this type of language. I understand you are frustrated, but the page works just fine for many other people. The page has been created and maintained at considerable effort by many contributors that share their knowledge freely and without compensation. Personally, I am very grateful that such a resource exists.

Answer 2

I think you are mistaken on the nature of the desired signal. It would seem quite absurd to design a filter for a signal that we don't know what it is. Often times, though, we do know what the desired signal is, e.g. in channel estimation we attempt to estimate the effects of a channel on a known signal. Or in active noise cancellation, such as the ECG example, we attempt to filter out the noise where we have the corrupted signal and an estimate of the noise. This is why the wiki page says "The adaptive filter would take input both from the patient and from the mains and would thus be able to track the actual frequency of the noise as it fluctuates and subtract the noise from the recording".

There are certain cases where a desired signal is not required, for example, in blind equalization. However, these blind algorithms come with other assumptions so that we can say if the assumptions are met, we've achieved our estimate of the desired signal.