Consider an unknown, quasi-periodic signal $x(t)$ with approximate slowly varying unknown period $T_x(t)$. A noisy, amplitude modulated version of this signal is observed:
$$r(t) = a(t) x(t) + n(t)$$
While $a(t)$ is unknown, its spectral content is concentrated in a band a bit lower than that of $x(t)$. Also, the bandwidth of $a(t)$ is lower than that of $x(t)$. There are instances when $a(t)$ is also quasi-periodic with approximate period $T_a(t) > T_x(t)$.
There are several possible goals:
- tracking real-time variations in $T_x(t)$ the approximate period of $x(t)$.
- recovery of $x(t)$
- recovery of both $x(t)$ and $a(t)$
What are some possible strategies to achieve (one or more of) these goals for either high or low SNR scenarios? Obviously for the case of sinusoidal $x(t)$, all of this corresponds to pretty standard amplitude demodulation from analog communications, but for my scenario, $x(t)$ is never a pure tone.
In response to comments, $x(t)$ is a quasi-periodic train of repeated biological transients which can be described by a number of harmonically related sines and cosines with over a fairly restricted band of frequencies. (It turns out that $x(t)$ is such that the lowest order harmonics are negligible.). The transient pulses are well-separated in time and do not overlap. The "local" period of this waveform is strictly bounded over some interval $[T_x^{\min}, T_x^{\max}]$. The exact shape of the transients is not known a-priori (but the above statements should hold irrespective of shape). The shape should remain roughly constant for many, many repetitions of the transient, but can change from one shape to another and do so rather quickly. However, such a shape change does not imply a sudden change in period.
It is probably reasonable to write $a(t)$ as
$$a(t) = 1 + m(t)\qquad |m(t)|< 0.5$$
$a(t)$ will never be negative, but I'm not sure if $|m(t)|<0.5$ is always true.
Unfortunately, the noise $n(t)$ is non-stationary and non-Gaussian. It overlaps spectrally with the desired signal $x(t)$.
An example of a realization of both the signal and the noise can be found here. In that particular example, the noise in green is atypically severe.
More detailed examples of the amplitude modulation effect are shown below. There are instances where it's relatively mild and other cases where it's far more pronounced. In general, it seems that one cannot assume that the amplitudes of any two consecutive peaks are very similar. However, in both cases there appears to be some repeating structure (quasi-periodicity?) in the amplitude modulation.
