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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.

mild AM

enter image description here

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  • $\begingroup$ You need to do pitch tracking and then use the result of your pitch detector to control a comb filter. $\endgroup$ Feb 11 at 22:09
  • $\begingroup$ @robertbristow-johnson Yes, I've been looking at pitch tracking techniques. Part of my concern is the presence of the amplitude modulation which degrades the quasi-periodicity of the received signal. $\endgroup$
    – rhz
    Feb 11 at 23:03
  • $\begingroup$ @rhz: can you clarify a few things? If you say "periodic" do you mean a sine wave or could it also be ANY periodic wave form (rectangle, triangle, gaussian pulses, etc). Is the frequency of the carrier bounded to a limited range around a center frequency or can it drift all the way done to 0 or up to a jillion Hz. Is the amplitude modulation bounded to a "normal" range $r(t) = (1+a(t))x(t), |a(t) < 0.5|$ or can you also get full carrier suppression? I think this is way too broad "as is" and in order to get useful answers you need to narrow it down to a more reasonable case. $\endgroup$
    – Hilmar
    Feb 12 at 3:36
  • $\begingroup$ @Hilmar -- updated question with more info. $\endgroup$
    – rhz
    Feb 12 at 4:45
  • $\begingroup$ @rhz: that helps. Sorry more questions. Does the repetition rate has an upper and a lower bound? Can the pulses overlap? Is the noise stationary? Are the pulses always the same or is the variability in terms of time domain shape, spectrum and/or duration? This is a tricky problem and the more you can constrain it, the better a solution you can get. It would reallly help if you posted one or a few pictures of "typical" waveforms. $\endgroup$
    – Hilmar
    Feb 12 at 5:31

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Okay, just FYI I've done this with audio signals. I have posted here the sufficient mathematics, but no code.

Here's the comb filter stuff.

Here is how to do precision delay.

Here is how to do pitch detection.

That lays out the math. There are tricks to save computational resources (MIPS). Coding it up isn't too awful bad, but it's a project.

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  • $\begingroup$ I will check this out. Does your proposal explicitly (or implicitly) handle the amplitude modulation effects? $\endgroup$
    – rhz
    Feb 12 at 4:47
  • $\begingroup$ If your modulating waveform $a(t)$ is much slower than $x(t)$, it's not really a problem. The issue is how much is $$r(t) \approx r\big(t+T_x(t)\big)$$ ? $\endgroup$ Feb 12 at 15:43
  • $\begingroup$ I've added plots that illustrate the amplitude modulation effect. $\endgroup$
    – rhz
    Feb 13 at 2:31
  • $\begingroup$ Okay, to the effect the amplitude $a(t)$ changes from one peak of $x(t)$, that difference in amplitude correspond to a normalized autocorrelation deviating from 1. $\endgroup$ Feb 13 at 8:13
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rb-j's suggestion is sensible, but I wonder if another similar problem might be of some help, too. In electronic warfare, there is the problem of pulse repetition interval (PRI) estimation.

I don't think the modulation part is explicitly dealt with in this report, but the report may be a good starting point for exploration.

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