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I have the following diagram for the adaptive seperation of a narrowband and wideband signal using LMS algorithim,

enter image description here

The way it was explained to us was that "The narrowband signal is correlatied over significant time whereas the broadband signal is not correlated over significant time" and in the delayed line, the ADF can only predict the narrowband.

That doesn't make much sense to me. Why is the narrowband correlated over significant time and the broadband isn't and also what's the point of the delay in this?

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A narrowband signal seems like (almost) periodic as indicated by

$$ x[n] = m[n] \sin( w_0 n) $$ where the message $m[n]$ has such a low bandwidth that the peak amplitude (the envelope) of the carrier sine wave changes very slowly compared to how fast the sine wave oscillates between those +/- envelope limits. This makes its autocorrelation sequence also to be long tailed and almost periodic. And it means that the narrowband signal is highly correlated for long lags (long delays).

The opposite is true for the wideband signal, its correlation sequence ceases very rapidly even for a small amount of lag.

The Wiener filter works by estimating the supplied desired response signal from its filter input signal, whose success depends on the amount of correlation between the desired signal and the filter input.

If there is large correlation between the desired signal and the filter input, then the filter will successfully estimate the desired signal and the error (which is desired signal minus the filter ouyput) signal will be small. On the other hand if there is not enough correlation between the two, then the estimation will be poor.

Based on this operational principle of the Wiener filter, and based on the fact that the narowband and the wideband signals have different correlation characteristics, from the given adaptive filter scheme, you can explain its operation as follows.

First, you see that the filter input signal and the desired response signals are the same (except the delay) and are a mixture of an auto-correlated narrowband component plus an uncorrelated wideband component.

The delay is essentially used to decorrelate the desired response and the filter input. However, by carefuly chosing the amount of delay (the lag), you can insure that the wideband component is fully decorraleted whilst the narrowband component is still corelated enough. If you chose the delay to too long, both components will lose their cross-correlation to the desired response. If you chose the delay too short, you would not be able to suppress the wideband component enough. Some intermediate delay will be sufficiently decorrelating the wideband component while still maintaining the cross-correlation between the narrowband components at the desired signal and the filter input.

When this is the case, the filter will successfully predict the correlated narrowband component in the desired response from the delayed version of it at its input, and it will therefore be subtracted from the supplied desired response signal which yields as the narrowband component as the error signal.

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  • $\begingroup$ "This makes its autocorrelation sequence also to be long tailed and almost periodic. And it means that the narrowband signal is highly correlated for long lags (long delays)." Just about this, the fact that the amplitude/envelope changes slowly, does that mean for a short amount of time nothing really changes and so it wouldn't be correlated and hence for a long delay, there would actual envelope changes that we could correlate to? I'm just a bit confused about the autocorrelation sequence of it? $\endgroup$ – AlfroJang80 Dec 12 '18 at 1:08
  • $\begingroup$ when things change slowly, they will be highly correlated for short lags. When things change fast, even a short lag would yield little correlation. $\endgroup$ – Fat32 Dec 12 '18 at 1:11
  • $\begingroup$ Ah. that makes more sense. So since the narrowband is changing slowly, by us taking a tapped delay sample of it and comparing it to the non-delayed version, we would see that its highly correlated. $\endgroup$ – AlfroJang80 Dec 12 '18 at 1:14
  • $\begingroup$ yes it's still correlated despite the delay. Note that delay tries to decorrelate the filter input and desired response. But it's not enough to decorrelate the narrowband component which is very strongly correlated for even long lags. But the wideband component is fully decorrelated. $\endgroup$ – Fat32 Dec 12 '18 at 1:17
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    $\begingroup$ Ah, so since the narrowband changes so slowly, we can use a long delay and keep it correlated while the wideband is completely uncorrelated, this becomes our desired signal and the ADF will only be able to predict what it's able to correlate between it's two inputs which is broadband+narrowband and correlated narowband and ends up only being able to get narrowband out, which is then subtracted from mixed signal to get undelayed broadband. Ah. that makes soo much more sense. Thank you so much. These filters always seemed like magic to me until now. $\endgroup$ – AlfroJang80 Dec 12 '18 at 1:25

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