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I have read that taking the natural logarithm of a signal $x$ in the time domain will introduce a spectral broadening and hence "up-sampling" process before the logarithm is required. I tried to test that by plotting the spectrum (for example: from $-B$ to $B$) for an arbitrary limited band signal from $-B/2$ to $B/2$ and taking the natural logarithm in the time domain but I didn't see any effect of the broadening in the frequency domain, I tried the same method to show the effect of up-sampling and down-sampling (decimation in time) and it was working.

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First note that when you use a logarithmic function you shall avoid negative arguments if it's output should be real valued. Then consider the following relation: $$ y[n] = \ln( 1 + x[n] )$$ where the input $x[n]$ is conditioned such that $ 0 < 1 + x[n] \leq2$ is maintained. Then a Maclaurin series expansion of this expression will be

$$ y[n] = \ln(1 + x[n]) = \sum_{k=1}^{\infty} \frac{(-1)^{k-1}}{k} x^k[n] .$$

For very small $x[n]$ the following approximation can also be taken: $$ y[n] \approx x[n] - x^2[n]/2 $$

Applying DTFT on this last line yields:

$$ Y(e^{j\omega}) \approx X(e^{j\omega}) - \frac{1/2}{2\pi} \left( X(e^{j\omega}) \star X(e^{j\omega}) \right) $$

The spectrum, $Y(e^{j\omega})$, of the logarithm of $1+x[n]$ will include multiple convolutions of $X(e^{j\omega})$ with itself. The relation becomes exact (with added terms which were ignored) as $k$ approaches infinity, showing that the spectrum will be broadened.

Indeed for any nonzero bandwidth input $x[n]$, the exact output $y[n]$ would have infinite bandwidth (or equivalently unavoidable aliasing) and no finite upsampling is enough to properly represent it. However in practice as $|x[n]|<1$ is maintained, its high powers, $x^k$ will tend to go zero quickly and can be ignored compared to some suitable threshold...

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I suppose that you will get more responses on this, but in case you don't, here's my takeaway: You do not want to operate on the time domain signal with a logarithm. It is a horrible distortion of the signal spectrum. The only exception that I know of is when you are trying to get information about the envelope (outline of the peaks) of the signal for the purpose of estimating signal power. Since signal power is often analyzed on a dB scale, it is a shortcut to getting the rms level in dB to take the absolute value, lowpass filter it to get the envelope, and take the log for expressing that in dB. Or square it and filter, then take the log. This approach is sometimes used in automatic gain control (AGC) circuits, where the log is done using a diode (due to the exponential I-V characteristic of a diode or transistor B-E junction). Gain is usually regulated on a log scale, as evidenced by the input range switches on oscilloscopes, volume controls on audio equipment, and RF attenuators. Controlling those is probably the only valuable application of the log of a time domain signal.

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