# Why does rectification of a digital signal double the frequency content?

I'm performing envelope detection in the digital domain. This consists of rectifying the signal (i.e. taking the absolute value) and low pass filtering it to extract the slower varying envelope signal. I've heard that rectification doubles the frequency content in the original signal, therefore this information has to be taken into account when designing the digital low pass filter.

I would like to understand the mechanism in which rectification doubles the frequency content of the original signal.

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Just look at the waveform.

If a sine wave goes through a complete cycle every t, and you full-wave rectify it, there will now be two upward humps in every t. Since each hump is a complete cycle, you've doubled the frequency:

More generally, a perfect full-wave rectification means the response function has even symmetry. This means that, for a sine wave input, the output will consist of only even-ordered harmonics, starting at the second harmonic, which is 2× the fundamental.

http://en.wikipedia.org/wiki/Even_and_odd_functions#Harmonics

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If the input is a single sine wave or a sum of odd-harmonic sinewaves with no DC offset, rectification will yield a signal with only even harmonics and a DC offset. Because the recficiation is non-linear, however, the presence of higher harmonics or a DC offset in the input signal may yield an output signal with odd harmonics. Note that because the rectified signal will likely contain harmonics that are not odd multiples of the second harmonic, one cannot rectify a signal again to redouble the frequency, even if one cancels the DC offset. –  supercat Jul 19 '11 at 15:39
@supercat: True. I edited it. –  endolith Jul 19 '11 at 17:20
The input need not be a sine wave for the output to consist only of even harmonics; any sum of odd-harmonic sinewaves will work. Basically, delaying such a wave by half the fundamental period will invert not just the fundamental, but all odd harmonic components regardless of phase; such inversion is then canceled out by rectification. –  supercat Jul 19 '11 at 18:35
3. Some hand-waving: Most real life rectifiers have exponential characteristics which can be approximated by their Taylor expansion. The first (and the biggest) non-linear term of this expansion is $Ax^2$ and $\sin^2x$ is $1 - \cos 2x \over 2$.