Meanwhile browsed the two paper and the thesis attached in the other thread. It all starting to make sense, very clever indeed, especially as an antenna filter.

However, I still don't get how to avoid overlapping all the multiple of LO bands down to zero.

My reasoning is like this (thinking first only about one phase/path and one low pass filter):

- a switch is a multiplier of Vin with either 0 or 1, therefore we can think about it as a normal mixer

- a normal mixer fed with f1 at one input and f2 at the other will produce two distinct tones f1+f2 and f1-f2 at output

- we plan to tune our LO such as f1-f2 will be zero

- the other tone, f1+f2, will be cut out anyway by the low pass filter following after multiplier

- since a square wave is not a pure tone, but a bunch of many harmonics, we can take the spectrum of the 0/1 waveform tone by tone and multiply each frequency with Vin

- since any 2 distinct frequencies are orthogonal to each other (doesn't "influence" or mix to each other in a linear circuit) we can just apply them all at once

- the spectrum of a pulsed square waveform is an infinite series of multiples of the base frequency, the amplitudes for each harmonic varies, but the overall envelope is a sin(x)/x with the same period as the width of one pulses

The spectrum for a square wave signal with duty cycle 50%, 25%, 12.5% and 6.125% (that will correspond to 2, 4, 8 and 16 phases/paths) will look like in the attached animated gif (made with real signals from the soundcard).

The first plot is the signal waveform at 4 different duty factors (from 50% to 6.125%), and the following two plots are the corresponding spectrum for these duty factors. Both spectrum plots are the same, just that one is with the spectral lines in Volts, and the other with the spectral lines in dB.

We see that the case for n=2 (50%) is the only one that has the most spaced apart harmonics, because it has only odd harmonics. All other duty factors will have both odd and even harmonics, and the bigger the n, the more equal in amplitude these harmonics will be. We note that all the harmonics are equally space at an f

_{LO} distance, and comparable in power (except for the missing one at each n

^{th} of LO multiple, where n is the number of paths/phases).

This is a concern because our multiplier will shift to zero not only the frequency of interest, LO, but any other multiple of LO frequency will be shifted to zero. Once we do that, there is no way to distinguish what signal came from which multiple of the LO frequency. So, this is a rather a comb pass any-LO-multiple-shifted-to-zero type of filter.

In other words, once we see a signal at the output, we have no clue from which frequency that signal came. It can be from any multiple of the LO. I am not sure yet how to get rid of all the other teeth of the comb and keep only the first one.

Is this type of filter used only where it doesn't matter from what multiple of the f

_{LO} our signal of interest is coming?