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Fourier transform of pulse sequence of varying pulse widths 
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#1
Feb2112, 02:28 PM

P: 40

I'm confused as to what to expect when I take, for example, the fourier transform of a sequence of 16 pulses of varying duty cycles, repeating. That is, after the 16th pulse, the entire sequence repeats.
My confusion is in the interaction of the frequency components of each pulse within the sequence, and the fact that these repeat at some periodic rate (the sequence repeats). I understand that a single pulse results in a sync function response in the frequency domain. And in a periodic pulse train, the period determines the separation between frequency bins, and the pulse width in the the width of the lobes in the Fourier series. What I don't get then is what happens when we have periodicity at the "group" level (every sequence), but not from one pulse to the next within a sequence (although the pulse width varies, but the period does not maybe that's the key). thanks! 


#2
Feb2112, 08:58 PM

P: 72

The periodicity of the 16 pulse sequence will determine the very lowest frequency in the spectrum, I believe. All other waveform artifacts will be expressed as harmonics. I'm not sure how applicable that is.
Apparently, if you consider the variation of the duty cycle as a phase modulation of some 50% duty cycle original signal you can calculate the effect on the fourier transform. The effect is the addition of sidebands around the features of the unmodulated transform. According to some website i found, for modulation by a sinusoidal signal, the amplitudes of the sidebands follow the Bessel functions. And thats where I get scared and turn around and flee. Yeah the fourier transform of a the Heaviside rectangle function is the sync function, but the transform of a periodic square wave is not related to sync, it is just an infinite series of delta spikes starting at the fundamental frequency. I imagine that if you want to visualize it, you could start with a graph of the delta functions,begining at some point and then decaying in amplitude as the frequency coordinate increases, like a comb with teeth that get shorter from left to right... and then in your mind replace every "tooth" with a big blurry mess. Yeah thats how I see the spectrum of your periodic duty cycle function. Can you just stick it into a oscilloscope and take a screen shot of the FFT? 


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