Spectrum of Pulses: Fast Rise/Fall Times & Discrete Peaks

[amfmrad]

I have a spectrum analyzer and pulse generator. I decided to see what the spectrum of a sequence of pulses was and found some surprises. The pulses have very fast rise and fall times (5 nanoseconds) with a pulse width of 30 nanoseconds. As a result I obtained the classical (Sinx/x)^2 power spectrum when the pulse period was long (>> 100 nanoseconds). However, as I increased the frequency of the pulse generator to up to 10 MHz (100 nanosecond pulse period) I saw discrete lines peaking at the envelope of the (Sinx/x)^2 spectrum. I then decided to model the spectrum and obtained the following equation which I would like to verify;

P(f)=(VT)^2 * [Sin(x)/x]^2 * [1/(1+y^2)] * [Sin(Nz)/Sin(z)]^2

P(f)=Power Spectrum at frequency f
V=Peak pulse voltage
T=Pulse Width (30 nanoseconds)
Tau=Exponential pulse rise time and fall time (5 nanoseconds)
Tp=Pulse Period (Pulse frequency=1/Tp) (40 nanoseconds minimum)
N= The number of pulses in the sequence (not critical but I use a number like 10 or 100)

x=pie*f*T with pie=3.14196...
y=2*pie*f*Tau
z=pie*f*Tp

I rechecked the equation and believe it to be correct. The calculated spectrum looks like what I see on the spectrum analyzer (discrete lines peaking at the single pulse spectrum) due to the last term in the equation [Sin(Nz)/Sin(z)]^2 which becomes unity at f = m/(2NTp) with m=0,1,2,.. However, this function does crazy things for other frequencies, which I can't explain?.

Incidentally, I have details of my calculations as well as graphs and pictures of the calculated and measured spectra but I don't know how to attach it?

Thanks to those interested.

Norman

 

[orthogonal]

If your spectrum is rising above the sinc^2 envelope then you must have some other source of energy corrupting your system. I would guess that you have some irregularity in your pulse width (called Duty Cycle Distortion (DCD)).

From the text of your post it sounds like you are sending a clock pattern but if that is the case then you would not see the sinc^2 envelope but discrete peaks. Could you please clarify? If you randomized the data pattern then you would see the sinc^2 envelope.

An excellent summary of the spectral content of signals can be found here:
http://pdfserv.maxim-ic.com/en/an/AN3455.pdf

 

[amfmrad]

Thanks for information and reference. You are correct that the spectrum never goes above the envelope of the single pulse spectrum (Sinx/x)^2. It all makes better sense now. I was hoping, however, to find a reference to the derivation and unusual property of the function (SinNz/Sinz)^2?

Norman