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# Bandwidth required for single gaussian shaped pulse

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#### erezbenari

##### Newbie level 3
Hi all,

I need to determine the bandwith required to transmit 1 nanosecond pulse-width gaussian-like single pulse.

what BW is required and how is that?

Thanks

The fourier transform of a gaussian pulse has a gaussian shape as well. If I remember right (please check in a text book), the half bandwidth will be 1/(2*pi*0.5 ns) = 317 MHz. To reproduce the waveform exactly, the bandwidth must be infinite. Practically, you can calculate the required bandwidth for a maximum pulse shape deviation. Limiting the bandwidth will change the gaussian shape towards a "ringing" sin(x)/x waveform.

P.S.: Guessed right, see Gaussian function - Wikipedia, the free encyclopedia
Taking the Fourier transform of a Gaussian function with parameters a, b = 0 and c yields another Gaussian function, with parameters ac, b = 0 and 1/c. So in particular the Gaussian functions with b = 0 and c = 1 are kept fixed by the Fourier transform (they are eigenfunctions of the Fourier transform with eigenvalue 1).

tony_lth

### tony_lth

Points: 2
Thanks man

In radar engineering, the simple calculation is BW=1/pulse width, so BW=1/1ns=1000MHz. Sounds a little bigger.
But weather radar always use pulse in us level, so the BW in several MHz.

nabla101

### nabla101

Points: 2
Hi,

can you tell me please why is that?
what is the frquency of a single pulse? how do I calculate it and how do I determine the BW in a rather more
"scientific" manner.

Thanks

For fixed freq CW with rectangular pulse modulation, the multiplication of BW and pulse width is 1.
The following table is from Radar Engineering book:
Pulse shape Match Filter PassBand Opt BW*PulseWidth MismatchLoss(dB)
Rectangular Rectangular 1.37 0.85
Rectangular Gaussian 0.72 0.49
Gaussian Rectangualr 0.72 0.49
Gaussian Gaussian 0.44 0
So if you use Gaussian match filter for receiver, the BW=0.44/1ns=440MHz.

first of all - thanks!

My application is probing a 1 ns generator with 200v output.
Assuming I'll use appropriate coupling device, the generator will feed the antenna.

Is there a mathematical proof or method I can derive the appropriate and necessary BW?
it has to be presented to a prof. in our physics dep. so I guess he would not like thumb-rules
and tables that I can not explain how values were derived.

Thnaks!!!

As I previously mentioned, to reproduce a gaussian pulse exactly, you need infinite bandwidth. So far the "scientific" solution, you are asking for.

For a more practical answer, I suggest to analyze the waveform distortions a gaussian pulse undergoes by applying different bandwidth limitations. Please consider, that the frequency filter shape (both magnitude and phase response) also matters for the resulting waveform.

The numbers presented by tony_lth give the result in terms of signal loss. In my opinion, using a gaussian filter shape can be an elegant trick for your presentation, because it keeps the waveform in time domain and the result can be easily obtained.

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