# relation between bandwidth and pulse width

Discussion in 'Homework Help' started by screen1988, May 20, 2013.

1. ### screen1988 Thread Starter Member

Mar 7, 2013
310
4
Here is a definition of bandwidth that I know:

Bandwidth is defined as a band containing all frequencies
between upper cut-off and lower cut-off frequencies."

But I also saw the formula:
$B=\frac{1}{\tau}$
This formula seems not fit with the definition above.
My question is:
Why bandwidth is related to pulse width that is B=1/τ where τ is the pulsewidth?

2. ### WBahn Moderator

Mar 31, 2012
21,857
6,347
How does this not fit with the other definition?

3. ### screen1988 Thread Starter Member

Mar 7, 2013
310
4
For example, I have a rectangular pulse train that has pulse width τ.

As in the picture, the pulse has a infinite number of frequency and I can't find its maximum and minimum frequencies.
I don't know why the formula above is found like that.

File size:
47.4 KB
Views:
1,172
4. ### WBahn Moderator

Mar 31, 2012
21,857
6,347
It has no "minimum" frequency because it has a DC component.

It has no maximum frequency because it has step discontinuities and so you need an infinite number of harmonics to converge on the waveform.

Also, you first said a pulse and now you are talking about a pulse train. Two different critters.

screen1988 likes this.
5. ### screen1988 Thread Starter Member

Mar 7, 2013
310
4
Do you mean that the formula $B=\frac{1}{\tau}$ is applied for a pulse?
I mixed up between pulse and pulse train. Their fourier transforms are different.
I have just found this picture and the formula above seem right if we ignore all frequencies higher than 1/T.

File size:
12.7 KB
Views:
1,138
6. ### WBahn Moderator

Mar 31, 2012
21,857
6,347
Yep, by convention, the bandwidth of a single rectangular pulse is taken to be the point of the first NULL in the spectrum. This may or may not be the best definition for a particular application, But it is convenient and the bulk of the information and energy is within that bandwidth.

screen1988 likes this.
7. ### t_n_k AAC Fanatic!

Mar 6, 2009
5,448
787
It might be of interest to screen1988 to investigate what would result if only that part of the spectrum up to and including the first spectral null point at $\frac{1}{\tau}$ was used to reconstruct the (reduced bandwidth) pulse signal.

I've tried this starting with the FFT on a 1024 sample simulated pulse with only the first 16 samples at high state [16 samples on then 1008 samples off]. I have then taken the inverse FFT with values in the rejected portion of the original FFT spectrum beyond the first null point set to zero. Since the conventional FFT produces a mirrored spectrum about the Nyquist frequency one must set rejected values to zero symmetrically about the Nyquist frequency.

It's an interesting outcome and demonstrates quite clearly why an infinite spectrum is required to accurately represent the time domain pulse.

screen1988 likes this.