# bandwidth limited pulse,what window function is it?

I want to know what kind of shape a multiple cycle sine wave pulse ( or maybe the correct term is burst ) should have to have the narrowest bandwidth possible at given pulse width. In the laser science field, modelocked lasers producing ultrashort pulses are said to make transform limited pulse. Transform limited and bandwidth limited are the same thing: it's a pulse with minimum spectral width possible at the specific length of that pulse. When I was looking at the amplitude envelope shape of these pulses, it reminded me of the window function like it's used for example in fft, and it looked kind of like gaussian (the sort of gaussian that starts and ends with zero):

My question is what amplitude envelope shape should that sinewave multiple cycle pulse have to have minimum spectral bandwidth for its duration? At first I thought, that's easy, it must be gaussian (the "confined gaussian" in wikipedia, zero at edges), but when I thought about it more I am not so sure anymore. Wouldn't for example a pulse with Nuttall shape be narrower spectraly? What about Hanning and Blackman? I know all these look almost the same but hey, the true transform/bandwidth limited pulse can be only one shape, so which is it? https://en.m.wikipedia.org/wiki/Bandwidth-limited_pulse

• What you're asking is, among all signals of a given duration (and no less), which one has smaller bandwidth? The answer is not straightforward because the spectrum will have a main lobe of certain width and sidelobes of certain power. In general you can't minimize both at the same time. You'll find a long list of possibilities here: en.wikipedia.org/wiki/Window_function
– MBaz
Commented Apr 6, 2017 at 1:54
• Yes yes! I think i confused you with with my bad writting,I absolutely dont care about the width of mainlobe or how high are the sidelobes closest to mainlobe,all I want is to know is that when I have short electromagnetic pulse or burst that is like 10 cycles long,what shape should it have to occupy least amount of bandwidth.Ultrashort laser pulses have certain shape when they say they are transform limited,what shape is that? en.m.wikipedia.org/wiki/Bandwidth-limited_pulse Commented Apr 6, 2017 at 6:43
• You're still a bit confused, I think. (1) This theory applies to any signal, including light-frequency EM signals. (2) The spectrum of the "burst" is the convolution of the spectrum of the sine (which is a delta) and the spectrum of the window, so you end up with the spectrum of the window. IOW, the BW of the burst is equal to the BW of the window. (3) There is no such thing as "the window with smallest bandwidth", because in theory they all have infinite BW; you need to decide the kind of sidelobe/mainlobe relationships that make sense for your application.
– MBaz
Commented Apr 6, 2017 at 14:18
• Thank you for clearing it up MBaz,I want window that will put as much power as possible into mainlobe and sidelobes close to mainlobe and as little as possible to the sidelobes far away from mainlobe.Lets say I have pulse that have center freqency 1000hz and duration 4 cycles and I want window that will put as much power into the freqency band from 750hz to 1250hz,what window would that be? Commented Apr 6, 2017 at 15:26
• I don't know for sure, but I think you can't go wrong with the one suggested by @SleuthEye. You can see the better-known windows in the wikipedia post and make your own choice :)
– MBaz
Commented Apr 6, 2017 at 16:36

From the great many types of window functions described on Wikipedia, the familiar rectangular window stand out has having a pretty small bandwidth. It's unfortunately also one with about the worst spectral leakage.

Now for a given maximum allowable spectral leakage level, you might want to have a look at the Dolph–Chebyshev window, which minimize the norm of the side-lobes for a given main lobe width (or equivalently minimizes the main lobe width for a given maximum side-lobe level). Note however that this window has impulse like discontinuities at the edges in the time-domain due to the constant side-lobe level in the frequency-domain. These discontinuities are more significant for larger side-lobe level as can be seen in the following graph:

• I like the approach of using that window and just delete the impulses at the end (or replace with adjacent) which results in a similar passband response (slightly degraded) but adds frequency roll-off for side-lobe levels that are more desirable for many applications (such as decimation where the sideline energy keeps folding in). Commented Apr 6, 2017 at 2:17
• So what shape is the transform limited pulse? Commented Apr 6, 2017 at 6:36
• You misunderstood me,but its my fault.I am not sure if these sidelobes and spectral leakage even exist in electromagmetic pulses,its not like I am cutting a signal into pieces with some window function in such way that it cuts the sine at non zero point causing distortion,these real world electric or photonic pulses always start and end with zero,you cant truncate photon in middle of cycle and cause distortion like a you can digital signal that you are cutting for fft analysis.I think I made big mistakes by talking about the whole window function thing,bandwidth limited pulse is what I meant Commented Apr 6, 2017 at 7:00
• I don't think I mentioned truncating photons. Let's say you have an integer number of cycles, then by shaping the amplitude of the cycles you are effectively multiplying the signal by a window. That shaped pulse would have some of its energy spread over nearby frequency, unlike an abstract infinite duration sinusoidal signal (which can only be approximated) Commented Apr 6, 2017 at 8:26
• yes thats exactly what I meant! Do you know what window type it is that gives narrowest bandwidth to the pulse duration? That Dolph Chebyshev window you mentioned have the sidelobes around the main lobe lowest in amplitude,but the sidelobes doesnt roll off as fast as lets Nutall,Blackman or Hann. en.wikipedia.org/wiki/Bandwidth-limited_pulse Commented Apr 6, 2017 at 9:22

You probably want to know the envelope f(t) of an electric field that gives the best time-bandwidth product. For RMS time and RMS bandwidth the solution is a Gaussian as has been found (rediscovered) by D. Gabor,"Theory of communication",J.Inst.Electr.Eng. London vol. 93 (1946) p. 429. The Gaussian f(t) is never zero. There is however a window function "confined Gaussian window" (CGW) that optimises the RMS time-bandwidth product under the condition of being zero outside a given interval (see http://dx.doi.org/10.1016/j.sigpro.2014.03.033 ).

Regards Daniel