What Does Phase of a Gaussian Pulse Mean in Wave Propagation?

[physicsst]

What is meant by phase of a pulse? (Gaussian or sech or any other pulse)
How can we mathemaitcally express it and what is the physical meaning of it?
I know about the phase of a wave but not sure about that of a pulse since it is a combination of waveforms of different frequencies.
Thanks

 

[PhilDSP]

Hi physicsst,

I'm not sure in what context what you refer to as "phase of a pulse" applies. There is the idea of an ideal pulse signal which has infinitely short duration coming from the source. As the signal traverses some dispersive media the pulse's duration will be lengthened and modified. One can then subtract the periodic components in the modified pulse and calculate what amounts to a filter that matches the media's characteristics.

A Fourier transform allows you to determine the periodic components in the pulse giving the amplitude and phase of each frequency component. But the phase is calculated from the beginning of the first data segment received. An interesting fact about Gaussian wave packets (Gaussian pulses) is that the Fourier transform is also Gaussian. I suppose certain pulse shapes can be identified by looking at a phase versus frequency plot that you would get from a FT. The change of the phase rather than its absolute value is what would be most telling.

 

[sophiecentaur]

I think the word 'time' rather than 'phase' would be more appropriate for a pulse. If you analyse a pulse into the frequency domain then the phase that you would need to introduce into all its components would need to vary over the bandwidth occupied (phase slope) if you wanted to delay the pulse in its original shape. This is because you would need to maintain the relative timings of the (continuous) sinewave components to be all the same to maintain the shape of a delayed pulse. So the phase, Φ(f) would need to be ft where f is the frequency and t is the required time delay.

 

[physicsst]

Hi physicsst,

I'm not sure in what context what you refer to as "phase of a pulse" applies. There is the idea of an ideal pulse signal which has infinitely short duration coming from the source. As the signal traverses some dispersive media the pulse's duration will be lengthened and modified. One can then subtract the periodic components in the modified pulse and calculate what amounts to a filter that matches the media's characteristics.

A Fourier transform allows you to determine the periodic components in the pulse giving the amplitude and phase of each frequency component. But the phase is calculated from the beginning of the first data segment received. An interesting fact about Gaussian wave packets (Gaussian pulses) is that the Fourier transform is also Gaussian. I suppose certain pulse shapes can be identified by looking at a phase versus frequency plot that you would get from a FT. The change of the phase rather than its absolute value is what would be most telling.

I am studying the propagation of pulse through a dispersive medium.And as you mentioned it is the change in phase that is important here.But I am confused about this phase thing.We are considering an initially chirped pulse so that there is a frequency variation.Since I've just started studying this,things are not very clear to me and that would have reflected in my question,sorry for that.What i,m thinking is,in the maxwells eqn of propgation if we take the envelope function as gaussian,what would be the phase?We have a functional form for the change in phase but we have to solve for the absolute value of phase.(sorry if that sounds stupid)anyway,thank you for your reply.

 

[sophiecentaur]

I am studying the propagation of pulse through a dispersive medium.And as you mentioned it is the change in phase that is important here.But I am confused about this phase thing.We are considering an initially chirped pulse so that there is a frequency variation.Since I've just started studying this,things are not very clear to me and that would have reflected in my question,sorry for that.What i,m thinking is,in the maxwells eqn of propgation if we take the envelope function as gaussian,what would be the phase?We have a functional form for the change in phase but we have to solve for the absolute value of phase.(sorry if that sounds stupid)anyway,thank you for your reply.

Where do you usually see a symbol for phase used? Is it not in the context of continuous waves? Why do you want to relate to to a pulse that is not continuous? If you stick to the implied definition of phase then you will have no problem. Don't try to apply the word where it is not consistent. The phases of the sinusoidal components of a pulse will all be changing differently so there is no 'phase' value which can be assigned. Use 'time' for a function of time.

 

[PhilDSP]

I am studying the propagation of pulse through a dispersive medium.And as you mentioned it is the change in phase that is important here.But I am confused about this phase thing.We are considering an initially chirped pulse so that there is a frequency variation.Since I've just started studying this,things are not very clear to me and that would have reflected in my question,sorry for that.What i,m thinking is,in the maxwells eqn of propgation if we take the envelope function as gaussian,what would be the phase?We have a functional form for the change in phase but we have to solve for the absolute value of phase.(sorry if that sounds stupid)anyway,thank you for your reply.

It sounds like you need to normalize the phase between the initial raw signal and the signal going through the medium. You can do that with the experimental results by lopping off all data prior to the initial rise in the slope of the pulse in both signals and then taking the FT or DFT of both. That is effecting shifting the time values to normalize the phase. Alternatively if you're comparing only Gaussian pulses you could normalize by shifting the time values so that the center of the Gaussian envelope matches in both cases. (That only works because the envelope is symmetric)

The same thing is done with the wave equation in both cases by setting the initial conditions. (Assuming you're referring to the differential equation for a wave)

 

[DarthSerious]

I am studying the propagation of pulse through a dispersive medium.And as you mentioned it is the change in phase that is important here.But I am confused about this phase thing.We are considering an initially chirped pulse so that there is a frequency variation.Since I've just started studying this,things are not very clear to me and that would have reflected in my question,sorry for that.What i,m thinking is,in the maxwells eqn of propgation if we take the envelope function as gaussian,what would be the phase?We have a functional form for the change in phase but we have to solve for the absolute value of phase.(sorry if that sounds stupid)anyway,thank you for your reply.

Any medium will have dispersion. Free space itself has linear dispersion (phase is linear with frequency). A previous reply mentioned phase [delay, or change] is related to a time delay. Yes, from a signals perspective, this is true. You are viewing it as propagation through a medium (of a given distance or time delay). Note, there is only a single relationship between phase and time (or distance) if the phase slope (versus frequency) is linear, as in free space. This is equivalent to a constant phase velocity. This is also equivalent to saying that the group velocity is constant or that group velocity dispersion (GVD) is zero. Usually, a medium is dispersive, in that phase slope isn't quite linear, and so a pulse will spread (in time, or distance) as it propagates.

There is usually some attenuation as well, and one can lump all these effects into a complex propagation "constant" or wavevector. But, I think you can figure that part on your own. Working with chirped pulses, you will likely be looking at higher-order dispersion terms (GVD is just the first) so, there are good resources out there...

You are really asking a more fundamental question. What is the phase of a [gaussian] pulse? The time and frequency domains are simply related through the Fourier transform. The transform of a gaussian (in time) is also a gaussian in frequency. Just look at Mathworld. Note, however, that the complex part of the transform integrates to zero. This generally true of all even waveforms. However, one usually doesn't have a gaussian (in time) centered at zero time, but instead at some finite t0 time. If you do that, you will find a linear phase offset (linear with frequency). This is equivalent to a port offset on a network analyzer, or simply a propagation distance in physics, or an application of the Fourier transform "shifting theorem". Multiplying the wave by exp(i*2pi*f*t0) will shift the waveform in time by t0, and add a linear slope offset to the phase spectrum.

Even more fundamental: What is the "phase" of your chirped gaussian pulse? Yeah, one can still have a gaussian amplitude ("envelope") and any phase spectrum you wish. The time domain waveform is then the inverse transform. One talks about a "transform-limited" pulse when the pulse width is as narrow as possible in time for the spectral content. Any kind of dispersion will broaden the pulse (and produce some chirp). The simplest kind of dispersion thus has only GVD with no higher order (cubic, quadratic, etc). That is, the phase versus frequency is a quadratic polynomial. The linear term can be zero if the pulse is centered at time=0, or it can have a finite linear term if the pulse is offset in time. But the second-order term will produce a waveform with the wave skewed toward one side of the envelope. Note, the amplitude is still gaussian, but the real component is lopsided.

This discussion is analogous to applying filter functions, either in the time domain or frequency domain. Just be careful with phase. Many folks ignore it altogether, only looking at the amplitude portion. The phase portion is essential to preserving the transforms and shape in the time domain. Be careful with the format of your data. Any real time waveform must have a hermitian spectrum. You need to be careful with digitized data, and know what your canned package is doing to the format of the data. Typically, the data is in a format analogous to a signed integer, where negative frequency wraps around to zero at the end of the waveform, e.g. FF is -1 for an 8-bit integer. Some packages truncate second half end of the data entirely (since it must be hermitian and "duplicates" the first half)...

I would suggest some reading on Fourier transforms, such as Ron Bracewell.

Good luck!