Fourier Transform of QAM

[MrAl]

Hi,

I am staying "tooned" as i watch South Park, so i guess you could say i am "well tooned"

Cant wait to see what you come up with here. I am brushing up on my theory a little too so i like seeing these questions/results also. What i think we need more of is 'applications' with results so we can compare answers.

 

[derick007]

steveB, MrAl

It would appear that for a periodic square wave, Fourier Series must be used rather than the Fourier Transform. Although the Fourier Series does not give an equation for the frequency domain, I have found I can interpret the spectral components from the Fourier Series. I would anticipate the envelope of the spectral components being the actual waveform in the frequency domain. It would appear that for a periodic square wave, the spectrum consists of line spectra at odd frequencies of the fundamental frequency. I compared 2 such waves, one only had positive amplitudes, but the other had both positive and negative. I presume the envelopes of both spectrums prove to be a sinc function ?

Regards, Derek

 

[MrAl]

Hi,

From post #15, you can get the Fourier Transform from the cn that you calculate with the integration:

F(jω)=2π Ʃ[n,-∞ to +∞] cn*δ(ω-n*ω0)

so the envelope is basically the cn multiplied by 2pi. The transform has the impulse function with it while the series wont have this.

If cn is the integral of e^(-j*w*n*t) from t=-T/4 to t=T/4 (square wave of amplitude 0 to 1) then the cn is:
sin(pi*n/2)/(pi*n)

and so the transform is:
F(jω)=2π Ʃ[n,-∞ to +∞] (sin(pi*n/2)/(pi*n))*δ(ω-n*ω0)

so the envelope is 2*pi*cn.

 

[derick007]

steveB, MrAl

I have found the frequency spectrum of :

s(t) . cos(2.pi.f.t + ps) where s(t) is a periodic square wave and ps = phase shift

to be a sinc function shifted in frequency by the carrier frequency f.

This is essentially a QAM signal apart from the fact the amplitude of the periodic square wave will vary with time. For 16 QAM, s(t) can have 3 different amplitudes which I do not think will change the frequency spectrum in terms of introducing additional frequency spectra ? Obviously varying the amplitude of s(t) will vary the amplitude in the frequency domain.

Can I take it then for higher orders of QAM, 64, 256, 512 etc. that the frequency spectrum is simply a sinc function shifted in frequency by the carrier ?

 

[MrAl]

Hi,

Care to show some of the work you did in calculating the modulation for the square wave of constant amplitude?

I am not totally sure what you are asking here. Are you actually asking that if the amplitude of the modulation signal changes that you expect there to be no additional frequencies introduced? I am not sure if i understand you right here, but if that's it then how do you explain a 3Hz signal changing the amplitude of the square wave so that the amplitude goes 1,2,3,2,1,2,3,2,1 etc.
Wouldnt you expect to see 3Hz frequency components in there? Of course a 3kHz wave should introduce a 3kHz component.
Also, when we calculate the transform for the square wave we assume that it has been turned on since the big bang. That means there's no step at t=0. Changing the amplitude means we'll be stepping every so often (both positive and negative), which probably has to be considered as well.

I am not sure if that kind of change is allowed or not, so you'll have to comment.

 

[derick007]

Hi MrAl

For the periodic square wave I used the Fourier Series to find an equation for the square albeit in the time domain. From this equation it was evident the frequency spectra would be line spectra at odd harmonics of the fundamental i.e. 1, 3, 5 etc which had ever decreasing amplitudes which changed sign alternatively i.e. at f the amplitude was +ve and 3f it was -ve.

The envelope of the spectra was a sinc function which as before with a single rectangular pulse I convolved with the cosine carrier to obtain the sinc function shited in frequency to the carrier frequency.

I am not totally sure about the 3 kHz signal, except to say that it is the periodic square wave which amplitude modulates the carrier.

I hope to send my work tomorrow - I need to scan it in work.

I am thinking of purchasing a copy of MATLAB - Simulink to help me understand this. Have you any experience of this ?

 

[steveB]

I am thinking of purchasing a copy of MATLAB - Simulink to help me understand this. Have you any experience of this ?

Matlab/Simulink is a great tool to use when learning and for solving real problems. The software is normally very expensive. If you are a student, you can get a student version for a very cheap price. The student version includes control toolbox, sig. proc. toolbox, symbolic toolbox and lots of other good stuff. There are very few limitations of the student version. The Matlab part is the same and the Simulink part is limited to 1000 blocks which is far more than you are likely to ever need.

If you don't want to buy the student version you can download Octave which is a very good Matlab clone and is free. Even though I use Matlab/Simulink for work, I often use Octave at home and have a version running on an Android tablet. It works great.

 

[MrAl]

Hi again,

Hey that sounds interesting Steve, i'll have to find/try that too. I didnt want to have to buy into Matlab either.

Derik, if the 3kHz signal causes the square wave to change amplitude (and something must be doing that if not a 3kHz wave) then the amplitude of the square wave changes and that means it's not constant which was your assumption.
We could easily envision a filter at the output where the square wave has created a constant DC value except for some small ripple. Then the amplitude of the square wave changes and this causes an exponential change in the DC amplitude. While the DC amplitude is changing, there must be some harmonics. After it settles back down the harmonics will only be those that appear for that square wave amplitude that you have calculated. Doesnt this sound reasonable?

 

[steveB]

Hey that sounds interesting Steve, i'll have to find/try that too. I didnt want to have to buy into Matlab either.

Mr. Al,

If you are thinking of this, then I would like to mention a warning about a possible complication. If you have an Android phone or tablet, then it's no problem. I made a nice little terminal with a blue tooth keyboard and an Android 7" tablet. It reminds me of the old days using a terminal with command line inputs. I even made the screen black with amber font to look like those old monochromatic CRT monitors.

However, when I searched for a compiled version of Octave for the Windows operating system, I only found instructions on how to compile it from the code, and I did not want to go down this road. However, a virus infection on my computer motivated me to switch to the Linux Mint operating system, and I found a compiled version of Octave for that.

I believe there is a version available for the Mac too.

If you use Windows, you may be able to find something already compiled, but I just thought I would warn you that I had trouble finding one.

 

[derick007]

MrAl, steveB

Apologies I haven't had time to scan my work.

For 16 QAM the phase shifted carrier only has 3 different amplitudes. I am assuming this can be modelled by a square amplitude modulating the carrier by multiplying one by the other ?

I have a 30 day trial of MATLAB - the home version certainly looks affordable.