How to demodulate QAM signal while the spectrum is inverted due to mixing process?

[speedracer]

hi! all,

i would like to know how i can demodulate a QAM signal while the spectrum is inverted due to mixing process?

i can extract both i channel and q channel fine, but the receiver needs to reverse the spectrum somehow.

thanx!

speedracer

 

[zorro]

Re: spectal inversion

Hi speedracer,

it is not necessary to reverse the spectrum. You can process the inverted spectrum, just taking into account that the phases are changed in its signs. For instance, a phase change of +90 degrees in the noninverted spectrum corresponds to a phase change of -90 degrees in the inverted spectrum, and conversely.
You invert the spectrum (change from inverted/noninverted) conjugating the complex quantities:
Change the sign of the Q channel and you get a spectral inversion.
Change the sign of the I channel and you get a spectral inversion plus a sign change (180 degrees rotation).
Instead of inverting the Q channel you could consider the complex-conjugated constellation in the decoding process, but it seems to be "cleaner" (for clarity and maintenability) to invert Q in the digital processing.
Regards

Z

 

[speedracer]

Re: spectal inversion

hi! zorror,

i am wondering how one can mathematically prove by swapping I and Q and multiply -1 to Q, the spectrum can be reversed back.

Thanks!

speedracer

 

[zorro]

Re: spectal inversion

Hi speedracer,

It is a mathematical property of Fourier transforms.
In what follows, the integrals extend over al the real axis;
P stands for PI
* stands for complex conjugated

The Fourier Transform of x(t) is X(f)=integ[x(t)exp(-j2Pft)dt]

The Fourier Transform of x*(t) is
integ{x*(t)exp(-j2Pft)dt} =
integ{x*(t)[exp(j2Pft)]*dt} =
integ{[(x(t)exp(j2Pft)dt]*} =
{integ[(x(t)exp(j2Pft)dt]}* = X*(-f)

The reversed spectrum is X*(-f), so the reversed spectrum in frequency domain corresponds to conjugation in time domain.
I hope this is clear. Regards

Z