Back to the README Previous step: QPSK modulation
Phase modulation is the most efficient modulation scheme [1] but also the most
complex to demodulate since an exact copy of the transmitted carrier must be
generated on the receiver. The classical means for cancelling the N-PSK
phase modulation is to consider the N-th power of the received signal. Indeed,
complex baseband signal s(t)=exp(j*2pi*df*t+j*phi(t)) with df the frequency
difference between emitter and receiver frequencies (to be cancelled for phase
demodulation) and the phase phi(t)=2*pi*k/N, k=[0:N-1], carrying the time-dependent
information, becomes after computing the N-th power:
s^N(t)=exp(j*2*pi*N*df*t+j*N*phi(t)) and since N*pi is N*2*pi*k/N=k*2*pi=0
since k is an integer, we end up with a clean carrier at N*df:
s^N(t)=exp(j*2*pi*N*df*t)
Either cancelling df or dividing the frequency of s^N(t) reproduces a clear copy
of the carrier and mixing with the incoming signal cancels the time varying
argument of the trigonometric function to only yield exp(j*phi(t)). These
operations are achieved with GNU Radio using the Costas loop whose bandwidth
is advised in the documentation to be about 2*pi/100 and whose order is the N in
N-PSK, i.e. the number of possible phase values encoding the signal. The
GNU Radio Companion flowchart is
Following the Costas loop, the Symbol Synchronization block avoids sampling the signal on the transitions of the signal connecting stable phase values but only samples a single phase estimate synchronized on the sharp transitions. In the following charts, from top to bottom are displayed the time-domain I and Q samples, middle-top is the spectrum of the signal, the signal squared and its fourth power, in the middle-bottom the resulting constellation plot after Costas loop and Symbol Synchronization with N clearly separated point clouds, and on the bottom the output of the Costas loop prior to Symbol Synchronization with the lines linking the point clouds.
BPSK modulation (N=2) with a Costas loop set to the second order:
Notice on the top chart how the imaginary part (red) was cancelled by the Costas
loop, leaving only a real part equal to exp(j*pi)=-1 or exp(j*0)=1, and on
the middle-top chart how the squared (red) and fourth power (green) both exhibit a
clean carrier while the raw spectrum (blue) has been spread by the modulation.
BPSK modulation (N=2) with a Costas loop (erroneously) set to the fourth order:
This time considering the N=4 with a BPSK modulation prevents the Costas loop from cancelling the imaginary part and both I and Q still carry some information.
QPSK modulation (N=4) with a Costas loop set to the fourth order:
This time, only the fourth power of the signal (green spectrum) cancels the modulation while both the raw and squared signal remain spread over the modulation bandwidth.
Using GNU/Octave, the N-th power of the signal s is computed with s.^N,
again leading to a clean carrier located at N*df when plotting abs(fftshift(fft(s.^N))).
The X-axis of this plot can be graduated with a frequency generated as linspace(-fs/2,fs/2,length(s))
with fs the sampling frequency.