In this post, we continue our study of the symmetries of CSP parameters. The second-order parameters–spectral correlation and cyclic correlation–are covered in detail in the companion post, including the symmetries for ‘auto’ and ‘cross’ versions of those parameters.
Here we tackle the generalizations of cyclic correlation: cyclic temporal moments and cumulants. We’ll deal with the generalization of the spectral correlation function, the cyclic polyspectra, in a subsequent post. It is reasonable to me to focus first on the higher-order temporal parameters, because I consider the temporal parameters to be much more useful in practice than the spectral parameters.
This topic is somewhat harder and more abstract than the second-order topic, but perhaps there are bigger payoffs in algorithm development for exploiting symmetries in higher-order parameters than in second-order parameters because the parameters are multidimensional. So it could be worthwhile to sally forth.
Continue reading “Symmetries of Higher-Order Temporal Probabilistic Parameters in CSP”
signals in the posted data set. I’ve now added another 
for a total of 
signals. The original signals are contained in Batches 1-5, the additional signals in Batches 6-28. I’ve placed these additional Batches at the end of the post to preserve the original post’s content.
PSK/QAM signals to the CSP Blog. These are the signals I refer to in the 
individual signal files:
fields. All temporal and spectral parameters (times and frequencies) are normalized with respect to the sampling rate. In other words, the sampling rate can be taken to be unity in this data set. These fields are described in the following list:
th signal is contained in the file signal_n.tim. The Batch 1 zip file contains signal_1.tim through signal_4000.tim.
-DQPSK, 16QAM, 64QAM, 256QAM, and MSK. These are denoted by the strings bpsk, qpsk, 8psk, dqpsk, 16qam, 64qam, 256qam, and msk, respectively.
.
.
. It can be any real number between 
and 
.
. So the BPSK signal in signal_1.tim has rate 
. If the downsample factor is zero in the truth-parameters file, no resampling was done to the signal.
dB. So the various SNRs are generated by varying the signal power.
:
. The carrier offset is 
and the excess bandwidth is 
. Because the signal type is 
and no conjugate cycle frequencies. But the square of the signal has cycle frequencies related to the quadrupled carrier:
seconds of data.
of the estimate is approximately 
is about 
, and the spectral resolution 
depends strongly on the particular estimator and its parameters. The resolution product 
was discussed in 
, cyclic moments and cyclic cumulants are usually identical. They differ when the signal contains one or more finite-strength additive sine-wave components. In the common case when such components are absent (as in our recurring numerical example involving 
. Most communication signal models possess odd-order moments and cumulants that are identically zero, so the first non-trivial order 
. Our estimation task is to estimate 
using a sampled-data record of length 
is the symbol index, 
is the symbol rate, 
is the carrier frequency (sometimes called the frequency offset), 
is the symbol-clock phase, and 
is the carrier phase. The finite-energy function 
is the pulse function (sometimes called the pulse-shaping function). Finally, the random variable 
is called the symbol, and has a discrete distribution that is called the constellation.
), the signal is even less realistic, and therefore more textbook.
, and I filter it using a linear time-invariant transformation with 
, producing the output 
, then what is the PSD of 
is the 
scales the input frequency component at 
. So it makes sense that the PSD at ![\displaystyle R_x(t, \boldsymbol{\tau}) = E \left[ x(t+\tau_1)x^*(t+\tau_2) \right] \hfill (1)](http://s0.wp.com/latex.php?latex=%5Cdisplaystyle+R_x%28t%2C+%5Cboldsymbol%7B%5Ctau%7D%29+%3D+E+%5Cleft%5B+x%28t%2B%5Ctau_1%29x%5E%2A%28t%2B%5Ctau_2%29+%5Cright%5D+%5Chfill+%281%29&bg=ffffff&fg=000&s=0 "\displaystyle R_x(t, \boldsymbol{\tau}) = E \left[ x(t+\tau_1)x^*(t+\tau_2) \right] \hfill (1)")
![\displaystyle R_{x^*}(t, \boldsymbol{\tau}) = E \left[ x(t+\tau_1)x(t+\tau_2) \right]. \hfill (2)](http://s0.wp.com/latex.php?latex=%5Cdisplaystyle+R_%7Bx%5E%2A%7D%28t%2C+%5Cboldsymbol%7B%5Ctau%7D%29+%3D+E+%5Cleft%5B+x%28t%2B%5Ctau_1%29x%28t%2B%5Ctau_2%29+%5Cright%5D.+%5Chfill+%282%29&bg=ffffff&fg=000&s=0 "\displaystyle R_{x^*}(t, \boldsymbol{\tau}) = E \left[ x(t+\tau_1)x(t+\tau_2) \right]. \hfill (2)")
are the non-conjugate and conjugate cyclic autocorrelation functions, respectively.