One of the things the machine learners never tire of saying is that their neural-network approach to classification is superior to previous methods because, in part, those older methods use hand-crafted features. They put it in different ways, but somewhere in the introductory section of a machine-learning modulation-recognition paper (ML/MR), you’ll likely see the claim. You can look through the ML/MR papers I’ve cited in The Literature ([R133]-[R146]) if you are curious, but I’ll extract a couple here just to illustrate the idea.
Continue reading “Are Probability Density Functions “Engineered” or “Hand-Crafted” Features?”Are Probability Density Functions “Engineered” or “Hand-Crafted” Features?
The Machine Learners think that their “feature engineering” (rooting around in voluminous data) is the same as “features” in mathematically derived signal-processing algorithms. I take a lighthearted look.
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.
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 
-th order 
, 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)](https://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&c=20201002 "\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)](https://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&c=20201002 "\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.