Spectral correlation density

The spectral correlation density (SCD), sometimes also called the cyclic spectral density or spectral correlation function, is a function that describes the cross-spectral density of all pairs of frequency-shifted versions of a time-series. The spectral correlation density applies only to cyclostationary processes because stationary processes do not exhibit spectral correlation.[1] Spectral correlation has been used both in signal detection and signal classification.[2][3] The spectral correlation density is closely related to each of the bilinear time-frequency distributions, but is not considered one of Cohen's class of distributions.

Definition

edit

The cyclic auto-correlation function of a time-series   is calculated as follows:

 

where (*) denotes complex conjugation. By the Wiener–Khinchin theorem [questionable, discuss], the spectral correlation density is then:

 

Estimation methods

edit
 
SCD Estimate of Common Communications Signals

The SCD is estimated in the digital domain with an arbitrary resolution in frequency and time. There are several estimation methods currently used in practice to efficiently estimate the spectral correlation for use in real-time analysis of signals due to its high computational complexity. Some of the more popular ones are the FFT Accumulation Method (FAM) and the Strip-Spectral Correlation Algorithm.[4] A fast-spectral-correlation (FSC) algorithm[5] has recently been introduced.

FFT accumulation method (FAM)

edit

This section describes the steps for one to compute the SCD on computers. If with MATLAB or the NumPy library in Python, the steps are rather simple to implement. The FFT accumulation method (FAM) is a digital approach to calculating the SCD. Its input is a large block of IQ samples, and the output is a complex-valued image, the SCD.

Let the signal, or block of IQ samples,   be a complex valued tensor, or multidimensional array, of shape  , where each element is an IQ sample. The first step of the FAM is to break   into a matrix of frames of size   with overlap.

 

where   is the separation between window beginnings. Overlap is achieved when  .   is a tensor of shape  , and   depends on how many frames were able to fit in  .

Next a windowing function   of shape  , like the Hamming window, is applied to each row in  .

 

where   is element-wise multiplication. Next the FFT is taken on each row in  

 

  is commonly known as the waterfall plot, or spectrogram. The next step in the FAM is for the phase to be corrected from delay of the FFTed frames.

 

where   is a tensor of shape   corresponding to each digital frequencies in the FFTs

 

Next the FFTs are autocorrelated to create a tensor   of shape  .

 

where   denotes complex conjugate. In other terms, if we let   be a matrix of shape  , we can rewrite   as

 

where H denotes Hermitian (conjugate and transpose) of a matrix. The next step is to take the FFT of   along the first axis.

 

  is the full SCD, but in the shape of a 3-dimensional tensor. What we aim for is a 2-dimensional tensor   (a matrix or image) of shape   where each entry corresponds to a particular frequency   and cyclic frequency  . All values of   in   can be arranged in the tensor  , and all values of   in   in the tensor  . Here,   and   are normalized frequencies.

 

 

where  . Now the SCD image   can be arranges in the form of a matrix with zeros where there are no values for a particular   pair in  , and entries from   where it is valid as per   and  .

Estimating the SCD by skipping the second FFT

edit

The full SCD is a rather large and computationally complex, mostly due to the second round of FFTs. Fortunately, from   an estimate   of the SCD can be calculated as

 

For even less computational complexity, we can compute   as

 

because averaging all values in an FFT window before or after an FFT are equivalent. Note that   will look like a 45 degree rotated version of the true SCD  .

References

edit
  1. ^ Gardner, W.A. (1986-10-01). "Measurement of spectral correlation". IEEE Transactions on Acoustics, Speech, and Signal Processing. 34 (5): 1111–1123. doi:10.1109/TASSP.1986.1164951. ISSN 0096-3518.
  2. ^ Yoo, Do-Sik; Lim, Jongtae; Kang, Min-Hong (2014-12-01). "ATSC digital television signal detection with spectral correlation density". Journal of Communications and Networks. 16 (6): 600–612. doi:10.1109/JCN.2014.000106. ISSN 1229-2370. S2CID 757095.
  3. ^ Hong, S.; Like, E.; Wu, Zhiqiang; Tekin, C. (2010-01-01). "Multi-User Signal Classification via Spectral Correlation". 2010 7th IEEE Consumer Communications and Networking Conference. pp. 1–5. doi:10.1109/CCNC.2010.5421830. ISBN 978-1-4244-5175-3. S2CID 17126519.
  4. ^ Roberts, R.S.; Brown, W.A.; Loomis, H.H. (1991-04-01). "Computationally efficient algorithms for cyclic spectral analysis". IEEE Signal Processing Magazine. 8 (2): 38–49. Bibcode:1991ISPM....8...38R. doi:10.1109/79.81008. ISSN 1053-5888. S2CID 1763992.
  5. ^ Borghesani, P.; Antoni, J. (October 2018). "A faster algorithm for the calculation of the fast spectral correlation". Mechanical Systems and Signal Processing. 111: 113–118. Bibcode:2018MSSP..111..113B. doi:10.1016/j.ymssp.2018.03.059. hdl:1959.4/unsworks_63608. ISSN 0888-3270. S2CID 125098069.

Further reading

edit
  • Napolitano, Antonio (2012-12-07). Generalizations of Cyclostationary Signal Processing: Spectral Analysis and Applications. John Wiley & Sons. ISBN 9781118437919.
  • Pace, Phillip E. (2004-01-01). Detecting and Classifying Low Probability of Intercept Radar. Artech House. ISBN 9781580533225.