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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

The cyclic auto-correlation function of a time-series ${\textstyle x(t)}$ is calculated as follows:

$R_{x}^{\alpha }(\tau )=\int _{-\infty }^{\infty }x\left(t-{\frac {\tau }{2}}\right)x^{*}\left(t+{\frac {\tau }{2}}\right)e^{-i2\pi \alpha t}\,dt$

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

$S_{x}^{\alpha }(f)=\int _{-\infty }^{\infty }R_{x}^{\alpha }(\tau )e^{-i2\pi f\tau }\,d\tau$

Estimation methods

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] One method to calculate the spectral correlation density of a discrete-time signal ${\textstyle x[n]}$ with sample period $T_{s}$ is to first calculate the sliding window discrete Fourier transform:

$X_{N'}(n,k)=\sum _{r=-N'/2}^{r=N'/2}a[r]x[n-r]e^{-j2\pi k(n-r)T_{s}}$ The variable N′ determines the frequency resolution of the spectral correlation density and the function ${\textstyle a[n]}$ is a window function of choice. Then, the spectral correlation density is calculated:

$S_{x}^{\alpha }(n,k)={\frac {1}{N}}\sum _{n=0}^{N-1}{\frac {1}{N'}}X_{N'}\left(n,k+{\frac {\alpha }{2}}\right)X_{N'}^{*}\left(n,k-{\frac {\alpha }{2}}\right)$

References

^ 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.
^ 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.
^ 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 (CCNC): 1–5. doi:10.1109/CCNC.2010.5421830.
^ 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. doi:10.1109/79.81008. ISSN 1053-5888.

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 {{Orphan|date=July 2016}}
-The '''spectral correlation density''', 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 process]]es because stationary processes do not exhibit spectral correlation.<ref>{{Cite journal|title = Measurement of spectral correlation|url = http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1164951|journal = IEEE Transactions on Acoustics, Speech and Signal Processing|date = 1986-10-01|issn = 0096-3518|pages = 1111–1123|volume = 34|issue = 5|doi = 10.1109/TASSP.1986.1164951|first = W.A.|last = Gardner}}</ref> Spectral correlation has been used both in [[Signal detection theory|signal detection]] and [[Signals intelligence|signal classification]].<ref>{{Cite journal|title = ATSC digital television signal detection with spectral correlation density|url = http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=7023288|journal = Journal of Communications and Networks|date = 2014-12-01|issn = 1229-2370|pages = 600–612|volume = 16|issue = 6|doi = 10.1109/JCN.2014.000106|first = Do-Sik|last = Yoo|first2 = Jongtae|last2 = Lim|first3 = Min-Hong|last3 = Kang}}</ref><ref>{{Cite journal|title = Multi-User Signal Classification via Spectral Correlation|url = http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5421830|journal = 2010 7th IEEE Consumer Communications and Networking Conference (CCNC)|date = 2010-01-01|pages = 1–5|doi = 10.1109/CCNC.2010.5421830|first = S.|last = Hong|first2 = E.|last2 = Like|first3 = Zhiqiang|last3 = Wu|first4 = C.|last4 = Tekin}}</ref> The spectral correlation density is closely related to each of the [[Bilinear time–frequency distribution|bilinear time-frequency distributions]], but is not considered one of Cohen's class of distributions.
+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 process]]es because stationary processes do not exhibit spectral correlation.<ref>{{Cite journal|title = Measurement of spectral correlation|url = http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1164951|journal = IEEE Transactions on Acoustics, Speech and Signal Processing|date = 1986-10-01|issn = 0096-3518|pages = 1111–1123|volume = 34|issue = 5|doi = 10.1109/TASSP.1986.1164951|first = W.A.|last = Gardner}}</ref> Spectral correlation has been used both in [[Signal detection theory|signal detection]] and [[Signals intelligence|signal classification]].<ref>{{Cite journal|title = ATSC digital television signal detection with spectral correlation density|url = http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=7023288|journal = Journal of Communications and Networks|date = 2014-12-01|issn = 1229-2370|pages = 600–612|volume = 16|issue = 6|doi = 10.1109/JCN.2014.000106|first = Do-Sik|last = Yoo|first2 = Jongtae|last2 = Lim|first3 = Min-Hong|last3 = Kang}}</ref><ref>{{Cite journal|title = Multi-User Signal Classification via Spectral Correlation|url = http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5421830|journal = 2010 7th IEEE Consumer Communications and Networking Conference (CCNC)|date = 2010-01-01|pages = 1–5|doi = 10.1109/CCNC.2010.5421830|first = S.|last = Hong|first2 = E.|last2 = Like|first3 = Zhiqiang|last3 = Wu|first4 = C.|last4 = Tekin}}</ref> The spectral correlation density is closely related to each of the [[Bilinear time–frequency distribution|bilinear time-frequency distributions]], but is not considered one of Cohen's class of distributions.
 == Definition ==

Revision as of 23:23, 15 March 2018

Definition

Estimation methods

References

Further reading