Rune Allnor
04-11-2007, 09:49 AM
Hi folks.
I have been testing the coherence formula
cxy(f) = |Cxy(f)|^2/(|Cxx(f)||Cyy(f)|) [1]
where Cxy(f) is the xy cross spectrum coefficient at frequency f and
Cxx(f) and Cyy(f) are the periodogram coefficients at frequency f of
x and y, respectively.
I compute the spectra as
- Compute the correlation sequences rxx[m], rxy[m] and ryy[m]
- Apply a covariance bias function b[m]
- Apply a window function w[m]
- Transform to frequency domain
- Apply the spectra to formula [1] above
I have tested this formula with signals x[n] = sin(2*pi*f*n) and y[n]
= cos(2*pi*f*n)
with various noise and other degradations. In other words, these are
narrowband
signals.
I find that the behaviour of the coherence formula depends very
strongly
on the window functions used in the algorithm above.
================================================
--- Case 1: Biased covariance + rectangular window
If one uses the "usual" covariance estimator with bias window
b[m]=(N-|m|)/N
and a rectancular window function w[m] = 1, then cxy == 1 for all f.
--- Case 2: Unbiased covariance + rectangular window
If one corrects for the covariance bias such that b[m] = 1, the
coherence
fluctuates wildly cxy(f) >> 1 outside the band where the sinusoidals
are defined, and approaches unity near the frequencies of the sines.
--- Case 3: Biased covariance estimator + non-rectanguar window
If one uses the bias function b[m]=(N-|m|)/N with any winow function
other than the rectangular window, the coherence estimate equals
unity
inside the bandwidth where the sinusoidals are, and/or lower,
0 <= cxy(f) <= 1, elsewhere.
--- Case 4: Unbiased covariance estimator + non-rectangular window
In these cases the coherence estimate is still unity near the
sinusoidals
but fluctuates wildly (cxy(f)>> 1) elsewhere.
==================================================
I would like to understand what is happening here. One might say that
"there is no problem since the coherence always equals unity where
it ought to; what happens in the noise bands is irrelevant anyway."
While this might be true, it is of no help when one approaches the
borderline conditions with weak signals in noise.
The above has to be well-known material, and ought to be described
and discussed somewhere. I would appreciate any pointers to
where such discussions might be found.
Rune
I have been testing the coherence formula
cxy(f) = |Cxy(f)|^2/(|Cxx(f)||Cyy(f)|) [1]
where Cxy(f) is the xy cross spectrum coefficient at frequency f and
Cxx(f) and Cyy(f) are the periodogram coefficients at frequency f of
x and y, respectively.
I compute the spectra as
- Compute the correlation sequences rxx[m], rxy[m] and ryy[m]
- Apply a covariance bias function b[m]
- Apply a window function w[m]
- Transform to frequency domain
- Apply the spectra to formula [1] above
I have tested this formula with signals x[n] = sin(2*pi*f*n) and y[n]
= cos(2*pi*f*n)
with various noise and other degradations. In other words, these are
narrowband
signals.
I find that the behaviour of the coherence formula depends very
strongly
on the window functions used in the algorithm above.
================================================
--- Case 1: Biased covariance + rectangular window
If one uses the "usual" covariance estimator with bias window
b[m]=(N-|m|)/N
and a rectancular window function w[m] = 1, then cxy == 1 for all f.
--- Case 2: Unbiased covariance + rectangular window
If one corrects for the covariance bias such that b[m] = 1, the
coherence
fluctuates wildly cxy(f) >> 1 outside the band where the sinusoidals
are defined, and approaches unity near the frequencies of the sines.
--- Case 3: Biased covariance estimator + non-rectanguar window
If one uses the bias function b[m]=(N-|m|)/N with any winow function
other than the rectangular window, the coherence estimate equals
unity
inside the bandwidth where the sinusoidals are, and/or lower,
0 <= cxy(f) <= 1, elsewhere.
--- Case 4: Unbiased covariance estimator + non-rectangular window
In these cases the coherence estimate is still unity near the
sinusoidals
but fluctuates wildly (cxy(f)>> 1) elsewhere.
==================================================
I would like to understand what is happening here. One might say that
"there is no problem since the coherence always equals unity where
it ought to; what happens in the noise bands is irrelevant anyway."
While this might be true, it is of no help when one approaches the
borderline conditions with weak signals in noise.
The above has to be well-known material, and ought to be described
and discussed somewhere. I would appreciate any pointers to
where such discussions might be found.
Rune