Graphics Programs Reference
In-Depth Information
the complex Fourier transform
X
(
f
) also contains information on the phase
relationship
W
(
f
) of the two signals:
The phase difference is important in calculating leads and lags between two
signals, a parameter often used to propose causalities between the two pro-
cesses documented by the signals. The correlation between the two spectra
can be calculated by means of the coherence:
The coherence is a real number between 0 and 1, where 0 indicates no cor-
relation and 1 indicates maximum correlation between
x
(
t
) and
y
(
t
) at the
frequency
f
. Signifi cant degree of coherence is an important precondition for
computing phase shifts between the two signals.
We use two sine waves with identical periodicities
=5 (equivalent to
f
=0.2) and amplitudes equal to two. The sine waves show a relative phase
τ
Cross PSD Estimate
Phase spectrum
20
4
f
1
=0.02
3
f
1
=0.02
15
2
1
10
Corresponding phase
angle of 1.2568, equals
(1.2568*5)/(2*π)=1.001
0
5
−1
0
−2
0
1
2
3
4
5
0
1
2
3
4
5
Frequency
Frequency
a
b
Fig. 5.8
Crossspectrum of two sine waves with identical periodicities τ=5 (equivalent to
f
=0.2) and amplitudes two. The sine waves show a relative phase shift of
t
=1. In the argument
of the second sine wave this corresponds to 2/5, which is one fi fth of the full wavelength
of τ=5.
a
The magnitude shows the expected peak at
f
=0.2.
b
The corresponding phase
difference in radians at this frequency is 1.2568, which equals (1.2568*5)/(2*) = 1.0001,
which is the phase shift of one we introduced at the very beginning.
