EEG and Brain Connectivity: A Tutorial - Bio-Medical Instruments, Inc.

The time series can be divided into two general categories: 1- the statistics of short-time
biological data and other short-time interval events such as economic, sociological, etc.
and 2- long time span events such as astronomical, meterological, geologic and
geophysical data . . . . . . . – to be continued
44- Appendix – B
Instantaneous Coherence and Phase Difference
Complex demodulation was used to compute instantaneous coherence and phasedifferences
(Granger and Hatanaka, 1964; Otnes and Enochson, 1972; Bloomfield, 2000).
This method first multiples a time series by the complex function of a sine and cosine at a
particular frequency followed by a low pass filter which removes all but very low
frequencies and transforms the time series into instantaneous amplitude and phase and an
“instantaneous” spectrum (Bloomfield, 2000). We place quotations around the term
“instantaneous” to emphasize that there is always a trade-off between time resolution and
frequency resolution. The broader the band width the higher the time resolution but the
lower the frequency resolution and vice versa (Bloomfield, 2000). For example, if we
multiply a time series {x t , t = 1, . . . , n} by sine ω 0 t and cos ω 0 t and then apply a low pass
filter F, we have
Z′
t
= F( xt
sinω
0t)
Z′
′= F( x cosω
0t)
t
2
[ ] 1 / 2
2
and ( Z ) + ( Z′
)
t
2 ′
t
t
is an estimate of the “instantaneous” amplitude of the
Z′
−1 t
frequency ω 0 at time t and tan is an estimate of the “instantaneous” phase at
′′
Z
t
time t.
The instantaneous cross-spectrum is computed when there are two time series {y t ,
t = 1, . . . , n} and {y’ t , t = 1, . . . , n} and if F [ ] is a filter passing only frequencies near
zero, then, as above
R
t
2
2
2
iω
[ ] [ ] [ ] 2 0t
y sinω t + F y cosω
t = F y e
= F
is the estimate of the
amplitude of frequency ω 0 at time t and
t
0
[ y ] ⎞
t
sinω
0t
[ y cosω
t] ⎟⎟
t
−1⎛
F
ϕ
t
= tan ⎜⎜
is an estimate of the phase of frequency ω 0 at
⎝ F
t 0 ⎠
time t and since
0
t