# Estimating the power in EEG frequency bands

Christian Hesse c.hesse at FCDONDERS.RU.NL
Fri Feb 23 01:44:59 CET 2007

```One further comment (please see below):

> Hi Thomas,
>> Following up on this conversation. It seems that the ‘induced
>> activity’ contains both phase-locked and non-phase-locked
>> activity, whereby the ‘evoked’ activity contains only phase-locked
>> activity. Is it then kosher to separate these components by linear
>> subtraction? For example, if we first compute the ‘induced’
>> activity by averaging power over individual trials, and from that
>> subtract the ‘evoked activity’ (calculated based on average
>> response) to get the induced activity without any phase-locked
>> activity?
>
> It is not correct to subtract because computing the induced and
> evoked power spectra involves squaring signal amplitudes (a non-
> linear operation), and hence, taking your terminology to refer to
> the instantaneous amplitudes of the signal components (this applies
> to any time-frequency tile)
>> Induced = Phase + Non-Phase
>>
>> And
>>
>> Evoked = Phase
>>
>> Then
>>
>> Non-Phase = Induced – Evoked
>>
>>
> what you actually get from spectral or time-frequency analysis is
> the power of your MEASURED signal
>
> Induced^2 = (Phase + Non-Phase)^2 = Phase^2 + 2*Phase*Non-Phase +
> Non-Phase^2
>
> Evoked^2 = Phase^2
>
> Then
>
> Induced^2 - Evoked^2 = 2*Phase*Non-Phase + Non-Phase^2  AND NOT Non-
> Phase^2
>
Note that the other crucial thing to consider here is that you are in
one case averaging power over trials over trials:

E[ (Induced^2)  ] =  E[ (Phase + Non-Phase)^2 ] = E[ (Phase^2 +
2*Phase*Non-Phase + Non-Phase^2) ] = E[ (Phase^2) ] E[ (Non-
Phase^2) ] + E[ 2*Phase*Non-Phase ]

this is why taking the square root of sqrt(Induced^2) does not give
(Phase + Non-Phase) but sqrt(E[ (Phase+Non-Phase)^2 ]).

in the evoked case you are taking the power of the average amplitude

Evoked^2 = E[ Phase ]^2  (---> note the ^2 on the outside of the sum)

so in subtracting you are actually assuming that E[Phase]^2 = E
[(Phase)^2] which is unlikely to be accurate the case in finite samples.

Hope I have not confused others (or myself) here.
Christian

----------------------------------------------------------------------
Christian Hesse, PhD, MIEEE

F.C. Donders Centre for Cognitive Neuroimaging
P.O. Box 9101
NL-6500 HB Nijmegen
The Netherlands

Tel.: +31 (0)24 36 68293
Fax: +31 (0)24 36 10989

Email: c.hesse at fcdonders.ru.nl
Web: www.fcdonders.ru.nl
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