Isolating frequency-specific oscillation from transient
The notation and terminology in this and the closely related paper could have been more simplified I think.

To understand what's going on there, we need to start from page 1583. Specifically:


$y(t)$ is basically your measured signal (the output from each neuronal population). A scaled version of that is what you would sense via some modality (in an ideal world).
$s(\omega, t)$ is the frequency-time representation of a $y(t)$. This is complex.
Given that this is about Event Related Responses (ERR):


$g(\omega, t)_T$ is the average absolute value of $s(\omega,t)$ after a number of trials. This is real valued.
$g(\omega, t)_e$ is the average of $s$ times the average of the conjugate of $s$ (both of which will be complex, the result of which however will be real).
These look similar but their magnitude is not.

Then the explanation of the breakdown of that "power" follows. What this means is that, based on the assumptions in the paper, we think that the neuronal activity is due to three things: a) Background activity, b) Activity because of connections between neuronal populations and c) Induced activity by the stimulus. Furthermore, these three components are assumed orthogonal. This means that if one "happens", the rest do not. This is important for later on.
WHERE/WHEN this holds then we could estimate the background activity by looking at the spectrum BEFORE the stimulus onset, we could estimate the activity due to the stimulus (the part that is phase locked to the stimulus), because we know the "shape" of the stimulus and therefore what is left should be due to structural connections. 


These are captured in $g(\omega, t))_i$ and $g(\omega,t)_T$.
Notice here that the baseline is only $g(\omega)$

So, when you arrive at equation 23, what you see is the explanation of that "adjustment" where:


$g(\omega,t)_T$ is the frequency-time representation of the averaged stimulus response after $T$ trials as before.
$g(\omega,t)_e$ is the frequency-time representation "power of average" as before
$\hat{\eta}$ now denotes the proportion of the frequency-time representation of $g(\omega,t)$ that is due to $g(\omega,t)_e$. The "trick" here is this, you have already assumed that the three components are orthogonal. WHERE/WHEN this holds, the "pseudoinverse" ($g(\omega, t)^+$) of a given $g(\omega, t)$ multiplied by the "shape" of the stimulus response $g(\omega,t)$ over the trials would be zero and you would be left with the "shape" of the stimulus response. So, that $\hat{\eta}$ business there (and the subtraction) is what allows you to do that "separation".



Perhaps it is now clear how to produce that $g(\omega,t)^+$. Provided that you have a given $g(\omega,t)_e$, invert it with pinv (or anything else equivalent to a pinv in another platform).

But, I wonder how effective would that be for what you are trying to do. Irrespectively of this specific technique, a "pulse" or a step or any abrupt change of a signal in the time domain usually has a broad representation in the frequency domain (whether that is Fourier or Wavelet, the point is that you need more components to synthesize the step). You are looking for "oscillations" in a 0-200ms interval. What is the frequency of the oscillations you are looking for?

Hope this helps