Temporal filtering amounts to removing the periodic waves not of interest to us from the power spectrum, and then summing the waves back again, using the inverse Fourier transform to create a new timecourse for the voxel.
42.
The " complex cepstrum " was defined by Oppenheim in his development of homomorphic system theory and is defined as the Inverse Fourier transform of the logarithm ( with unwrapped phase ) of the Fourier transform of the signal.
43.
One can view the multiplier operator " T " as the composition of three operators, namely the Fourier transform, the operation of pointwise multiplication by " m ", and then the inverse Fourier transform.
44.
So, cyclic codes are vectors in the field GF ( q ) and the spectrum given by its inverse fourier transform is over the field GF ( q ^ m ) and are constrained to be zero at certain components.
45.
From the basic " k "-space formula, it follows immediately that we reconstruct an image I ( \ vec { x } ) simply by taking the inverse Fourier transform of the sampled data, viz.
46.
IFT, i . e ., that the cepstrum is the " inverse Fourier transform of the log-magnitude Fourier spectrum " . ( the difference between squaring or taking the absolute value amounts to an overall factor of 2)
47.
For non-realtime filtering, to achieve a low pass filter, the entire signal is usually taken as a looped signal, the Fourier transform is taken, filtered in the frequency domain, followed by an inverse Fourier transform.
48.
Some methods shift the peak in Fourier space and apply non-linear optimization to maximize the correlogram peak, but these tend to be very slow since they must apply an inverse Fourier transform or its equivalent in the objective function.
49.
As we increase the length of the interval on which we calculate the Fourier series, then the Fourier series coefficients begin to look like the Fourier transform and the sum of the Fourier series of begins to look like the inverse Fourier transform.
50.
By taking the inverse Fourier transform of \ tilde { A } ( \ omega, z + h ) one obtains A \ left ( t, z + h \ right ); the pulse has thus been propagated a small step h.
