Fourier Transform: Synthesis

A single sine wave

A single sine wave with twice the frequency and half the amplitude of the first

                                                                       

The sum of the first and second sine waves.  

Fourier Transformt = 0:0.001:0.6;

x = sin(2*pi*50*t)+sin(2*pi*120*t);

y = x + 2*randn(size(t));

plot(1000*t(1:50),y(1:50))

title('Signal Corrupted with Zero-Mean Random Noise')

xlabel('time (milliseconds)')

Y = fft(y,512);

Pyy = Y.* conj(Y) / 512;

f = 1000*(0:256)/512;

plot(f,Pyy(1:257))

title('Frequency content of y')

xlabel('frequency (Hz)')

example from Matlab documentation

fft

Fourier Transform

1/f^1, “pink” noise versus 1/f^0 “white” noise

Fourier Transform

Neural oscillations are patterened, relate to each other,

**but NOT by an integer ratio**

Fourier Transform

• Power at some frequencies depends on behavioral state.

(note: here, power is whitened)

Fourier Transform

Deviations from 1/f in time are significant…see self-criticality

Fast Fourier Transformt = 0:0.001:0.6;

x = sin(2*pi*50*t)+sin(2*pi*120*t);

y = x + 2*randn(size(t));

plot(1000*t(1:50),y(1:50))

title('Signal Corrupted with Zero-Mean Random Noise')

xlabel('time (milliseconds)')

Y = fft(y,512);

Pyy = Y.* conj(Y) / 512;

f = 1000*(0:256)/512;

plot(f,Pyy(1:257))

title('Frequency content of y')

xlabel('frequency (Hz)')

What is a Spectrogram?

example from Matlab documentation

fft

Power Spectra• Example of fast-Fourier

with small temporal windows over long behavioural epochs (p. 106)

• Example of Fourier averaged over 1s time window (plot on left) and smoothed using smaller, overlapping windows to reveal dynamics(main plot)

fixation period face on