Get documents about "convolution
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Impulse response function h(t), or hi
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0 5 10 15 20
time (s)
Stimulus: s = {1,0,0,0,0,….}
Two events
2.5
2
1.5
1
0.5
0
0 5 10 15 20
time (s)
Stimulus: s = {1,0,0,0,1,0….}
Two events
2.5
2
1.5
1
0.5
0
0 5 10 15 20
time (s)
Stimulus: s = {1,0,0,0,.5,0….}
Arbitrary experimental design
7
6
5
4
3
2
1
0
0 50 100 150 200 250
time (s)
Stimulus: s = {0,0,1,0,1,1,0,0,1,……}
3.5
3
2.5
Response at time i=18
2 s4h14 + s6h12 + s10h8
1.5
1 Response at time i
s1hi-1 + s2hi-2 + …+ si-1h1
0.5
0
0 5 10 15 20 25 30 35 40
time (s)
i
Ri s j hi j R sh
j 1
Stimulus: s = {0,0,0,1,0,1,0,0,0,1,0,0,…}
2 2
1.5 1.5
1 1
0.5 0.5
0 0
0 20 40 60 80 0 20 40 60 80
time (s) time (s)
0.2 0.2
i 0.15 0.15
Ri s j hi j
j 1 0.1 0.1
0.05 0.05
0 0
0 20 40 60 80 0 20 40 60 80
time (s) time (s)
6
4
2
0
0 10 20 30 40 50 60 70 80
time (s)
Design Matrix Impulse response Response
X h R
x =
Xh R
if
Xh R
then
ˆ ( X T X ) 1 X T R
h
Time-contrast separability
Boynton, G.M., et al., J Neurosci, 1996. 16(13): p. 4207-21.
Shift-invariance
response to 6 sec pulse shift and duplicate
3 3
0 0
-3 -3
0 10 20 30 40 0 10 20 30 40
response to 12 sec pulse predict 12 sec from 6 sec pulses
3 3
0 0
Intensity
-3 -3
0 10 20 30 40 0 10 20 30 40
Time (sec)
Boynton, G.M., et al., J Neurosci, 1996. 16(13): p. 4207-21.
Shift-invariance
3 predicts 6
gmb
3
Response to pulsed stimulus
0 Prediction from shift and sum
-3
0 10 20 30 40
3 predicts 12 6 predicts 12
3 3
0 0
-3 -3
0 10 20 30 40 0 10 20 30 40
3 predicts 24 6 predicts 24 12 predicts 24
3 3 3
Intensity
0 0 0
-3 -3 -3
0 10 20 30 40 0 10 20 30 40 0 10 20 30 40
time (sec)
Boynton, G.M., et al., J Neurosci, 1996. 16(13): p. 4207-21.
Additivity
single
stimulus
paired
same time
SOA
Stimuli:
0.5 c/deg, 8 Hz counterphase
1sec duration 30 deg
48 repetitions per condition
30 deg
single
stimulus
paired
same
Nonlinearity
after subtracting
response to first pulse:
single
stimulus
paired
same
paired
orthogonal
after subtracting
response to first pulse:
What does convolution have to do with sinusoids
and Fourier Transforms?
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0 50 100 150 200 250
time (s)
Convolution of sinusoid is a sinusoid:
Convolution changes the amplitude and phase
of a sinusoid – but NOT the frequency
2 2 2
1.5 1.5 1.5
1 1 1
0.5 0.5 0.5
0 0 0
0 15 30 45 60 0 15 30 45 60 0 15 30 45 60
time (s) time (s) time (s)
2 2 2
1.5 1.5 1.5
1 1 1
0.5 0.5 0.5
0 0 0
0 15 30 45 60 0 15 30 45 60 0 15 30 45 60
time (s) time (s) time (s)
sin(2jt ) * h(t ) b j sin(2jt j )
Fourier transform:
n
s (t ) a j sin(2jt j )
w1
Convolution of sinusoid is a sinusoid:
sin(2jt ) * h(t ) b j sin(2jt j )
Combined:
n
s (t ) * h(t ) a j b j sin(2jt j j )
w1
Convolution of a function can be expressed via
amplitude scaling and phase shifts of the function’s
Fourier components.
The amplitude scaling and phase shifts of the function’s Fourier
components are determined by the Fourier transformation of the
impulse response function.
FFT of h(t)
1 350
0.9
300
0.8
250
0.7
200
0.6
Amplitude
Phase
0.5 150
0.4
100
0.3
50
0.2
0
0.1
0 -50
60 30 15 7.5 3.75 60 30 15 7.5 3.75
Period (sec) Period (sec)
The convolution theorem:
FFT of s(t) S(w)
stimulus: s(t)
FFT of h(t)
HDR: h(t) H(w)
convolution multiplication
IFFT
response: f(t)*h(t) S(w) H(w)
stimulus fft of stimulus
3 0.12
2.5 0.1
2 0.08
amplitude
1.5 0.06
1 0.04
0.5 0.02
0 0
0 40 80 120 6 12 18 24 30 36
Time (sec) Cycles/scan
response fft of response
0.1
1.2
1 0.08
0.8
amplitude
0.06
0.6
0.04
0.4
0.02
0.2
0 0
0 40 80 120 6 12 18 24 30 36
Time (sec) Cycles/scan
Example: simple block design
10
9
8
7
6
5
4
3
2
1
0
0 50 100 150 200 250
time (s)
Stimulus: s = {1,1,1,…,0,0,0,…,1,1,1,…,0,0,0,…}
stimulus fft of stimulus
3 0.8
2.5
0.6
2
amplitude
1.5 0.4
1
0.2
0.5
0 0
0 40 80 120 160 200 240 6 12 18 24 30 36
Time (sec) Cycles/scan
response fft of response
0.8
1.2
1
0.6
0.8
amplitude
0.6 0.4
0.4
0.2
0.2
0 0
0 40 80 120 160 200 240 6 12 18 24 30 36
Time (sec) Cycles/scan
fMRI amplitude for different stimulus
frequencies and contrasts
fMRI amplitude at different frequencies
for a 30 second period stimulus
Time (s)
