# convolution

Shared by:
Categories
Tags
-
Stats
views:
0
posted:
9/22/2012
language:
Unknown
pages:
25
Document Sample

```							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  sh
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.

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(2jt ) * h(t )  b j sin(2jt   j )
Fourier transform:
n
s (t )   a j sin(2jt   j )
w1

Convolution of sinusoid is a sinusoid:

sin(2jt ) * h(t )  b j sin(2jt   j )

Combined:
n
s (t ) * h(t )   a j b j sin(2jt   j   j )
w1

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)

```
Related docs
Other docs by ajizai
True scary creatures.ppt - bishopcook09