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convolution

VIEWS: 10 PAGES: 25

									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.
                          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(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)

								
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