184
7 Integration of Functionals, PCM and Stochastic IntegralEquations
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Z(−0.5, −0.5)
Z(0.0, −0.5)
Z(0.5, −0.5)
Z(−0.5, 0.0)
Z(0.0, 0.0)
Z(0.5, 0.0)
Z(−0.5, 0.5)
Z(0.0, 0.5)
Z(0.5, 0.5)
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=
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0.25000 0.25000 0.00000 0.25000 0.25000
0.062500 0.062500 0.00000 0.37500 0.37500
0.00000 0.00000 0.00000 0.25000 0.25000
0.062500 0.37500 0.062500 0.062500 0.37500
0.015625 0.093750 0.015625 0.093750 0.56250
0.00000 0.00000 0.00000 0.062500 0.37500
0.00000 0.25000 0.25000 0.00000 0.25000
0.00000 0.062500 0.062500 0.00000 0.37500
0.00000 0.00000 0.00000 0.00000 0.25000
0.00000 0.00000 0.00000 0.00000
0.00000 0.062500 0.062500 0.00000
0.00000 0.25000 0.25000 0.00000
0.062500 0.00000 0.00000 0.00000
0.093750 0.015625 0.093750 0.015625
0.062500 0.062500 0.37500 0.062500
0.25000 0.00000 0.00000 0.00000
0.37500 0.00000 0.062500 0.062500
0.25000 0.00000 0.25000 0.25000
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z 00
z 01
z 02
z 10
z 11
z 12
z 20
z 21
z 22
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.
(7.53)
We will apply the notion of the ‘impulse response,’ that is well-known in
linear system theory, to the solution of (7.53). The left-hand side of (7.53)
is replaced by nine ‘source impulses,’ which are simply the basis vectors,
[1, 0, · · · , 0], · · · , [0, 0, · · · , 1], and one then computes the nine responses,
[z 00 , · · · , z 22 ], to each of these basis vectors. Then the response to a general
source vector is simply the superposition of the impulse responses weighted by
the appropriate coefficients. This follows because of the expansion of an arbitrary
vector in terms of the ‘impulse basis’:
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Z(−0.5, −0.5)
Z(0.0, −0.5)
Z(0.5, −0.5)
Z(−0.5, 0.0)
Z(0.0, 0.0)
Z(0.5, 0.0)
Z(−0.5, 0.5)
Z(0.0, 0.5)
Z(0.5, 0.5)
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= Z(−0.5, −0.5)
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1
0
. . .
0
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+ · · · + Z(0.5, 0.5)
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0
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. . .
1
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(7.54)
Hence, the solution of (7.53) is given by