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5 Methods for Discrete Processing and Analysis of Biosignals

Fig. 5.16: Digital impulse response.

This convolution sum is generally expressed by the operator "∗" as in analogue sig-

nals, i.e. for two discrete signals x1 and x2 it is defined as.

x1(n) ∗x2(n) :=

∞

∑

ν=−∞

x1(ν)x2(n −ν) .

(5.49)

Explanatory Example

A causal time-invariant system has the finite impulse response

g(0) = 1 ;

g(1) = 0.75 ;

g(2) = 0.5 ;

g(3) = 0.25 ;

g(i) = 0

for i > 3 .

If a causal input signal x(n) consists of the values

x(0) = −1 ;

x(1) = 0.5 ;

x(2) = −0.25 ;

x(i) = 0

for i > 2 ,

then for the finite output signal y(n) according to Equation 5.48 we obtain:

y(0) = x(0)g(0)

= −1

y(1) = x(0)g(1) + x(1)g(0)

= −0.25

y(2) = x(0)g(2) + x(1)g(1) + x(2)g(0) = −0.375

y(3) = x(0)g(3) + x(1)g(2) + x(2)g(1) = −0.1875

y(4) =

x(1)g(3) + x(2)g(2) = 0

y(5) =

x(2)g(3) = −0.0625 ,

or by a scalar product using matrices:

[[[[[[[[[

[

y(0)

y(1)

y(2)

y(3)

y(4)

y(5)

]]]]]]]]]

]

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

y

=

[[[[[[

[

g(0)

0

0

g(1)

g(0)

0

g(2)

g(1)

g(0)

g(3)

g(2)

g(1) 0

g(3)

g(2)

0

0

g(3)

]]]]]]

]

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

Dr{g}

󳐂^[[

[

x(0)

x(1)

x(2)

]]

]

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

x

=

[[[[[[[[[

[

−1

−0.25

−0.375

−0.1875

0

−0.0625

]]]]]]]]]

]

.