5.3 Methods for Analysis and Processing of Discrete Biosignals
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173
The vectors x and y contain the values for the input and output signals respect-
ively. The matrix Dr{g} is a strip-triangular matrix and contains per column the
values of the impulse response, whose values are summarised in the vector g :=
[g(0), g(1), . . . , g(N −1)]. The values of the matrix Dr{g} in the i-th column are
shifted downwards by one index compared to the values in the i −1-th column.
The output signal y(n) has a total of L = M + N −1 values for a finite impulse
response g(n) with N values and a finite input signal with M values. In our example,
N = 4 and M = 3, so that one obtains the value L = 3 + 4 −1 = 6 for its length. A
corresponding graphical representation is shown in Figure 5.17.
In general, for a finite input signal with N values and an impulse response with
M values, the discrete-time convolution can be expressed in matrix form by
[[[[[[[[[[[
[
y(0)
y(1)
y(2)
...
...
y(L −1)
]]]]]]]]]]]
]
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
y
=
[[[[[[[[[[[[
[
g(0)
0
0
g(1)
g(0)
0
...
g(1)
g(0)
g(N −1)
...
g(1)
0
g(N −1)
...
0
0
g(N −1)
]]]]]]]]]]]]
]
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
Dr{g}
[[[
[
x(0)
...
x(M −1)
]]]
]
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
x
(5.50)
For periodic signals, the convolution requires not the response of a system to one unit
impulse, but that of an periodic impulse sequence of unit impulses with a certain
period length Np. The impulse response then gives rise to the pulse response as shown
in Figure 5.18 (centre).
If the period length Np is greater than the length N of the impulse response g(n),
then the impulse response gp(n) is simply obtained from the string of impulse re-
sponses g(n) repeated after Np samples. If this is not the case, the individual impulse
responses overlap and the impulse response cannot be determined again from the im-
pulse response simply by fading out a period, see Figure 5.18 (below).
With a periodic input signal, the convolution sum (Equation 5.48) does not have to
be formed from n = −∞to ∞. Here, as with the Fourier series, one period is sufficient,
i.e.
y(n) =
Np−1
∑
ν=0
x(ν)gp(n −ν) =
Np−1
∑
μ=0
x(n −μ)gp(μ) ,
μ = n −ν
(5.51)
or in matrix form:
[[[[[
[
y(0)
y(1)
...
y(Np −1)
]]]]]
]
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
y
=
[[[[[
[
gp(0)
gp(Np −1)
⋅⋅⋅gp(1)
gp(1)
g(0) ⋅⋅⋅
⋅⋅⋅gp(2)
...
...
...
gp(Np −1)
gp(Np −2)
⋅⋅⋅gp(0)
]]]]]
]
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
Zykl{gp}
[[[[[
[
x(0)
x(1)
...
x(Np −1)
]]]]]
]
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
x
.
(5.52)