Mathematical Notations
A.1 Moving Median
Moving median .lters are frequently used for outlier removal in signal pro-cessing. The moving median works similar to FIR or IIR .lters, but extracts the median rather than the sum. Median .lters can have individual weight for each tap, but this is not exploited here.
mm(x, n, K)= mediannk=n.K [x(k)] (A.1)
A.2 Windowed RMS
A windowed RMS is used for estimating signal energy at a limited segment of the input signal. A windowed RMS with window size K of a signal with length N gives an output of length N . K.
1 n
wrms(x, n, K)= (x(k) . xˉ)2 (A.2)
K
k=n.K
A.3 Wavelets
The wavelet theory discussed here is intentionally cut short. For better un-derstanding of wavelet theory and applications, the reader should refer to the
relevant
literature,
such
as
[27],
[19]
and
[8].
131
A.3.1 Continuous Wavelet Transform
Wavelet theory shows that a signal f(t) can be represented as a weighted sum of functions ψj,k(t) (Eq.
A.3).
The
integer
indices
j and k represents scaling and translation of some arbitrary wavelet base function ψ(t) (Eq. A.4).
The better choice of ψ(t) depends on the application. For de-noising and compression, a wavelet is chosen which con.nes noise and signal in di.erent parts of the dj,k coe.cient
matrix
(Eq.
A.5).
For
event
detection,
a
wavelet
is chosen which gives the event an easily recognizable signature.
f(t)= dj,kψj,k(t) (A.3) j,k
ψj,k(t)=2j/2ψ(2jt . k) (A.4)
dj,k =<f(t),ψj,k(t) > (A.5)
The above de.nitions will for a .nite length signal give a coe.cient matrix with .nite length in the translation (k) dimension. The number of possible scalings (j) is however in.nite. This means that reconstructions is not pos-sible unless coe.cients for an in.nite number of scales are computed.
A.3.2 Discrete Wavelet Transform
The Discrete Wavelet Transform (DWT) overcomes this by expanding over only a limited number of scales, leaving the remaining part of f(t) in an approximate vector ak. This way the detail matrix dj,k and the approximate vector ak will together hold all information necessary to reconstruct the orig-inal
signal
(Eq.
A.6).
The number of scales must be chosen so that some intelligent partition of signal features is obtained between the approximate vector and the di.erent scale vectors of the detail matrix.
f(t)= akφ(t . k)+ dj,kψj,k(t) (A.6) k j,k
<ψ(t),φ(t) >=0 (A.7)
The wavelet function ψ(t) and the scaling function φ(t), used for extract-ing the approximate, must be orthogonal (Eq. Eq. A.7).
Consequently, the DWT can only be calculated using wavelet base functions for which an orthogonal scaling function exist.
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