5.3 Methods for Analysis and Processing of Discrete Biosignals

|

187

power density

Spectogram with narrow window

signal

Fig. 5.27: Spectrogram as in Figure 5.26 but with a narrow Gaussian window with a temporal width of

0.5 seconds.

The discrete Fourier transform according to Equation 5.23

F(l) =

N−1

∑

i=0

fp(i)e^−j2πil/N

=

N−1

∑

i=0

fp(i) cos (^2πil

N ^{) + j}

N−1

∑

i=0

fp(i) sin (^2πil

N ^{) ,}

can in fact also be understood as a filter bank, where for each frequency to be determ-

ined with fl = l⋅fa/N with fa = 1/Ta (sampling frequency) and l = 1, . . . , N−1 a filter is

used. This is because the calculation of the real and imaginary parts corresponds to a

cross-correlation (cf. subsubsection 5.3.1.2) between the signal fp(i) and the functions

cos(2πil/N) and sin(2πil/N) with

—Rxy1(l, m) := ¹

N

N−1

∑

i=0

fp(i)

⏟⏟⏟⏟⏟⏟⏟

x(i)

cos (^{2π(i + m)l}

N

)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

y1(i+m)

und

—Rxy2(l, m) := ¹

N

N−1

∑

i=0

fp(i)

⏟⏟⏟⏟⏟⏟⏟

x(i)

sin (^{2π(i + m)l}

N

)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

y2(i+m)

,

m = 0.

The cross-correlation can equally also be interpreted as a filtering of two signals, since

a correlation corresponds to a convolution, see subsection 5.3.2. The transformations

are therefore "filter banks" that apply a specific filter to the signal under investigation

for each pixel (e.g. a specific frequency). Unfortunately, the wavelet-filter", such as