5.3 Methods for Analysis and Processing of Discrete Biosignals

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response is not only discrete but also periodic as in the time domain. The relation-

ship between the time and frequency range of a periodic and discrete signal and its

spectrum is determined by the discrete Fourier-transformation(DFT) according to

g(n) = ¹

N

N−1

∑

m=0

G(m)e^{j2π nm}

N ,

G(m) =

N−1

∑

n=0

g(n)e^{−j2π nm}

N

(5.111)

where G(m) is the sampled value of the desired transfer function at the mth digit and

g(n) is the sampled value of the impulse response at the nth digit. N corresponds to the

number of samples per period in the time domain as well as the number of frequency

points in the frequency domain.

In the filter synthesis, the periodic impulse response g(n) is determined with the

help of the DFT from the values of the transfer function G(m) sampled per period and

interpreted as the coefficient ci of a non-recursive (FIR) filter (see Figure 5.35 and Equa-

tion 5.102). If, in addition, the filter is to have a constant group delay, the magnitude of

the impulse response g(n) must be symmetrical to ^N

2 ^{if N is even, or N−1}

2

(for N odd),

ie:

|g(n)| = |g(N −n)|

n = 0, . . . ,

{

{

{

N

2

for N even

N−1

2

for N odd

.

(5.112)

Here the group delay is ^NT

2 ^{, and the magnitude of the transfer function can be calcu-}

lated according to Equation 5.105 to Equation 5.108.

If the impulse response g(n) is to be real, it follows from the Fourier transform

that the real part of the spectrum must be mirror symmetrical to the coordinate origin

and the imaginary part must be point symmetrical to it. However, since the spectrum

repeats periodically with multiples of the sampling frequency fa/2, the spectral part

from −fa/2 to 0 is equal to the spectral part from fa/2 to fa. However, the DFT does not

consider spectral components for negative frequencies but for the whole range from 0

to the sampling frequency fa. Therefore, the range from fa/2 to fa can be obtained from

the symmetry condition for the spectral range at the negative frequencies. It therefore

follows:

G(N −m) = G^∗(m)

m = 0 to N ,

(5.113)

whereby, according to Equation 5.111, the back transformation in the DFT can be sim-

plified into the time domain. Thereby, with the help of splitting the DFT sum into two

equal halves with bracketing out the function factor e^{j2π N}

2

n

N in the two partial sums,

the later substitution m^󸀠= N −m and from the symmetry condition according to Equa-