200

|

5 Methods for Discrete Processing and Analysis of Biosignals

5.

The transformation equations between the z- and p- domains are then obtained

as.

p = ¹

Ta

ln z ≈²

Ta

z −1

z + 1

und

z ≈

1 + p/ ²

Ta

1 −p/ ²

Ta

,

(5.96)

resp.

G(z) ≈G (p = ²

Ta

z −1

z + 1^{) .}

Explanatory example

The analogue RC low-pass already used in the above pulse invariance method accord-

ing to Figure 5.39 with a cut-off frequency fg = 200 Hz and the component values

R = 800 kΩand C = 1 nF according to Figure 5.39 is to be replaced by a digital low-

pass filter with the same cut-off frequency and a sampling frequency of 1 kHz accord-

ing to the bilinear transformation method. The associated transfer function G(p) of

the low-pass filter with A1 = p1 = −ωg =

1

RC ^{is given by}

G(p) =

A1

p −p1

=

ωg

p + ωg

.

(5.97)

After inserting the bilinear transformation according to Equation 5.96, it follows:

G(z) ≈G (p = ²

Ta

z −1

z + 1^{) =}

ωB

2

Ta

z−1

z+1 ^{+ ωB}

=

ωB

2fa+ωB ^{(1 + z)}

ωB−2fa

2fa+ωB ^{+ z}

= ^{c0 + c1z}

d0 + z

,

(5.98)

and further by comparing the coefficients:

c0 = c1=

ωB

2fa + ωB

=

2π ⋅200 Hz

2 ⋅1 kHz + 2π ⋅200 Hz⁼

π

π + 5 ^{= 0.385869 . . .}

(5.99)

d0

= ^{ωB −2fa}

2fa + ωB

= ^{2π ⋅200 Hz −2 ⋅1 kHz}

2 ⋅1 kHz + 2π ⋅200 Hz^{= π −5}

π + 5 ^{= −0.22826 . . . .}

(5.100)

However, digital filters can also be designed directly in the discrete-time domain

without a diversion via the design of analogue filters. Of the available methods, two

will be described in more detail here: i) the direct discrete-time synthesis using the

window method, representative of methods for the design of non-recursive (FIR)-filters

and ii) the frequency sampling method as an example for the general synthesis of re-

cursive (IIR-) or non-recursive (FIR)-filters. Additional procedures can be taken from

various publications in this field (e.g. B. [56, 62, 70]).

Direct discrete-time synthesis using window method

A non-recursive digital filter N-th order (FIR filter) according to Figure 5.35 has no feed-

back and according to Equation 5.79, because of which the feedback coefficients di are

all equal to zero, has the transfer function

G(z) = ^{c0 + c1z + ⋅⋅⋅+ cN−1zN−1 + cNzN}

z^N

.

(5.101)