250

|

6 Applications and Methods in Biosignal Processing

Fig. 6.23: Amount-frequency-response |ATP(f)| for the low-pass part of the bandpass according to

Figure 6.22.

The high pass is realised by subtracting the output of a first order low pass from the

output of an all pass, which delays the signal by an additional 16 samples. This corres-

ponds to 80 ms at a sampling frequency of 200 Hz. It has no gain (or a gain of 1) and a

cut-off frequency of approximately 5 Hz, see Figure 6.24. Thereby one obtains for the

transfer function AHP(z)

AHP(z) = ^{−1/32 + z−16 −z−17 + z−32/32}

1 −z⁻¹

(6.15)

and the corresponding magnitude frequency response

|AHP(ωTa)| = ^√1 + ( ^{sin (16ωTa)}

32 sin (ωTa/2)⁾

2

−

sin (16ωTa)

16 tan (ωTa/2) ^.

(6.16)

Analogous to the procedure for the low-pass filter, the following is obtained for the as-

sociated algorithm for the output signal yHP(n) as a function of the input signal xHP(n):

yHP(n) = yHP(n −1) −xHP(n)/32 + xHP(n −16) −xHP(n −17) + xHP(n −32)/32 .

(6.17)

Differentiate

The QRS complex in the ECG has steep edges, especially at the R wave. In order to

emphasise these, the output signal of the bandpass filter is now digitally filtered using

a fifth-degree non-recursive filter according to the numerical approximation

̃yDif(n) = (2xDif(n + 2) + xDif(n + 1) −xDif(n −1) −2xDif(n −2))/8

(6.18)

and differentiated. However, since the digital filter must be causal in a real-time ap-

plication, a delay of 2 clock instants (equivalent to 2 ⋅5 ms = 10 ms) must be applied