5.3 Methods for Analysis and Processing of Discrete Biosignals

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time t / ms

ideal low pass g(t)

window function w(t)*100

Fig. 5.42: Impulse response g(t) of the ideal low-pass and rectangular window w(t) enlarged by a

factor of 100 for better representation.

have (cf. Figure 5.42). As with the impulse invariance method, the impulse response of

the associated digital FIR filter at the sampling times nTa (n = 0 to 8) should have the

same values as the impulse response of the analogue filter, except for a scaling factor

of the magnitude. This corresponds to a rectangular window w(t) which cuts off the

values of the impulse response of the ideal filter according to N ⋅Ta = 8 ⋅1 ms = 8 ms

(see Figure 5.43). This gives the filter coefficients to

ci = g(i) = Ta gan(iTa) = ^{sin(ωgTa(i −N/2))}

π(i −N/2)

,

i = 0 to N .

(5.109)

Here, the analogue impulse response gan(t) is additionally multiplied by the sampling

interval Ta so that the impulse response of the digital filter becomes dimensionless.

Furthermore, for our example it follows:

ci = ^{sin(0.4π(i −4))}

π(i −4)

,

i = 0 bis 8 .

Thus, for the filter coefficients ci:

c0 = c8 = −0.07568 ,

c1 = c7 = −0.06237 ,

c2 = c6 = 0.09355 ,

c3 = c5 = 0.3027 ,

c4 = 0.4 .

Since the filter order- N is even and the impulse response is mirror symmetric to to, it

follows according to Equation 5.105 for the transfer function:

G(z = e^jωTa) = {c4 + 2

3

∑

i=0

ci cos [ω(4 −i) ms]} e^{−jω4 ms} .

(5.110)

From this example it can be seen that even with a higher filtering order N not nearly

the rectangular magnitude frequency response of an ideal low-pass filter is achieved.