Biosignal Processing - Introduction to Biomedical Engineering - page 711

EXAMPLE PROBLEM 11.21
An electromyographic (EMG) signal contains energy within the frequencies 25 and 100 Hz.
Design a filter to remove unwanted noise.

Solution
We need to design a band-pass filter with passband frequencies 25 and 100 Hz. First determine
the cutoff frequencies in rad/s. Since W c ¼ 2p f c ,

W 1 ¼ 50p

W 2 ¼ 200p

Next, we find the impulse response of the corresponding low-pass and high-pass filters:

h HP ð t Þ¼dð t Þ W 1
p

sinc ð W 1 t Þ¼dð t Þ 50 sinc 50p t ð Þ

h LP ð t Þ¼ W 2
p

sinc ð W 2 t Þ¼ 200 sinc 200p t ð Þ

The band-pass filter impulse response is

h BP ð t Þ¼ h BP ð t Þ * h LP ð t Þ¼ dð t Þ ½ 50 sinc 50p t ð Þ * 200 sinc 200p t Þ ð

The described ideal analog filters provide a conceptual reference for various filter design
applications. In practice, real analog filters cannot be implemented to achieve the strict
specifications of the ideal filter because the impulse response of ideal filters is of infinite
duration (extends from ). Thus, the ideal filters require that one integrate over

an infinite amount of time to produce an output. Typically, most analog filters are designed
with simple electronic circuits. Various approximations to the ideal low-pass, high-pass,
and band-pass filter can be derived that are well suited for a variety of applications,
including signal analysis of biomedical signals.

1 to þ1

11.6.5 Digital Filters
Digital systems are described by difference equations, just like analog systems are
described by differential equations. Difference equations are essentially discretized differ-
ential equations that have been sampled at a particular sampling rate. The general form
of a real-time digital filter/difference equation is

M N

y ð k Þ¼

0 b m x ð k m Þ 1 a m y ð k m Þ

ð 11 : 44 Þ

m ¼ m ¼

where the discrete sequence x ( k ) corresponds to the input and y ( k ) represents the output

sequence of the discrete system. For instance, if M ¼ 2 and N ¼ 2, then

y ð k Þ¼ b 0 x ð k Þþ b 1 x ð k 1 Þþ b 2 x ð k 2 Þ a 1 y ð k 1 Þ a 2 y ð k 2 Þ