Discrete and Digital Signals and Systems - Digital Signal Processing: An Introduction with MATLAB and Applications - page 102

0

−20

−40

−60

Ω

0

Ω c π 2 π

Fig. 2.26 Frequency response of a LPF designed by remez (dotted curve for ad-hoc frequency-
sampling design)

2.6.6 Applications of FIR Digital Filters

2.6.6.1 Communication Channel Equalization

Telephone and wireless channels often behave like band-limited filters with fre-
quency response H(f). This filtering is in general an undesirable frequency
dependent distortion which needs to be eliminated - it can be removed via the use
of an equalizer as explained below.
The transfer function of the channel can be estimated by sending a training
signal [e.g., a unit pulse function d(n) along the channel and then measuring the
impulse response h(n) and the corresponding transfer function H ð f Þ¼Ff h ð n Þg ].
An equalizer (or inverse filter) with a frequency response H e (f) = 1/H(f) is then
applied to counteract the channel distortion. Normally one chooses an FIR filter
(also called a transversal filter), and the frequency sampling approach is often used
to design H e (f).

2.6.6.2 The Moving Average Filter

Sometimes it is necessary to smooth data before making a decision or interpre-
tation about that data. Averaging is a commonly used smoothing technique and
this averaging can be implemented with simple FIR filters. For example, the stock
market prices fluctuate from day to day, and even sometimes hour to hour, To
observe trends some form of smoothing or averaging is necessary. Typically, one
takes an average of the stock price over several days before deciding the actual
trend of prices. Note that the global average is misleading in these cases and
cannot be used—rather, local or moving average should be used. (See Fig. ( 2.27 )).
Since data x(n) is assumed to be continuously flowing in for the stock market
scenario, local averaging is done at each instant n. If, for example, averaging is
performed over three consecutive samples, the output becomes: