Understanding IIR (Infinite Impulse Response) Filters - An Intuitive Approach
by Dan Lavry, Lavry Engineering
People less familiar with mathematics or network theory, often tend to assume that digital signal processing is beyond
their understanding. For engineers, mathematics is a tool. Most problems are kept within some range of "common sense"
understanding. This article present some of the more intuitive aspects of IIR filters.
Introduction:
FIR filters offer great control over filter shaping and linear phase performance (waveform retention over the pass band). At
times, the required number of taps (coefficients) exceeds available hardware compute power, or acceptable input to output
time delay. Another type of digital filter, a close cousin to analog filters, is the IIR. Analog filters utilize analog components
to filter continues signals and IIR's use numerical processing to filter sampled data signals, yet they are both based on
"pole and zero" theory, yielding Butterworth, Chebyshef, Bessel and the other familiar filter response curves.
IIR filters offers a lot more "bang per tap" then FIR's. Computational efficiency and relatively short time delay make them
desirable, especially when linear phase is of lesser importance.
FIR's are "forward" structures. Signal samples are sent forward from tap to tap. M
any taps translate to long delays, and
increase in the
number of arithmetic operations. IIR's use feedback. The signal path is no longer a straight delay. Much of
the filtering action depends on a feedback path. P
ortions of the output are feedback to be recomputed "over and over" by
the same few coefficients
. Each feedback tap contributes to shaping of many samples without the cost of additional
arithmetic.
Digital feedback:
A digital oscillator may be constructed by means of simple arithmetic. let us set an initial output value to say 1. If we
change the output by multiplying it by -1 at each sample clock time, we end up with a sequence of 1, -1,1,-1 and so on.
Let us change the multiplying coefficient to -1/2. The output sequence becomes 1,-1/2,1/4,-1/8.
.. yielding a wave form
known as
"exponentially decaying oscillations", or "damped ringing". If we choose a positive coefficient of 1/2, the
sequence becomes
1,1/2,1/4,1/8.
.. producing an exponential decay. Both ringing and decay remind us of behavior of some physical
phenomena, such as encountered in tuned circuits and capacitor discharge circuits, thus providing some basis for
emulating analog circuits by computational means.
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1
0.8
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1
1.2
coefficient = -1/2
V1
n
n
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0.2
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coefficient =
1/2
V2
n
n
Speed of decay is is controlled by the coefficient values. Comparing the plots bellow (coefficient values -.8 and .8) to the
previous plots
(coefficient values -.5 and .5)
shows slower decay.
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