Page 348 - Programming Microcontrollers in C

Digital Signal Processor Operations 333

a function were delivered into the input of a filter, the output of the
filter would be the filter weighting function.
As we move to the digital realm, the Dirac Delta function is re­
placed by the Kroniker Delta function. This simple function, δ , is 0
k
for all values of k except for k = 0 where its value is 1. If this func­
tion is sent through a digital filter, the output observed is the weighting
function of the filter.
In both cases, analog and digital, the frequency response of the
filter is the Fourier transform of the filter weighting function. This
duality between the frequency response and the time domain response
makes it possible to design filters to accomplish what is really de­
sired. Usually, the filter specification is best established in the
frequency domain. The designer knows what frequencies are to be
passed or rejected by the filter. There has been a long history in the
field of passive network synthesis devoted to the “approximation
problem.” How does one specify a filter to meet accurately a desired
frequency response? This problem has led to many sophisticated
approaches to the specification by mathematics of a frequency re­
sponse to meet the system need. More important, these frequency
responses have a nature that can be realized by a finite collection of
passive electrical components—resistors, capacitors, and inductors.
In other words, these frequency responses are realizable.
A series of mathematical transformations exist that can be used
to transform frequency responses directly to filter weighting func­
tions. We will not go into these transformations here, but will refer
you to Elliott for practical means to specify the weighting functions
7
for digital filters. A more general text on this subject is by Antoniou. 8
The time domain response calculation is called a convolution. If there is
a signal x applied to a filter with a weighting function h there will be an
k k
output from the filter at each time sample, and this output will be called
y . The relations between these parameters are given by the equation
k
n–1
y = x h
k ∑ k–i i
i=0

7 Elliott, Douglas F., Handbook of Digital Signal Processing and Applications:
Academic Press Inc. 1987
8 Antoniou, Andreas, Digital Filter Analysis and Design: McGraw Hill, 1979