Discrete and Digital Signals and Systems - Digital Signal Processing: An Introduction with MATLAB and Applications - page 92

Fig. 2.16 A finite impulse
response (FIR) digital filter
and its implementation (with
N = 2M ? 1) being odd

Digital FIR Filter

x ( n ) h ( n ) y ( n )

y ( n ) = x ( n ) * h ( n )

z −1 z −1 z −1 x ( n )

h o h 1 h 2 h 2M

y ( n )

ifft y ð n Þ¼ Y 1 ð k Þ:

Y ð k Þ¼ X ð k Þ H ð k Þ!

remembering that it is necessary to zero-pad both the input signal and impulse
response up to N x ? N h - 1 samples so as to avoid undesirable circular convo-
lution effects.

2.6.4.3 FIR Filtering of Long Data Sequences

In many applications (e.g., speech processing), the signal x(n) can be very long. In
this case, one cannot perform FIR filtering using the above technique efficiently
since the fft computation time would be very large. Also, saving all of the input
data would put a significant strain on the computer memory. Furthermore, real-
time processing would be impossible since it would be necessary to wait until the
end of the signal to start processing. Instead one can exploit the fact that an FIR
filter is a linear system, and obeys the superposition property. Hence, one can
partition the data record x into a sequence of blocks, each of relatively small length
K, noting that K [[ M, M being the filter length. Each of these blocks can be
filtered separately and the resulting outputs combined to yield an overall output.
The process is called the overlap-add method and is illustrated in Fig. ( 2.17 ).
MATLAB the above overlap-add method could be implemented on MATLAB
using the instruction fftfilt(x,h) , where x is the signal and h is the FIR filter
impulse response.

2.6.4.4 Pole-Zero Diagram and Stability of FIR Filters

The FIR transfer function in ( 2.21 ) can be written as:

½ h 0 z N 1 þ h 1 z N 2 þþ h N 1 ;

1
z N 1

H ð z Þ¼ Y ð z Þ= X ð z Þ¼

which shows that an FIR filter with impulse response of length N has N - 1 zeros
and a multiple pole of order N - 1 at the origin (z = 0) (Since this multiple pole is
inside the unit circle, it poses no stability problems).