U. Glaeser

FIGURE 27.21 Symmetrical FIR filter.
Using modern DSP’s FIR filters can easily be implemented with the MACD instruction, which realizes
multiplication, accumulation, and data move. When used with repeat next instruction (RPT) MACD
becomes a single-cycle instruction once the RPT pipeline is started. The theoretical maximum length L
of the FIR filter can be computed as
© 2002 by CRC Press LLC
(27.44)
where T s is the sampling period, T c is the instruction cycle, and k is the number of converted channels.
Assuming sampling rate of 48 ksamples/s, 25 ns instruction cycle of the DSP and 6 output channels, we
can realize FIR filters with the maximum length of about 138 [25].
Another method for the implementation of FIR filters in DSPs consists in the use of two further new
features of modern DSPs, namely the circular addressing and the FIRS instruction. This possibility can
be effectively used, if the filter has a symmetric impulse response h(n) (cf., Fig. 27.21), i.e., if the filter
output signal is given by
(27.45)
The FIRS instruction can add two data values (stored in a circular buffer) in parallel with the multiplication
of this result by a filter coefficient. Once the repeat pipeline is started, this instruction becomes
also a single-cycle instruction. A computational complexity is in this case reduced by half and makes it
possible to realize FIR filters with the double length as compared with the programming technique
previously described[48].
IIR Filters
x(
k)
+
-1
z
-1
z
-1
z
+ + +
-1
z
× () 0 × h () 1 × h()
2
×
h h(
N / 2 -1)
N/2−1
∑
+
L
=
Although FIR filters have important advantages as linear phase, stability, robustness, easy design, and
implementation, their infinite impulse response (IIR) counterparts will have complexity (order) reduced
by some orders of magnitude. Therefore, IIR filters are advantageous over and above FIR filters in
particular applications. IIR filters are typically designed starting with an analogue reference filter and
then performing the bilinear transformation [8]. Denote by H(s) transfer function of the analog reference
filter. Then the resulting IIR filter transfer function H(z) is calculated as
Ts -------
Tck -1
z
-1
z
y(
k)
y( n)
= h( n)
{ x( n– k)
+ x[ n– ( N– 1 – k)
] }
k=0
H( z)
=
H s
() 2 z 1
s = ----
Ts –
---------z
+ 1
-1
z
(27.46)