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282 Chapter 6 IMPLEMENTATION OF DISCRETE-TIME FILTERS
%xlabel(’Digital Frequency in \pi units’,’fontsize’,10);
ylabel(’Magnitude Response’,’fontsize’,10);
title(TITLE,’fontsize’,10,’fontweight’,’bold’);
subplot(2,2,3); plot(w/pi,magHhat1,’r’,’linewidth’,2); axis([0,1,0,0.7]);
xlabel(’Digital Frequency in \pi units’,’fontsize’,10);
ylabel(’Magnitude Response’,’fontsize’,10);
title(TITLE1,’fontsize’,10,’fontweight’,’bold’);
subplot(2,2,4); plot(w/pi,magHhat2,’r’,’linewidth’,2); axis([0,1,0,0.7]);
xlabel(’Digital Frequency in \pi units’,’fontsize’,10);
ylabel(’Magnitude Response’,’fontsize’,10);
title(TITLE2,’fontsize’,10,’fontweight’,’bold’);
The plots are shown in Figure 6.33. The top row shows plots for the direct
form, and the bottom row shows those for the cascade form. As expected, the
magnitude plot of the direct form is severely distorted for 16-bit representation,
while those for the cascade form are preserved even for 11-bit word length. □
□ EXAMPLE 6.28 An 8th-order bandpass filter was obtained using the elliptic filter design approach.
This and other design methods will be discussed in Chapter 8. The
MATLAB functions needed for this design are shown in the following script.
This design produces direct-form filter coefficients b k and a k , using 64-bit
floating-point arithmetic, which gives the precision of 15 decimals and hence
can be considered as unquantized coefficients. Table 6.1 shows these filter
coefficients.
Represent the unquantized filter coefficients using 16-bit and 8-bit word
lengths. Plot the filter log-magnitude responses and pole-zero locations for both
the infinite and finite word-length coefficients.
TABLE 6.1 Unquantized IIR filter coefficients used in Example 6.28
k b k a k
0 0.021985541264351 1.000000000000000
1 0.000000000000000 −0.000000000000004
2 −0.032498273955222 2.344233276056572
3 0.000000000000000 −0.000000000000003
4 0.046424673058794 2.689868616770005
5 0.000000000000000 0.000000000000001
6 −0.032498273955221 1.584557559015230
7 0.000000000000000 0.000000000000001
8 0.021985541264351 0.413275250482975
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