Signal Processing - July 2016 - 101

More discussion about
parameters a, n,and b
The purpose of the weight coefficient a
in (11), (13), (15), and (17) is to trade off
the performance between compensation
in passband and attenuation in stopband.
Usually, the initial solution of a lowpass
filter h(n) designed by the MATLAB
'fdatool' function (or other computer-aided design software) has a good
attenuation performance in stopband,
therefore, the CCF design should place
emphasis on the compensation performance, in other words, a should be closer to 1, for possible initial choices in the
range 0 < a < 1. To further determine
the value, a standard binary search can
be employed. According to our filter
design experience, several attempts
(usually, fewer than ten) are enough to
find out an acceptable a, as seen in
Tables 1-4.
Generally, the step size n used in the
steepest descent algorithm affects the
convergence time for the iterative process. For a given cost function, a proper
n should be considered. A standard
binary search can still be used to find
out a proper step size n. In the initial
attempt, n should be small enough to
guarantee the convergence of the iterative process. When considering that the
filter design is an offline task, the result
precision is more important than the
time to find the solution. All of these
reasons point toward choosing a small
n just like in the section "Weighted Cost
Function Considering Decimation."
When choosing a weighting function in the section "Weighted Cost
Function Considering Decimation,"
the exponential function w(~) = b~
was chosen because of its simplicity. How should one determine b?
For example, in the best results of
Table 3, there are two local maxima:
2.8325 × 10 −4 dB at 0 Hz and 4.1974

0.24
Magnitude (dB)

0.18
0.12
0.06
0.00
-0.06
-0.12
-0.18
-0.24
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
Frequency (kHz)
FRM of Original CIC Filter
FRM of CCF
Total FRM Before Decimation
Total FRM After Decimation

FIgure 5. The FRMs of CIC filter, CCF, and total FRM of compensated CIC filter before and after
decimation in section "Weighted Cost Function Considering Decimation."

4
Magnitude (×10-4 dB)

CIC filter after decimation is smaller
than that before decimation and 2)
compared with the best results in the
section "Direct Cost Function Considering Decimation," after decimation, the
maximum ripple of the compensated
CIC filter in the passband is reduced to
about 3.8 × 10 −4 dB.

2
0
-2
-4
Total FRM Before Decimation
Total FRM After Decimation

-6
-8

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
Frequency (kHz)

FIgure 6. Total FRMs of compensated CIC filter before and after decimation in the section "Weighted Cost Function Considering Decimation."

× 10 −4 dB at 3.43 KHz, i.e., 2.8325
× 10 −4 dB at ~ = 0 rad and 4.1974
× 10 −4 dB at ~ = 0.3367 rad. To make
the two local maxima as same as possible, we set w(0.3367)/w(0) = b0.3367 =
(4.1974 × 10 − 4 /2.8325 × 10 − 4) b ≈
1.4819 b, where b = 2.5 is an empirical constant. Then we obtain b ≈ 48.9.
Finally, after some fine-tuning, b is set
as 47, and now the two local maxima
are almost same. Without carrying out
the design method in the section "Direct
Cost Function Considering Decimation,"
the local maxima are not available. In
this case, since b is always greater than
1, we have to search a proper b with
the standard binary search. Incidentally, other weighting functions with a
IEEE Signal Processing Magazine

July 2016

similar weighting behavior can also be
applied here.
Note that, a, n, and b can be determined independently because there is
almost no coupling among them. If a
priority order is asked for searching or
determining their values, our advice is
first n, then a, and last b.

Conclusions
In this article, a novel combined compensation filter design method is presented
for the compensation of the CIC filter in
a Δ-R ADC. It provides the possibility to
compensate the CIC filter with a relatively low-order FIR filter without the loss of
performance. Compared with those
results without the compensation filter, a
101

Table of Contents for the Digital Edition of Signal Processing - July 2016