Signal Processing - January 2016 - 155

the prototype filter to meet our design
specifications in step 4.
Step 4
Check if the frequency response H cas2 (~)=
H sha2 (~) H sha1 (~) H pro (~) meets the det 2 or ~t p2
sign specifications. If either R
does not meet the design target, we first
make a one-step adjustment for ~ pl and
then manually adjust Rl so that H cas2 (~)
meets the design specifications. From Tat 2 = 3.2 dB
ble 3, we already know that R
and ~t p2 = 0.095r rad/sample for K 1 = 6
and K 2 = 3. In other words, the cut-off
frequency ~t p2 is 0.005r rad/sample bet 2 is 1.2 dB
low the design target, while R
larger than the design target. We first extend the resulting cut-off frequency by increasing ~ pl to 0.13r rad/sample and
then manually adjust Rl to around 4 dB.
t 2 and ~t p2 for various
The resulting R
choices of Rl = 5 are shown in Table 4.
Because the design specifications are just
met when Rl = 5 dB, we finally set the parameters of the prototype filter to Rl = 5
dB and ~ pl = 0.13r rad/sample.
Step 5
S u b s t i t u t e e jw = z i n t o H pro (~),
H sha1 (~), and H sha2 (~), to obtain
N

H pro ^ z h =

/ k = 0b (k) z -k
M
1 - / a ^ k h z -k
k=1

(17)

[TABLE 3] THE RELATION BETWEEN K 2 AND ASSOCIATED Rt 2
t p2 WHEN K 1 IS SET TO 6.
AND ~

5

Rt 2 (dB)
4.7

t p2 (rad/sample)
~
0.090 r

4

3.9

0.092 r

3

3.2

0.095 r

2

3.0

0.096 r

1

2.9

0.097 r

K2

[TABLE 4] THE RELATION BETWEEN Rl AND ASSOCIATED Rt 2
t p2 . K 1 IS SET TO 6 AND K 2 IS SET TO 3.
AND ~

3.8

Rt 2 (dB)
2.3

t p2 (rad/sample)
~
0.098 r

4.0

2.2

0.098 r

4.2

2.1

0.099 r

4.4

2.1

0.099 r

4.6

2.1

0.099 r

4.8

2.0

0.099 r

5.0

2.0

0.100 r

Rl (dB)

x (n) and y (n) . We can swap the order
of filtering because both filters are linear
and time invariant.
The trick shown in steps 1-6 lies in
first determining the filter orders and
then fitting the design specifications
through trimming the prototype filter frequency response. This is a simple yet efficient way to solve (16). Steps 1-3 try to

and
H sha ^ z h = 1 ! z -K ,

10

(18)
0

respectively.

y ^ nh = b ^0h x ^ nh + b ^1h x ^ n - 1h
+ g + b ^ N h x (n - N)
+ a ^1h y ^ n - 1h + a ^2h y ^ n - 2h
(19)
+ g + a^ M h y ^ n - M h
and
(20)

respectively. The input and output
sequence of each filter are denoted by

-10
Magnitude (dB)

Step 6
By inspection, we can express the timedomain equations for the prototype filter
and the shaping filter as

y ^ n h = x (n) ! x (n - K),

find a suitable shaping filters through an
initial guess for the prototype filter. After
the two shaping filters order are set, we
proceed to trim the prototype filter so
that the design specifications are met. If
we want to achieve a sharper transition
band, we can cascade more shaping filters. We present two examples regarding
the design of a LPF and a BPF. Note that

Chebyshev Type-1 Filter
Singly Cascaded Filter
Doubly Cascaded Filter

-20
-30
-40
-50
-60

0

0.5

1

1.5
2
ω (rad/sample)

2.5

3

[FIG5] The magnitude frequency responses of H cas1 (~) and H cas2 (~) for the design
specifications R = 2 dB and ~ p = r/10 rad/sample. Also shown in the figure is a
Chebyshev type-1 digital filter of the same specifications.

IEEE SIGNAL PROCESSING MAGAZINE [155] jANuARy 2016
Table of Contents for the Digital Edition of Signal Processing - January 2016
