Presented at ICSP2018(14th International Conference on Signal Processing) Also served as a session chair of Session 1A1. Date : Aug 13, 2018 Place : Beijing, China DOI) 10.1109/ICSP.2018.8652330 URL) https://ieeexplore.ieee.org/document/8652330 [ URL of the paper/preprint ] https://www.researchgate.net/publication/327130082_A_Method_to_Derate_the_Rate-Dependency_in_the_Pass-Band_Droop_of_Comb_Decimators

1. GCT Semiconductor, Inc.http://www.gctsemi.com
A Method to Derate the Rate-Dependency in the
Pass-Band Droop of Comb Decimators
Ealwan Lee
GCT Semiconductor, Inc.
Session : Digital Signal Processing (01A1-04)
Aug 13, 2018 (13:30 ~ 15:20)

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GCT Semiconductor, Inc.ICSP 2018
Table of Contents
 Introduction (pp.2 ~ 3)
 Problem description
✓ Comb decimator
✓ Pass-band droop
 Decomposition of the pass-band droop
✓ Dependency on order(N) and rate(M)
 Proposed method : 3-tap FIR derating filter
 Idea (p. 3)
 Mathematical formulation & derivation (pp.4 ~ 5)
 Characteristics
 Application : minimal pass-band change can be obtained in
 Filter sharpening of comb decimation filter (pp. 6 ~ 7)
 Bifurcate-zero comb decimation filter (pp.8 ~ 9)
 Compensator (p. 10)
✓ More brief closed-form expression
 Conclusion (p. 11)

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1. Description of the Problem
 Cascaded-Integrated-Comb decimator
 Used as a front-end decimator in multi-rate IFIR filter design.
 Pass-band droop is dependent on both filter order(N) and decimation rate(M).
✓ Enforces re-design of the subsequent filters, compensation filter, for different M.
H3,4(z)
H3,(z)
H3,4(z)
C/2 H3,(z)
/2 (Hz)
/2 (Hz)
deviation of
pass-band
1 - C/2
+M/2
𝐻 𝑁,𝑀 𝑧 =
1
𝑀
⋅
1 − 𝑧−𝑀
1 − 𝑧−1
𝑁
sincN function
in frequency domain

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2.1. Basic Concept of the Proposed Method
 Does not try to compensate the overall pass-band droop
 It is the role of conventional compensator, CN,M,L(z).
 Keep the pass-band deviation only constant against the change of M.
 The first order dependency on Τ𝜔 𝑀 2
in denominator should be eliminated.
𝐻 𝑁,𝑀 𝑒 𝑗𝜔/𝑀
=
sin 𝜔
2
𝑀 ∙ sin 𝜔
2𝑀
𝑁
=
ൗsin 𝜔
2
𝑁 𝜔
2
𝑁
ൗsin 𝜔
2𝑀
𝜔
2𝑀
𝑁
≈
ൗsin 𝜔
2
𝑁 𝜔
2
𝑁
1 − 1
3!
∙ 𝜔
2𝑀
2
+ ∆
𝑁
≈
ൗsin 𝜔
2
𝑁 𝜔
2
𝑁
1 − 𝑁
3!
∙ 𝜔
2𝑀
2
+ ∆
𝑑𝐻 𝑁,𝑀 = 𝐻 𝑁,𝑀(𝑒 𝑗
𝜋
2𝑀) − 𝐻 𝑁,32(𝑒 𝑗
𝜋
2∙32) Reference point = 32 ~ 
As M ,
pass-band droop 
deviation from the reference 
Taylor expansion
and
1st order approx.

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2.2. Derating Filters
 3-tap FIR symmetric filter is adopted for derating filter, DN(z).
 Unlike conventional compensation filter, derating filter is located in integral stage.
 Filter tap coefficient should be rational to be implemented in integer arithmetic.
 Compensation Filter becomes insensitive to the changes of M.
z-1 z-1
DN(z) M
+ +
- -
z-1 z-1
X(z) Y(z)
CN,L (z)
1 − 𝑧−1 −𝑁
1 − 𝑧−𝑀 −𝑁
𝑀
𝐷 𝑁 𝑧 =
1 + 𝑏 𝑁 ⋅ 𝑧−1
+ 𝑧−2
2 + 𝑏 𝑁
𝐷 𝑁 𝑒 𝑗𝜔/𝑀
=
𝑒−𝑗𝜔/𝑀
2 + 𝑏 𝑁
∙ 𝑏 𝑁 + 2 ∙ cos 𝜔 ∙ 𝑀
≈
𝑒−𝑗𝜔/𝑀
2 + 𝑏 𝑁
∙ 𝑏 𝑁 + 2 −
𝜔2
𝑀2
+ ∆
L

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2.3. Derivation of the Filter Tap Coefficient
 bN of DN(z) is chosen to remove the first order dependency of the denominator.
 Range of the filter order for the the proposed method to be effective is 1~6.
✓ Constraint : DN(z) should be low-pass filter not to degrade the suppression level of the overall filter.
 The order of dependency on Τ𝜔 𝑀 2
becomes 2 from 1.
 It virtually flattens the curve of dGN,M compared with dHN,M.
𝐺 𝑁,𝑀(𝑧) = 𝐻 𝑁,𝑀(𝑧) ∙ 𝐷 𝑁(𝑧)
𝐺 𝑁,𝑀 𝑒
𝑗𝜔
𝑀 =
ൗsin 𝜔
2
𝑁 𝜔
2
𝑁
1 − 𝑁
3!
∙ 𝜔
2𝑀
2
+ ∆
∙
𝑒−
𝑗𝜔
𝑀
2 + 𝑏 𝑁
∙ 𝑏 𝑁 + 2 −
𝜔2
𝑀2
+ ∆
≈
sin 𝜔
2
𝑁
𝜔
2
𝑁 ∙ 𝑒−
𝑗𝜔
𝑀 ∙
𝑏 𝑁
2 + 𝑏 𝑁
∙ 1 −
𝜔2
𝑀2
2
+ ∆
𝑏 𝑁 =
24
𝑁
− 2
Independent of M Order of dependency on M -2: 1 -> 2

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3.1. Comb Decimator with Filter Sharpening
 Suggested by A. Kwentus et al. in 2000[6].
 On the half-way to our work from the first CIC filter by E. B. Hogenauer in 1988[2].
 The order of proto-type comb decimator should be even order.
✓ Minimal order of comb decimator with filter sharpening = 6 (= 3*2).
 Derating is applied to each component apart.
M
X(z) Y(z)
M
-
+
z-1
3
1
1 − 𝑧−1 4
𝑧−1
1 − 𝑧−1 2 𝐷6(𝑧)
𝐷4(𝑧)
1 − 𝑧−1
𝑀
2
1 − 𝑧−1
𝑀
4
𝐹2.𝑀 𝑧 = 3 ∙ 𝑧−1
∙ 𝐻2,𝑀
2
𝑧 − 2 ∙ 𝑧−𝑀
∙ 𝐻3,𝑀
3
𝑧
2
Delay for impulse response alignment
0
0.05
0.1
0
0.05
0.1
0 1 2 3 4 5 6
0
0.05
0.1
𝑧−𝑀
∙ 𝐻2,𝑀
2
(𝑧) = 𝑧−𝑀
∙
1
𝑀
∙
1 − 𝑧−𝑀
1 − 𝑧−1
4
𝑧−1
∙ 𝐻2,𝑀
3
(𝑧) = 𝑧−1
∙
1
𝑀
∙
1 − 𝑧−𝑀
1 − 𝑧−1
6

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3.1. Comb Decimator with Filter Sharpening(cont’d)
 Significant change in overall filter impulse response.
 Both Pass-band droop and Pass-band deviation is significantly reduced.
 Close to that of 2nd order conventional CIC decimator.
Deepened null
with Multiple zeros
Flattened pass-band
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
-80
-70
-60
-50
-40
-30
-20
-10
0
Frequency (Hz)
Magnitude(dB)

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3.2. Modified Comb Decimator with Bifurcate Zeros
 Suggested by f. harris et al. in 2016[11]
 A simple method to distribute/bifurcate multiple stop-band zeros of comb decimator.
 Derating is applied to each component.
M
X(z) Y(z)
M
-
+
z-1
1
1
1 − 𝑧−1 𝑁−2
𝑧−1
1 − 𝑧−1 2
𝐷 𝑁(𝑧)
𝐷 𝑁−2(𝑧)
1 − 𝑧−1
𝑀
𝑁−2
𝐹3.𝑀 𝑧 = 𝑧−1
∙ 𝐻 𝑁,𝑀(𝑧) − 𝛼 ∙ 𝑧−𝑀
∙ 𝐻 𝑁−2,𝑀(𝑧)
1 − 𝑧−1
𝑀
2
𝛼
Delay for impulse response alignment
𝑧−𝑀
∙ 𝐻 𝑁−2,𝑀(𝑧) = 𝑧−𝑀
∙
1
𝑀
∙
1 − 𝑧−𝑀
1 − 𝑧−1
𝑁−2
𝑧−1
∙ 𝐻 𝑁,𝑀(𝑧) = 𝑧−1
∙
1
𝑀
∙
1 − 𝑧−𝑀
1 − 𝑧−1
𝑁
𝛼 = 1/64
0
0.05
0.1
0
0.05
0.1
0 1 2 3
0
0.05
0.1
0 1 2 3
N=3 N=2

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3.2. Modified Comb Decimator with Bifurcate Zeros(cont’d)
 Minor change in overall filter impulse response.
 No significant changes in pass-band characteristics(both droop and deviation).
 Monotonic LPF property (no stop-band zeros on unit circle) guarantees
 the property of bifurcate zeros in the stop-band.
 𝐷0 𝑧 = 0 + 1 ∙ 𝑧−1
+ 0 ∙ 𝑧−2
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
-80
-70
-60
-50
-40
-30
-20
-10
0
Frequency (Hz)
Magnitude(dB)
Bifurcated multiple zeros

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3.3. Compensator in Explicit Closed Form
 3-tap or 5-tap FIR filter is used for compensation filter in differentiator part.
 It behaves as a high pass filter in pass-band region.
 Generally, filter design program is re-run if M changes.
 Effect on the explicit closed form of maximally flat compensator.
 Suggested by A. Fernandez-Vazquez et al in 2012[13]
 Apparent dependency of the tap coefficient on M-2 exits in the formula.
 Adoption of DN(z) in integral stage simply removes the dependency and reduces word-length.
𝑐 𝑁,𝐿 𝑧 𝑀
= 𝑐0 + 𝑐1 ∙ 𝑧−𝑀
+ 𝑐2 ∙ 𝑧−2𝑀
𝑐0 = 𝑐2 = −
𝑁
32
∙
1 − 𝑀−2
1 − 2−2
𝑐1 = 1 − 𝑐0 + 𝑐2
ǁ𝑐0 = ǁ𝑐2 = −
𝑁
24
𝑀 → ∞
ǁ𝑐0 < 𝑐0 < 0

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Conclusion
 Investigation
 Extra pass-band droop is observed as a function of decimation ratio(M).
 Proposed Method
 Does not try to compensate the overall pass-band droop
✓ Add additional pass-band droop
 3-tap FIR filter group, DN(z), is proposed to derate the rate dependency of the pass-band.
✓ Integer arithmetic is still available for efficient hardware implementation.
 Application : Minimal Pass-Band Change can be obtained in
 Filter sharpening of comb decimation filter
 Bifurcate-zero comb decimation filter
 Compensator
✓ More brief closed-form expression for filter tap coefficients with reduced word-length.
 Implication
 Can be applied to most comb decimation filters if properly decomposed.