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Kusch, Rüdiger: Decimation Lowpass Filters for Sigma-Delta Modulators - A Comparative
Study
Hamburg: Diplomica GmbH, 2002
Zugl.: Braunschweig, Technische Universität, Studienarbeit, 1998
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Abstract
The purpose of this thesis is to compare several filter topologies used for the
decimation of sigma-delta modulated digital signals. The goal is to present
optimized filter architectures with regard to an efficient VLSI implementation.
A fifth-order 1-bit sigma-delta modulator using local feedback techniques
will be considered as the front-end A/D converter. The subsequent digital
filter reduces the sampling rate by a factor of 32.
The decimation filter
must guarantee a narrow transition band between 0.5 and 0.55 and stopband
attenuation of 100dB.
Chapter 1 provides a brief introduction into the principles of digital
signal processing. The considerations are focused on FIR filters due to the
requirements for acoustic applications.
Chapter 2 illustrates the proposed overall structure and the design flow.
The objective of chapter 3 is to present the principles of oversampling data
converters using sigma-delta techniques. The 5V fifth-order -modulator
with 90dB dynamic range (SNR+THD) will be presented, which has been fab-
ricated in 1.2µm CMOS technology. For the sake of simplicity and robustness,
a 1-bit quantizer will be used.
Chapter 4 deals with typical hardware realizations of digital filters. Apart
from the "brute force" implementation of the multirate filter with identical
filters running in parallel, also the LUT-based approach for small filter orders
ii
will be presented. Due to the advantages of compact implementation, the
bit-serial approach and the bit-serial multiplier are investigated in detail.
In chapter 5 the straightforward one-stage multirate FIR filter will be
introduced. To satisfy the specifications, a 4096 tap lowpass FIR filter will
be designed.
The influence of coefficient quantization is investigated and
furthermore the "block scaling" method, to represent small values, is presented.
The single-stage implementation becomes the more unattractive the higher the
filter specifications are.
Chapter 6, therefore, focuses the investigations on cascaded structures. The
first stage is realized as a comb or sinc
K
filter and decimates by a factor of 8
or 4. The frequently used conventional comb filter will be used but also a new
architecture will be described. The new structure is based on the conventional
comb filter with filter sharpening techniques to improve the frequency behavior.
The unavoidable passband droop must be compensated for by the following
lowpass FIR filter. In order to compare several filter realizations, three examples
are considered. These are the comb-FIR cascade, the sharpened comb-FIR
cascade and the sharpened comb-half band filter cascade. Finally, the FIR
filter realization using periodically time-varying coefficients (FIR-PTV filter)
will be considered.
iii
Decimation Lowpass Filters for Sigma-Delta
Modulators - A Comparative Study
Thema der vorliegenden Studienarbeit ist der Vergleich und die Aufwandsab-
sch¨atzung verschiedener digitaler Dezimationsfilter f¨
ur den Einsatz bei A/D-
Wandlern nach dem Sigma-Delta Prinzip.
Den Anfang macht eine Einf¨
uhrung in die Grundlagen der digitalen Filtertech-
nik sowie der Sigma-Delta Modulation. Anschlieend werden die M¨oglichkeiten
der Harwareimplementierung prinzipiell vorgestellt. Im Hinblick auf eine VLSI
Implementierung, ist jeweils der Hardwareumfang abgesch¨atzt worden.
Als Referenz dient das unkaskadierte FIR Dezimationsfilter. Die hohen An-
forderungen, ein schmales ¨
Ubergangsband (0.5; 0.55) und eine Sperrd¨ampfung
von 100dB, machen ein Filter der Ordnung 4096 n¨otig. Das Frequenzverhalten
wurde mit Routinen aus den Matlab Toolboxen bestimmt. Es ist der Ein-
fluß einer Koeffizientenquantisierung mittels Simulation gezeigt worden. Eine
minimale Koeffizientenwortl¨ange von 22 Bit konnte ermittelt werden. Das block-
weise Skalieren
von kleinen Koeffizienten wurde an einem Beispiel verdeutlicht.
Die Realisierung des Filters ist in einer Multiraten-Architektur vorgeschlagen
worden.
Den Hauptteil der Studie stellen die Filterkaskaden dar. Es wurde das Frequenz-
verhalten f¨
ur drei Kaskaden ermittelt. Die erste Stufe ist in jedem Fall ein
sinc
K
(Comb) Filter. Ferner wurde eine modifizierte Comb-Filter Struktur
untersucht, mit der eine Frequenzgangformung m¨oglich ist. F¨
ur beide Struk-
turen wurde der Implementierungsaufwand abgesch¨atzt. Das nachfolgende FIR
lowpass Filter kompensiert den "passband droop" im Signalband. Der zu er-
wartende Substratbedarf l¨at sich an Hand der Filterl¨ange absch¨atzen. Ferner
wurden die Vorteile eines Halbband-Filters bei der Dezimation f¨
ur diese An-
wendung aufgezeigt. Die Realisierung des FIR Filters ist mit konventionellen
MAC Bausteinen m¨oglich. Eine alternative Realisierungsform ist die FIR-PTV
Struktur (periodical time-varying coefficients), welche abschlieend beschrieben
wurde.
iv
Acknowledgments
I would like to take this opportunity to thank all of the people who have helped
me to complete this work.
First, I want to express my deepest thanks to my major Professor, Dr.
Godi Fischer, for the support and guidance during the course of my studies.
Special thanks to Alan J. Davis
1
for his assistance in preparation of this
thesis.
I would also like to thank Professor Leland B. Jackson
2
for helpful discus-
sions regarding the stability of recursive filters.
Finally, I wish to express appreciation to my family and friends for their
support and encouragement.
1
Alan J. Davis is with the Naval Undersea Warfare Center, Newport, RI
2
University of Rhode Island, Department of Electrical Engineering
v
List of Symbols
x(n)
Input Sequence
y(n)
Output Sequence
D
Decimation Ratio
OSR
Oversampling Ratio
f
p
Passband Frequency
f
s
Stopband Frequency
f
sa
Sampling Frequency
f
o
Output Frequency
f
N y
Nyquist Frequency
N
Filter Order
p
,
s
Passband Ripple, Stopband Ripple
A
s
Stopband Attenuation
A
f b
Maximum Alias Rejection
A
pd
Passband Droop
p
Permissible Errors in the Passband
s
Permissible Errors in the Stopband
a
i
Filter Coefficients (FIR Part)
b
j
Filter Coefficients (IIR Part)
h
i
[h(n)]
Impulse Response
Q
Quantization Step
E
Q
Quantization Error
n
2
0
Noise Power
SNR
Signal-to-Noise Ratio
G
0
Gain Term
P
0
Leak Term
t
sl
Slewing Time
L
PTV Filter Order
c
m
PTV Filter Coefficients
C
out
PTV Filter Constant
K
in
, K
out
PTV Filter Constant Terms
b
sign
, b
0
Bits to Encode PTV Coefficients
vi
Contents
Abstract
ii
Acknowledgments
v
List of Symbols
vi
Table of Contents
vii
List of Tables
xi
List of Figures
xiii
1 Introduction
1
1.1 The z-Transform . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2 Digital Filter Fundamentals . . . . . . . . . . . . . . . . . . . .
2
1.3 Decimation Filters . . . . . . . . . . . . . . . . . . . . . . . . .
5
1.3.1
Multistage Decimation Filters . . . . . . . . . . . . . . .
9
1.4 Comb Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
1.4.1
Cascaded Comb Filters . . . . . . . . . . . . . . . . . . .
15
1.4.2
Sharpened Comb Filter . . . . . . . . . . . . . . . . . . .
18
1.5 Anti-Aliasing . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
1.5.1
Alias Rejection using Comb Filters . . . . . . . . . . . .
21
1.6 Finite Word-Length Effects
. . . . . . . . . . . . . . . . . . . .
23
1.6.1
Number Representation . . . . . . . . . . . . . . . . . .
23
1.6.2
Roundoff Noise . . . . . . . . . . . . . . . . . . . . . . .
24
1.6.3
Truncation and Rounding Errors . . . . . . . . . . . . .
24
vii
1.7 PTV(D)-Filter . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
1.7.1
Radix-r Signed Digit Number Representation . . . . . .
28
1.7.2
Quantization Error . . . . . . . . . . . . . . . . . . . . .
31
1.7.3
Radix-3 SD Representation . . . . . . . . . . . . . . . .
31
1.7.4
Radix-4 SD Representation . . . . . . . . . . . . . . . .
32
1.7.5
The Design Flow for a PTV Filter . . . . . . . . . . . . .
32
2 Design Environment
33
2.1 -Converter Structure . . . . . . . . . . . . . . . . . . . . . .
33
2.2 Digital Filter Design Flow . . . . . . . . . . . . . . . . . . . . .
34
2.3 Proposed Realization . . . . . . . . . . . . . . . . . . . . . . . .
34
3 Oversampling A/D Converters
35
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
3.2 Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
3.2.1
Stability . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
3.2.2
Signal-to-Noise Ratio (SNR) . . . . . . . . . . . . . . . .
38
3.3 Nonideal Effects . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
3.4 DelSi - Simulation Tool . . . . . . . . . . . . . . . . . . . . . . .
45
3.5 The IFLF5 -Modulator . . . . . . . . . . . . . . . . . . . . .
45
3.5.1
The Topology . . . . . . . . . . . . . . . . . . . . . . . .
45
3.5.2
Fully Differential SC Integrator . . . . . . . . . . . . . .
46
3.5.3
Input overload treatment . . . . . . . . . . . . . . . . . .
46
4 Hardware Realization
49
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
4.2 LUT-Based Serial Distributed Multiplication . . . . . . . . . . .
50
4.3 Multirate Decimation Filter . . . . . . . . . . . . . . . . . . . .
52
4.4 The Bit-Serial Approach for the FIR Filter Implementation in
FPGAs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
4.5 Modified FIR Filter in Direct Form . . . . . . . . . . . . . . . .
60
viii
4.6 The Basic Building Blocks . . . . . . . . . . . . . . . . . . . . .
60
5 Design of a One-Stage FIR Filter
61
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
5.2 Coefficient Quantization . . . . . . . . . . . . . . . . . . . . . .
65
5.2.1
Technique to reduce Quantization Noise . . . . . . . . .
70
5.3 Hardware Implementation . . . . . . . . . . . . . . . . . . . . .
73
6 Designing a Multistage FIR Filter
79
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
6.2 Proposed Structure for this Design . . . . . . . . . . . . . . . .
80
6.2.1
Specifications . . . . . . . . . . . . . . . . . . . . . . . .
80
6.2.2
Two-Stage Decimation . . . . . . . . . . . . . . . . . . .
80
6.2.3
Three-Stage Decimation . . . . . . . . . . . . . . . . . .
82
6.3 The Comb - FIR Filter Cascade . . . . . . . . . . . . . . . . . .
83
6.3.1
Realization of the First Stage . . . . . . . . . . . . . . .
83
6.3.1.1
Comb Filter 5th order . . . . . . . . . . . . . .
87
6.3.1.2
Comb Filter 6th order . . . . . . . . . . . . . .
88
6.3.1.3
Comb Filter 7th order . . . . . . . . . . . . . .
90
6.3.1.4
Comb Filter 8th order . . . . . . . . . . . . . .
91
6.4 The Comb - FIR Filter Cascade
Design Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . .
92
6.4.1
Filter Properties . . . . . . . . . . . . . . . . . . . . . .
92
6.4.2
Hardware Requirements . . . . . . . . . . . . . . . . . .
93
6.4.3
An Modification of the Comb - FIR Filter Cascade . . .
99
6.4.3.1
Filter Properties . . . . . . . . . . . . . . . . .
99
6.4.3.2
Hardware Requirements . . . . . . . . . . . . . 102
6.4.4
Sharpened Comb Filter . . . . . . . . . . . . . . . . . . . 103
6.5 The Sharpened Comb Filter - FIR Compensator Cascade
Design Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.5.1
Filter Properties . . . . . . . . . . . . . . . . . . . . . . 107
ix
6.5.1.1
Quatizated Coefficients . . . . . . . . . . . . . . 113
6.5.2
Hardware Requirements . . . . . . . . . . . . . . . . . . 115
6.6 The Half-Band Filter . . . . . . . . . . . . . . . . . . . . . . . . 118
6.6.1
Determination of the Coefficients . . . . . . . . . . . . . 119
6.7 The Comb - Half-Band Filter Cascade
Design Example 3 . . . . . . . . . . . . . . . . . . . . . . . . . . 123
6.7.1
The Half-Band Filter Section . . . . . . . . . . . . . . . 123
6.7.2
The Front-End realized as Sharpened Comb Filter . . . . 131
6.8 The FIR Filter realized as a PTV Filter Structure . . . . . . . . 132
6.8.1
Description of the Topology . . . . . . . . . . . . . . . . 132
6.8.2
Hardware Requirements . . . . . . . . . . . . . . . . . . 136
6.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
7 Conclusions
139
A Filter Coefficients and Matlab Files
141
A.1 Filter Coefficients for Design Example 2 . . . . . . . . . . . . . 141
A.2 m-file coeff trunc.m . . . . . . . . . . . . . . . . . . . . . . . . . 146
A.3 m-file coeff round.m . . . . . . . . . . . . . . . . . . . . . . . . 146
A.4 m-file design sinc2sh.m
. . . . . . . . . . . . . . . . . . . . . . 147
A.5 m-file design sinc2shcomp.m
. . . . . . . . . . . . . . . . . . . 147
A.6 m-file design sinc3.m . . . . . . . . . . . . . . . . . . . . . . . . 150
A.7 m-file design hbfir.m . . . . . . . . . . . . . . . . . . . . . . . . 153
A.8 m-file dec2radix.m . . . . . . . . . . . . . . . . . . . . . . . . . 155
Bibliography
158
x
List of Tables
1.1 Filter Specifications, with Filter Order N . . . . . . . . . . . . .
5
1.2 Alias Attenuation for the Comb Cascade . . . . . . . . . . . . .
16
1.3 Parameters and Bit-Precision for the PTV Decimator . . . . . .
29
1.4 Hardware Requirements for a Radix-2/-3 encoded FIR-PTV Fil-
ter, q is the word length of the representation and w
in
is the input
signal word length
. . . . . . . . . . . . . . . . . . . . . . . . .
30
1.5 Quantization Step for Radix-r represented Coefficients . . . . .
31
1.6 Encoding for the Coefficients in Radix-3 Representation . . . . .
31
1.7 Encoding for the Coefficients in Radix-4 Representation . . . . .
32
4.1 Required Hardware for a LUT-Based FIR Filter Realization,
where m is the coefficient length, k the output data length and
N the filter order . . . . . . . . . . . . . . . . . . . . . . . . . .
51
4.2 Required Hardware for a Multirate Decimation Filter, where m
is the coefficient length, N the filter order and p the number of
parallel processing paths . . . . . . . . . . . . . . . . . . . . . .
52
4.3 Number of Levels for O(i) Operands
. . . . . . . . . . . . . . .
56
4.4 Required Hardware for the Bit-Serial Structure with final Column
Adder realized with Carry-Save Adders, where m is the coefficient
length and N the filter order (symmetric coefficients) . . . . . .
57
4.5 Hardware Requirements for the Bit-Serial Architecture in Trans-
posed Form, where m is the coefficent lenght and N the filter
order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
4.6 Basic Logic Blocks . . . . . . . . . . . . . . . . . . . . . . . . .
60
5.1 Filter Specifications . . . . . . . . . . . . . . . . . . . . . . . . .
63
xi
5.2 Quantization Step . . . . . . . . . . . . . . . . . . . . . . . . . .
65
5.3 Required Hardware for the Multirate Decimation Filter without
Multiplexing, m=22, N=4096, p=128 . . . . . . . . . . . . . . .
75
5.4 Required Hardware for the Multirate Decimation Filter using
Barrel Shifter and Time Devision Multiplexing, m=22, N=4096
76
6.1 Parameters for the Two-Stage Filter Cascade f
sa
= 5MHz . . .
81
6.2 Parameters for the Three-Stage Filter Cascade f
sa
= 5MHz
. .
82
6.3 Maximum Alias Attenuation and Passband Droop of the conven-
tional Comb Filter (f
p
= 78.125kHz, f
sa
= 5MHz)
. . . . . . .
85
6.4 Maximum Alias Attenuation and Passband Droop of the modified
Comb Filter (f
p
= 78.125kHz, f
sa
= 5MHz) . . . . . . . . . . .
86
6.5 Hardware Requirements for the length-8, length-4 standard
Comb Filter K
1
=6, K
2
=5 . . . . . . . . . . . . . . . . . . . . .
98
6.6 Maximum Alias Attenuation and Passband Droop of the sharp-
ened Comb Filter f
p
= 78.125kHz, f
sa
= 5MHz . . . . . . . . . 105
6.7 Summary of the Filter Parameters for this Example . . . . . . . 109
6.8 Hardware Requirements of the Sharpened Comb Filter . . . . . 116
6.9 Possible Constellations using Comb and Half-Band Filters for an
overall Decimation of 32 . . . . . . . . . . . . . . . . . . . . . . 122
6.10 Requirements in Logic Blocks for the FIR-PTV Filter . . . . . . 136
6.11 Summarization of the Filter Topologies considered in this Chapter138
xii
List of Figures
1.1 Block Diagram of a Digital System . . . . . . . . . . . . . . . .
2
1.2 FIR Filter in Direct Form . . . . . . . . . . . . . . . . . . . . .
3
1.3 Single Stage Digital Filter and D to 1 Decimator . . . . . . . . .
6
1.4 Direct Form of a FIR Filter with Decimation . . . . . . . . . . .
7
1.5 Magnitude Spectrum in the Decimation Process . . . . . . . . .
8
1.6 Direct Form of a FIR Filter with Symmetric Coefficients . . . .
8
1.7 Cascaded Decimation Filter . . . . . . . . . . . . . . . . . . . .
10
1.8 Frequency Decimation in a Cascaded Filter Structure . . . . . .
10
1.9 Magnitude Response of a Comb Filter with D=16 . . . . . . . .
13
1.10 Comb Filter with Decimation . . . . . . . . . . . . . . . . . . .
14
1.11 Passband Droop . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
1.12 Cascaded Comb Filter with Decimation . . . . . . . . . . . . . .
15
1.13 Magnitude Response of a Cascaded Comb Filter with Decimation 16
1.14 Block Diagram of the two Stage Comb Filter . . . . . . . . . . .
16
1.15 Block Diagram of the length-(D+1) Comb Filter Kth order . . .
17
1.16 Frequency Response of the length-(D+1) Comb Filter 6th order
18
1.17 Sharpened Comb Filter K = 4, D = 8 . . . . . . . . . . . . . . .
20
1.18 Block Diagram of the Sharpened Comb Filter . . . . . . . . . .
20
1.19 Redrawn Block Diagram of the Sharpened Comb Filter . . . . .
20
1.20 Alias Prevention in Oversampled A/D Converters, sampled signal
without (a) and with (b) band-limitation . . . . . . . . . . . . .
22
1.21 Alias Rejection with Comb Filters . . . . . . . . . . . . . . . . .
22
1.22 Truncating and Rounding . . . . . . . . . . . . . . . . . . . . .
25
1.23 Block Diagram of the FIR-PTV Filter . . . . . . . . . . . . . .
28
xiii
1.24 Scaling to exploit the digit range . . . . . . . . . . . . . . . . .
28
1.25 Overall FIR-PTV Filter . . . . . . . . . . . . . . . . . . . . . .
29
1.26 Single Add/Subtract Cell for a Ternary Coefficient Set . . . . .
30
2.1 Block Diagram of the Oversampling A/D Converter . . . . . . .
33
2.2 Digital Filter Design Flow . . . . . . . . . . . . . . . . . . . . .
34
2.3 Block Diagram of the proposed Configuration . . . . . . . . . .
34
3.1 Spectral Noise Density of the fifth-order modulator, Quantization
Noise Level . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
3.2 Integrator Realization in SC-Technique . . . . . . . . . . . . . .
40
3.3 Feedback Amplifier . . . . . . . . . . . . . . . . . . . . . . . . .
41
3.4 Ideal Integrator and Integrator with finite Amplifier Gain . . . .
42
3.5 Circuit Diagram of the Integrator . . . . . . . . . . . . . . . . .
43
3.6 Desired Amplifier Response versus real Step Response . . . . . .
45
3.7 Block Diagram of the IFLF5 -Modulator . . . . . . . . . . .
46
3.8 Power Spectrum of the IFLF5 -Modulator . . . . . . . . . .
47
3.9 Modulator Output Spectrum (Overall Frequency Range) . . . .
47
3.10 Modulator Output Spectrum . . . . . . . . . . . . . . . . . . . .
47
3.11 Modulator Output Spectrum after Lowpass Filtering . . . . . .
47
3.12 Circuit Diagram of the IFLF5 -Modulator . . . . . . . . . .
48
4.1 LUT-Based Serial Distributed Arithmetic . . . . . . . . . . . . .
51
4.2 Architecture of the Multirate Decimation Filter . . . . . . . . .
53
4.3 The Registered Adder/Subtractor . . . . . . . . . . . . . . . . .
53
4.4 The Bit-Serial Architecture with Symmetric Coefficients . . . .
54
4.5 Bit-Serial two's Complement Multiplier . . . . . . . . . . . . . .
55
4.6 Multilevel Carry-Save Adder . . . . . . . . . . . . . . . . . . . .
56
4.7 FIR Filter in Transposed Form . . . . . . . . . . . . . . . . . .
58
4.8 Bit-Serial Adder/Subtractor . . . . . . . . . . . . . . . . . . . .
59
4.9 FIR Filter Realization using Bit-Serial Arithmetic based on the
Transposed Form . . . . . . . . . . . . . . . . . . . . . . . . . .
59
xiv
4.10 Modified Direct Form of a FIR Filter with Decimation and Sym-
metric Coefficients . . . . . . . . . . . . . . . . . . . . . . . . .
60
5.1 Block Diagram of the Structure . . . . . . . . . . . . . . . . . .
63
5.2 One-Stage Decimator for this Example . . . . . . . . . . . . . .
63
5.3 Specifications for this Application . . . . . . . . . . . . . . . . .
63
5.4 Desired Frequency Response using Chebyshev Window, N = 4096 64
5.5 Passband Ripple using Chebyshev Window . . . . . . . . . . . .
64
5.6 Frequency Response with 16 Bit Quantization,f
sa
= 5MHz,
OSR=32 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
5.7 Frequency Response with 16 Bit Quantization, Transition Band
67
5.8 Frequency Response with 18 Bit Quantization, f
sa
= 5MHz,
OSR=32 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
5.9 Frequency Response with 18 Bit Quantization, Transition Band
67
5.10 Frequency Response with 20 Bit Quantization, f
sa
= 5MHz,
OSR=32 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
68
5.11 Frequency Response with 20 Bit Quantization, Transition Band
68
5.12 Frequency Response with 22 Bit Quantization, f
sa
= 5MHz,
OSR=32 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
68
5.13 Frequency Response with 22 Bit Quantization, Transition Band
68
5.14 Frequency Response with 24 Bit Quantization, f
sa
= 5MHz,
OSR=32 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
5.15 Frequency Response with 24 Bit Quantization, Transition Band
69
5.16 Block Scaling of the target Coefficients . . . . . . . . . . . . . .
71
5.17 Filter Coefficients after 16 Bit Quantization . . . . . . . . . . .
72
5.18 Filter Coefficients in Floating Point Arithmetic . . . . . . . . .
72
5.19 FIR Filter with Quantized Coefficients, N = 198, q = 16bit . . .
72
5.20 Architecture of a Multirate Decimation Filter (brute force ap-
proach)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
5.21 Time Multiplexing of an Add and Accumulate Unit . . . . . . .
76
xv
5.22 Multirate
Filter
Realization
using
Multiplexed
Add/Accumulators and a Barrel Shifter . . . . . . . . . . . . . .
77
5.23 4×4 Barrel Shifter . . . . . . . . . . . . . . . . . . . . . . . . . 78
6.1 Two-Stage Decimation . . . . . . . . . . . . . . . . . . . . . . .
81
6.2 Three-Stage Decimation . . . . . . . . . . . . . . . . . . . . . .
82
6.3 Block Diagram of the Comb - FIR Cascade . . . . . . . . . . . .
83
6.4 5th order Comb Filter with Decimation D = 8 . . . . . . . . . .
87
6.5 Cascade of four length-8 and one length-10 comb Filter . . . . .
87
6.6 6th order Comb Filter with Decimation D = 8 . . . . . . . . . .
88
6.7 Cascade of five length-8 and one length-9 comb Filter . . . . . .
88
6.8 Cascade of five length-8 and one length-10 comb Filter . . . . .
89
6.9 7th order Comb Filter with Decimation D = 16 . . . . . . . . .
90
6.10 Cascade of six length-16 and one length-22 comb Filter . . . . .
90
6.11 8th order Comb Filter with Decimation D = 16 . . . . . . . . .
91
6.12 Cascade of seven length-16 and one length-22 comb Filter . . . .
91
6.13 Block Diagram of the Comb Filter Cascade followed by the FIR
Compensator . . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
6.14 Composite Frequency Response at the output of the second Comb
Filter (1 refers to f
1
/2=312.5kHz)
. . . . . . . . . . . . . . . .
94
6.15 The desired FIR Compensator Frequency Response for the two
stage Comb Cascade . . . . . . . . . . . . . . . . . . . . . . . .
94
6.16 Frequency Response of the Comb Stages and Compensation Filter 94
6.17 Overall Frequency Response normalized to f
1
/2=312.5kHz . . .
94
6.18 Realization of the Comb Filter . . . . . . . . . . . . . . . . . . .
96
6.19 (a) Block Diagram of the recursive Comb Filter (b) Block Dia-
gram of the Moving Average Filter . . . . . . . . . . . . . . . .
96
6.20 Block Diagram of the Add/Subtract Unit . . . . . . . . . . . . .
97
6.21 Overall Structure of the Comb Filter, Decimation is not shown .
97
6.22 Block Diagram of the two-stage Comb Cascade with FIR Com-
pensator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
xvi
6.23 Frequency Response at the output of the first Comb Filter nor-
malized to f
sa
= 5MHz
. . . . . . . . . . . . . . . . . . . . . . 100
6.24 Composite Frequency Response at the output of the second Comb
Filter (1 refers to f
2
=312.5kHz) . . . . . . . . . . . . . . . . . . 100
6.25 Frequency Response of the FIR Compensator versus desired Fre-
quency Response . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.26 Frequency Response of the Comb Stages and Compensation Fil-
ter
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.27 Overall Frequency Response of the Filter Cascade for this Exam-
ple, N
F IR
= 645 . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.28 Block Diagram for the modified Comb Cascade . . . . . . . . . 103
6.29 Sharpened Comb Filter length-8 K = 4 . . . . . . . . . . . . . . 105
6.30 Sharpened Comb Filter length-16, K = 6 . . . . . . . . . . . . . 105
6.31 Block Diagram of the Sharpened Comb-FIR Filter Cascade . . . 110
6.32 Desired Frequency Response of the Second Filter Stage . . . . . 110
6.33 Frequency Response of H
21
(z), N = 11 . . . . . . . . . . . . . . 110
6.34 Frequency Response of H
22
(z), N = 136 . . . . . . . . . . . . . . 111
6.35 Composite Frequency Response H
2
(z)=H
21
(z)· H
22
(z) . . . . . . 111
6.36 Used 4th order Sharpened Comb Filter with a following Decima-
tion D = 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.37 Frequency Response of the Sharpened Comb Filter versus FIR
Compensator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.38 Frequency Response of H
31
(z), N
31
=46 . . . . . . . . . . . . . . 112
6.39 The Passband Region of H
31
(z) . . . . . . . . . . . . . . . . . . 112
6.40 Frequency Response of FIR Filter H
32
(z) . . . . . . . . . . . . . 112
6.41 Composite Frequency Response of H
3
(z) = H
31
·H
32
(z) . . . . . 112
6.42 Frequency Response of H
2
(z) with 12 Bit Quantizated Coefficients113
6.43 Frequency Response of H
3
(z) with 12 Bit Quantizated Coefficients113
6.44 Passband Ripple of H
3
(z) with 12 Bit Quantizated Coefficients . 114
6.45 Block Diagram of the Sharpened Comb Filter . . . . . . . . . . 116
xvii
6.46 Redrawn Block Diagram of the Sharpened Comb Filter for this
Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.47 Pipelined Adder for the Comb Filter Implementation . . . . . . 117
6.48 Half-Band Filter
. . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.49 Comb - Half-Band Filter Cascade . . . . . . . . . . . . . . . . . 119
6.50 Determination of the impulse Response of the Half-Band Filter . 121
6.51 Frequency Response of the Half-Band Filter Example . . . . . . 121
6.52 Block Diagram of the Sharpened Comb Filter with Half-Band
Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
6.53 Overall Frequency Response of the first Half-Band Filter . . . . 124
6.54 Passband Section of the first Half-Band Filter . . . . . . . . . . 124
6.55 Plotted Coefficients of the first Half-Band Filter, N
HB1
= 28 . . 125
6.56 Overall Frequency Response of the second Half-Band Filter . . . 126
6.57 Passband Section of the second Half-Band Filter . . . . . . . . . 126
6.58 Plotted Coefficients of the second Half-Band Filter, N
HB2
= 44 . 128
6.59 Overall Frequency Response of the third Half-Band Filter . . . . 128
6.60 Passband Section of the third Half-Band Filter . . . . . . . . . . 128
6.61 Plotted Coefficients of the third Half-Band Filter, N
HB3
= 600 . 130
6.62 Passband Ripple of the overall Half-Band Filter . . . . . . . . . 130
6.63 Frequency Response of the Sharpened Comb Filter, K=3, D=4 . 131
6.64 Passband Droop of the Sharpened Comb Filter, K=3, D=4 . . . 131
6.65 FIR-PTV
Filter,
Periadically-Time
Varying
Coefficients
c
0
, ···,c
L-1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
6.66 Overall FIR-PTV Filter . . . . . . . . . . . . . . . . . . . . . . 134
6.67 Desired Frequency Response of the Target Filter, N = 478 . . . 135
6.68 Coefficients of the Target Filter, N = 478 . . . . . . . . . . . . . 135
6.69 Radix-3 Encoded Coefficients for a Target Filter with 100dB
Stopband Attenuation, N = 478, D = 8 . . . . . . . . . . . . . . 135
6.70 Radix-3 Encoded Coefficients for a Target Filter with 100dB
Stopband Attenuation, N = 478, D = 16 . . . . . . . . . . . . . 135
xviii
6.71 Radix-3 Encoded Coefficients for a Target Filter with 40dB Stop-
band Attenuation, N = 188 . . . . . . . . . . . . . . . . . . . . 136
6.72 Single Add/Subtract Cell for a Ternary Coefficient Set . . . . . 137
xix
Chapter 1
Introduction
1.1 The z-Transform
The z-transform is used to describe linear time-invariant systems (LTI) for
discrete-time signals as the Laplace transform does for the analysis of continous-
time signals in LTI systems. The transform simplifies the signal analysis and
makes it possible to characterize a LTI system. The z-transform for a known
sequence x(n) where - n is defined by
X(z) =
n=-
x(n) z
-n
(1.1)
where z is a complex variable. The z-transform of a sequence can be viewed
as a unique representation of the signal sequence x(n) in the complex z-plane.
Knowing the pole-zero locations, the system can be estimated with regard to
stability. Herein the unit circle plays an important role.
The z-transform is an infinite power series and converges everywhere in the z-
plane only if x(n) is of finite duration. The z-transform converges everywhere
outside a circle of radius R
1
if the sequence x(n) is causal, what means x(n)=0
for 0 N
1
n . X(z) converges inside a circle of radius R
2
, if the sequence
x(n) is noncausal, or in a more formal expression for - n N
2
< 0 is
x(n)=0. Finally, if x(n) is defined over - N
1
n N
2
, then X(z)
converges between these circles.
1
1 Introduction
A digital system is generally described by its transfer function
H(z) =
Y(z)
X(z)
.
(1.2)
Furthermore, a digital system with feedback and an additive noise source as
shown in Figure 1.1 is described by the signal transfer function (STF) and noise
transfer function (NTF). The STF is given by
H
STF
(z) =
H
t
(z)
1 - H
t
(z)
,
Q(z) = 0
(1.3)
and the NTF is determined by
H
NTF
(z) =
1
1 - H
t
(z)
,
X(z) = 0
(1.4)
Figure 1.1 shows the block diagram of a digital system with feedback.
t
H (z)
Q(z)
Y(z)
X(z)
+
+
+
+
Figure 1.1: Block Diagram of a Digital System
1.2 Digital Filter Fundamentals
A digital filter is completely characterized by its difference equation. The dif-
ference equation describes the relationship between input and output.
y(n) = a
0
· x(n) + a
1
· x(n - 1) + a
2
· x(n - 2) + ··· + a
N
· x(n - N)
- b
1
·y(n-1)-b
2
·y(n-2)-···-b
L
·y(n-L)
(1.5)
where x(n) is the discrete-time input signal and y(n) the output sequence.
The transfer function of a digital filter is generally given by
H(z) =
L
j=0
b
j
z
-j
1 -
N
i=1
a
i
z
-i
(1.6)
2
1 Introduction
H(z) =
Y(z)
X(z)
=
1 + a
1
z
-1
+ a
2
z
-2
+ ··· + a
N
z
-N
1 + b
1
z
-1
+ b
2
z
-2
+ ··· + b
L
z
-L
.
(1.7)
The transfer function of the nonrecursive FIR filter is
H(z) =
Y(z)
X(z)
= 1 + a
1
z
-1
+ a
2
z
-2
+ ··· + a
N
z
-N
(1.8)
A FIR filter is characterized by the impulse response written as a finite convo-
lution sum
y(n) =
N-1
k=0
h(k)x(n - k)
(1.9)
where x(n) is the input signal, h(n) the impulse response of the filter and y(n)
the output signal. The FIR filter is sometimes also called a convolution filter,
because of the method of realization. Viewing the FIR filter from the time-
domain, the system is also called moving-average filter. Obviously, the transfer
function of a FIR filter in the frequency-domain is given by
H(z) =
N-1
n=0
h(n)z
-n
(1.10)
Due to its nonrecursive structure, FIR filters are always stable and provide lin-
h0
x(n)
z
h1
z
h2
z
z
y(n)
-1
-1
-1
-1
hm
Figure 1.2: FIR Filter in Direct Form
ear phase if the coefficients are symmetric. They are well suited for applications
in which an arbitrary magnitude response is desired and frequency distortion
due to nonlinear phase must be avoided, e.g. in applications of speech process-
ing or acoustics in general.
The performance of a FIR filter is bounded firstly by the available filter taps and
secondly by finite word length effects. Let us first consider the trade-off between
the width of the transition band, stopband attenuation and filter length. The
filter length is a function of the allowed stopband and passband ripple (
s
,
p
)
3
1 Introduction
and the width of the transition band f . The peak-to-peak passband ripple in
decibels is given by
A
p
= 20 · log
10
1 +
p
1 -
p
[dB]
(1.11)
and the stopband attenuation is determined by
A
s
= 20 · log
10
s
[dB]
(1.12)
where
p
is the passband ripple and
s
the stopband ripple.
The order of a digital lowpass filter is determined by the empirical Kaiser rela-
tion [1].
N =
D
(
p
s
)
f
(1.13)
with
D
(
p
s
) = log
10
s
[a
1
(log
10
p
)
2
+ a
2
log
10
p
+ a
3
]
+
[a
4
(log
10
p
)
2
+
a
5
log
10
p
+
a
6
]
(1.14)
and
p
:
passband ripple
s
:
stopband ripple
a
1
=
0.005309
a
2
=
0.07114
a
3
=
-0.4761
a
4
=
-0.00266
a
5
=
-0.5941
The transition band is normalized relative to the input sampling frequency and
given by
f =
f
s
- f
p
f
sa
(1.15)
where f
p
is the passband frequency, f
s
the stopband frequency and f
sa
the
sampling frequency. A more useful equivalent equation to determine the filter
length is
N = -
20 log
10
p
s
- 13
14.6 · f
+ 1
(1.16)
4
1 Introduction
As equation (1.16) demonstrates, the filter length highly depends on the width
of the transition band f . Table 1.1 shows this trade-off for a FIR lowpass
filter with 100dB stopband attenuation. The higher the filter specifications are,
f
p
[1]
f
s
[1]
F
p
F
s
f
N [100dB]
0.5
0.6
78125
93750
0.003125
1908
0.5
0.55
78125
85937.5
0.0015625
3814
0.5
0.53
78125
82812.5
0.0009375
(6358)
0.5
0.525 78125 82031.25 0.00078125
(7628)
Table 1.1: Filter Specifications, with Filter Order N
the higher the filter order. In a multistage filter structure, every single stage has
its own specifications. Figure 1.7 shows a multistage filter cascade and (1.16)
becomes
N
i
= -
20 log
10
p,i
s,i
- 13
14.6 · f
i
+ 1
(1.17)
for the ith filter stage. The transition band becomes
f
i
=
f
s,i
- f
p,i
f
sa,i
(1.18)
where f
s,i
is the stopband frequency, f
p,i
the passband frequency and f
sa,i
the
input sampling rate for stage i, respectively.
A digital filter is furthermore characterized by the required computation in
multiplications per second [2]
R =
N · f
sa
2D
[multiplications/sample]
(1.19)
where f
sa
denotes the sampling frequency, N the filter length and D the deci-
mation ratio. The above presented one-stage FIR filter (N=3814, f
sa
=5MHz)
requires 297 968 750 multiplications per second.
1.3 Decimation Filters
In many digital signal processing applications, the sampling rate has to be re-
duced. The process of decimation, obtaining a signal with a lower sampling
5
1 Introduction
rate, is also called sampling rate conversion. In applications using oversampling
techniques, decimation will furthermore reduce the quantization noise. Figure
1.3 shows the block diagram of a single-stage decimator to illustrate the dec-
imation process. In an efficient architecture, the decimator is located before
the coefficient multiplier as depicted in Figure 1.4. Hence, the multiplication
is performed at the reduced sampling rate of f
s
/D. Since having symmetric
coefficients, further savings in complexity are possible. Figure 1.6 shows this
approach.
Decimation by D means that every Dth output sample is required. This means
for the filter realization, that only every Dth sample need to be computed. In
order to avoid aliasing, the signal must be band-limeted to = /D. Figure
1.5 shows the magnitude spectrum of the band-limeted signal. The original
spectrum is periodic in = 2. The downsampled signal is described by
Y(e
jD
) =
1
D
D-1
i=0
X(e
j(-2i/D)
)
(1.20)
The resulting downsampled spectrum is periodical in = 2/D. This can be
regarded as a new sampling rate of =D·. In Figure 1.5 is a new axis with
depicted.
The designed decimation filter must have two major properties. First, it must
s
s
D
f
@
*
x(n)
@ f
x (n)
y(m)
D
Down-Sampler
H(z)
Filter
Figure 1.3: Single Stage Digital Filter and D to 1 Decimator
fulfill the demands in attenuating the out-of-band signals and the modulator
quantization noise. Second, the noise of the filter itself must be sufficiently low.
The noise caused within the filter is essentially coefficient quantization noise
and roundoff noise.
6
1 Introduction
The downsampling process is described with the following equations. The se-
quence at the output of the filter is given by (ref. Figure 1.3)
x
(n) =
N
i=0
h(i)x(n - i)
(1.21)
The final decimated signal is hence
y(m) = x
(mD)
(1.22)
with n = m·D. The output, depending on the input signal, can be written as
y(m) =
N
i=0
h(i)x(mD - i)
(1.23)
D
D
D
D
z
z
z
x(n)
z
h
0
h1
h2
hm
y(m)
-1
-1
-1
-1
Figure 1.4: Direct Form of a FIR Filter with Decimation
7
1 Introduction
2
D
f
Ny
X
D
X
2f = f
Ny s
2
D
2
f
Ny,n
f /D
s
f
s,n
'
Figure 1.5: Magnitude Spectrum in the Decimation Process
z
z
z
z
x(n)
D
D
D
D
D
D
D
1
h
0
h
2
h
y(m)
z
z
z
hm/2 -1
-1
-1
-1
-1
-1
-1
-1
Figure 1.6: Direct Form of a FIR Filter with Symmetric Coefficients
8
1 Introduction
1.3.1 Multistage Decimation Filters
When large downsampling ratios must be realized, the filter requirements can
be great. Large oversampling ratios and large decimation ratios cause a very
narrow transition band with regard to the overall frequency range [0; f
sa
]. This
leads to a prohibitively large filter length, as mentioned before. Moreover, in a
single-stage filter structure, a large word length is required to avoid quantization
noise and roundoff errors. In a multistage architecture sampling frequency is
decimated in steps. Every single stage has its own specifications. Due to the
lower input sampling rate at the intermediate stages, f
i
is not as narrow as in
the one-stage case. Figure 1.7 shows a two-stage filter structure. The frequency
bands are subdivided in
passband:
0 f f
p,i
(1.24)
and transition band:
f
p,i
f f
s,i
=
f
i
2
(1.25)
where i is the stage index.
At the output of the ith stage, the sampling frequency becomes
f
i
=
f
i-1
D
i
.
(1.26)
Due to an overall decimation of
D =
D
i
,
(1.27)
the final output sampling frequency will be
f
o
=
f
s
D
i
=
f
s
D
.
(1.28)
Figure 1.8 illustrates the steps of decimation. The frequency breakpoints in the
final stage are the same as in the one-stage case. The only diffenence is that
the final stage has a lower input frequency, which means a smaller filter order.
The most important advantages of a cascaded FIR structure are: [2]
9
1 Introduction
D
H (z)
@
D
1
2
1
H (z)
2
x(n)
@ f
@
f
D
s
s
1
y(m)
D D
1 2
f s
Figure 1.7: Cascaded Decimation Filter
f
sampl
1
D
f =
1
f =
D
2
1
f
2
f
p,1
f
s,1
s,2
f
f
f
s,3
f
p,3
p,2
1st-stage
2nd-stage
3rd-stage
passband
transition band
Figure 1.8: Frequency Decimation in a Cascaded Filter Structure
10
1 Introduction
· Reduced overall filter order and therefore reduced storage requirements
· Significantly reduced computation to implement the architecture
· Reduced finite word length effects (roundoff noise, bit-sensitivity)
1.4 Comb Filters
If we realize a decimation filter cascade, the first stage should be suitable for the
"bulk" of decimation. The first stage operates at a high frequency while good
silicon utilization should be achieved. Especially for this purpose comb filters
(sometimes called sinc
K
filters, where K is the order of the comb filter) are
well suited. They have good properties for decimation purposes and a simple
structure [25]. The advantages of the comb filter compared with the FIR filter
are:
· No multipliers are required
· A simple structure can be designed
· No coefficient storage
· The architecture is independent to the decimation ratio
· The early decimation leads to lesser dynamic power consumption for the
following stages
The comb can be treated as notch filter with zeros at
k
=
2·k
D
for k = 1, 2, ...,
D/2. The comb filter is a recursive filter whose coefficients b
j
are equal to one.
One obtains a conditionally stable liner phase filter with length D. The transfer
function is
H(z) =
D-1
n=0
z
-n
=
1 - z
-D
1 - z
-1
(1.29)
The transfer function of the comb filter is derived from the moving average filter
by rewriting it in a recursive form. In order to derive equation (1.29), let us
11
1 Introduction
consider the moving average process
y(n) =
D-1
i=0
x(n - i)
(1.30)
and rewrite it as
y(n) =
D-2
i=-1
x(n - i) + x(n) - x(n - D)
(1.31)
= y(n - 1) + x(n) - x(n - D).
(1.32)
Obviously, equation (1.29) and (1.32) describe the same system. This expres-
sion leads to equation (1.29). Every new output sample can be determined by
adding the previous output sample to the new input value and subtracting the
input value, that occured D samples ago. We obtain a significant reduction in
computation time with the recursive structure [33]. The computation of a new
output sample requires t
c
=
1
2·f
sa
seconds. The conventional moving average
filter consisting of D delay units needs a computation time of t
c
=
1
D·f
sa
seconds
for every new output sample, where f
sa
is the input sampling frequency.
Due to its recursive structure, the comb filter is only conditionally stable. Fur-
thermore, a DC or a low frequency input signal will cause initial values because
of the IIR part. Figure 1.9 shows the magnitude response of a comb filter with
D=16. The magnitude converges to zero at multiples of 2/D. That is the most
important property for using comb filters in a decimation filter. If we choose the
filter length equal to the decimation ratio, the attenuation in the aliased bands
can be sufficiently high. Of course the attenuation depends on the filter order.
The aliased bands are those frequency bands which will be folded back after
decimation. This property is sometimes called 'natural anti-aliasing' . Figure
1.21 illustrates the effect of comb filter anti-aliasing. Another reason that makes
comb filter interesting is that no multipliers are required. The disadvantage, on
the other hand, is the relatively low attenuation. We are not able to achieve
sufficient alias rejection with one comb filter. A cascade of comb filters is nec-
essary. The disatvantage of the multiple-comb filter is the inherent passband
droop. Figure 1.11 makes this clear. The passband droop increases with the
12
1 Introduction
-60
-40
-20
0
16
H (e )
= 1
D = 16
D
2
= k
k = 1 ... D/2
Magnitude Response
Frequency
j
2/D
1/2
Normalized
Frequency
Figure 1.9: Magnitude Response of a Comb Filter with D=16
order and slightly with the decimation ratio, as can be seen in Table 6.3. Con-
sidering the overall perfomance, we cannot neglect the loss in magnitude in the
passband section. A subsequent FIR or IIR filter stage is required to correct
this deviation. Those filters are therefore often called compensation filters [28].
The system function of the conventional comb filter in the z-domain is
H(z) =
1
D
D-1
k=0
z
-k
=
1
D
·
1 - z
-D
1 - z
-1
(1.33)
and in the frequency domain
H(e
j
) =
1
D
·
sin( · D/2)
sin(/2)
(1.34)
with
=
2 · f
f
sa
(1.35)
Figure 1.10 shows the block diagram of a comb filter. In order to reach a
more efficient implementation, the basic structure can be redrawn using the
commutative rule, as shown in Figure 1.10. The differentiation (1 -z
-1
) is now
performed at a lower sampling rate. With this modification, we are able to
reduce the number of registers and the processing rate [27].
13
1 Introduction
1 - z
-D
-1
1 - z
D
y(m)
x(n)
x(n)
1 - z -1
1
D
-1
y(m)
(a)
(b)
1 - z
Figure 1.10: Comb Filter with Decimation
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
-30
-25
-20
-15
-10
-5
0
5
Normalized Frequency [Omega]
Attenuation (dB)
Merged comb
4
Filter (1 - z
-08
)
3
(1 - z
-08
)
aliased band
Figure 1.11: Passband Droop
14
1 Introduction
1.4.1 Cascaded Comb Filters
With a single comb filter, sufficient stopband attenuation is not achievable.
Therefore, a cascaded comb structure is often applied, as mentioned before.
The cascaded integrators are usually followed by the intermediate downsampler
and finally by the FIR section. Figure 1.12 shows this approach. Equation
(1.36) is the transfer function of the cascaded comb filter.
H(z) =
1
D
·
1 - z
-D
1 - z
-1
K
(1.36)
The transfer function in the frequency domain is, respectively
H(e
j
) =
1
D
·
sin( · D/2)
sin(/2)
K
(1.37)
with
=
2 · f
f
sa
(1.38)
where f
sa
denotes the sampling frequency. Equation (1.37) describes a lowpass
filter with linear phase. Cascading must be continued until the desired stop-
band attenuation at =
2
D
-
p
is reached. Figure 1.13 shows the magnitude
-1
1 - z
D
y(m)
x(n)
-1
y(m)
K
K
(1 - z )
1
Figure 1.12: Cascaded Comb Filter with Decimation
responses of a two stage and five stage comb filter. The worst case aliasing will
occur at
f b
= 2/D -
p
. Hence, the worst alias attenuation is given by
A
alias
(
fb
) = 20 · log
sin(
fb
· D/2)
D · sin(
fb
/2)
K
|f
p
(1.39)
where f
p
is the passband frequency.
With (1.39) we can determine the maximum alias rejection as a function
of the decimation ratio D, the filter order K and the passband frequency.
Unfortunately, the passband droop increases with the number of comb filters
15
1 Introduction
D
K = 1
K = 2 K = 3
K = 4
K = 5
K = 6
K = 7
p
2
26dB
52dB
79dB
105dB 131dB 157dB 183dB 0.96875
4
23dB
46dB
68dB
91dB
114dB 137dB 159dB 0.46875
8
17dB
34dB
51dB
68dB
85dB
102dB 119dB 0.21875
16
10dB
21dB
31dB
42dB
52dB
63dB
73dB
0.09375
Table 1.2: Alias Attenuation for the Comb Cascade
0
0.5
1
1.5
2
2.5
-140
-120
-100
-80
-60
-40
-20
0
20
Frequency (MHz)
Attenuation (dB)
Sinc
2
and Sinc
5
Filter, D = 16, F
s
= 5MHz
two stages
five stages
Figure 1.13: Magnitude Response of a Cascaded Comb Filter with Decimation
z -1
-1
z
+
1:D
1/D
1/D
f /D
s
-1
z
z -1
+
+
+
+
+
-
-
+
+
+
+
x(n)
fs
y(m)
(1 - z )
1
(1 - z )
-1
-1
IIR Part
FIR Part
Figure 1.14: Block Diagram of the two Stage Comb Filter
16
1 Introduction
K.
Figure 1.14 shows the realization of a 2nd order comb filter cascade.
Applying the commutative rule, the structure is split into a FIR and IIR part,
corresponding to Figure 1.12.
Within the class of comb filters, a modification of the structure should be
mentioned. Basically, this is the conventional merged filter structure with an
additional delay in the forward path [27]. Figure 1.15 shows this approach. The
transfer function for a single stage is given by
H(z) =
1 - z
-D
1 - z
-1
+ z
-D
=
1 - z
-(D+1)
1 - z
-1
(1.40)
The transfer function for a Kth-order cascade is given by
H(z) =
1 - z
-(D+1)
1 - z
-1
·
1 - z
-D
1 - z
-1
K-1
.
(1.41)
This is basically a conventional (K-1)th-order comb filter supperposed with the
modified comb filter (1.40) with single order. Figure 1.16 shows the frequency
response for an example. An additional narrow notch is inserted on the left of
the
k·2
D
notch. The shown example is a length-(D+1) comb filter with an order
of K=6 and a decimation ratio of 8. The attenuation to the left of the notch at
=2/D is slightly increased. In some cases, the specifications can be achieved
by a lower order. The length-(D+1) filter is always single order. The changes
in the overall frequency behavior depend strongly on the order K.
1 - z-1
z-1
D
D
1/D
1/D
1
1 - z -1
x(n)
1
1 - z -1
K-1
+
1 - z -1
K-1
y(m)
Figure 1.15: Block Diagram of the length-(D+1) Comb Filter Kth order
The transfer function for the example shown in Figure 1.16 is
17
1 Introduction
H(z) =
(1 - z
-8
)
5
(1 - z
-9
)
(1 - z
-1
)
6
(1.42)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-140
-120
-100
-80
-60
-40
-20
0
20
Normalized Frequency [Omega]
Attenuation (dB)
Merged comb
6
Filter (1 - z
-08
)
5
(1 - z
-09
)
aliased band
Figure 1.16: Frequency Response of the length-(D+1) Comb Filter 6th order
Multistage filter design using comb filters means usually an architecture of
one or more comb filters followed by a FIR or IIR compensation filter. Figure
6.3 shows the block diagram of a comb - FIR filter cascade with specifications
for the design example. For audio applications, where a linear phase is required,
FIR compensators are mandatory.
1.4.2 Sharpened Comb Filter
In recent years a modified comb structure denoted as sharpened comb filter
has also been used in decimation filters. The properties are described in [30],
[31]. With the sharpened comb filter, we are able to reduce the passband droop
18
Ende der Leseprobe aus 182 Seiten
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Details
- Arbeit zitieren
- , 1998, Decimation Lowpass Filters for Sigma-Delta Modulators, Hamburg, Bedey Media GmbH, https://www.diplom.de/document/221679
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