Decimation Lowpass Filters for SIGMA-DELTA Modulators - A Comparative Study

Contents

Abstract

Acknowledgments

List of Symbols

Table of Contents

List of Tables

List of Figures

1 Introduction
1.1 The z-Transform
1.2 Digital Filter Fundamentals
1.3 Decimation Filters
1.3.1 Multistage Decimation Filters
1.4 Comb Filters
1.4.1 Cascaded Comb Filters
1.4.2 Sharpened Comb Filter
1.5 Anti-Aliasing
1.5.1 Alias Rejection using Comb Filters
1.6 Finite Word-Length Effects
1.6.1 Number Representation
1.6.2 Roundoff Noise
1.6.3 Truncation and Rounding Errors
1.7 PTV(D)-Filter
1.7.1 Radix-r Signed Digit Number Representation
1.7.2 Quantization Error
1.7.3 Radix-3 SD Representation
1.7.4 Radix-4 SD Representation
1.7.5 The Design Flow for a PTV Filter

2 Design Environment
2.1 ΣΔ-Converter Structure
2.2 Digital Filter Design Flow
2.3 Proposed Realization

3 Oversampling A/D Converters
3.1 Introduction
3.2 Fundamentals
3.2.1 Stability
3.2.2 Signal-to-Noise Ratio (SNR)
3.3 Nonideal Effects
3.4 DelSi - Simulation Tool
3.5 The IFLF5 ΣΔ-Modulator
3.5.1 The Topology
3.5.2 Fully Differential SC Integrator
3.5.3 Input overload treatment

4 Hardware Realization
4.1 Introduction
4.2 LUT-Based Serial Distributed Multiplication
4.3 Multirate Decimation Filter
4.4 The Bit-Serial Approach for the FIR Filter Implementation in FPGAs
4.5 Modified FIR Filter in Direct Form
4.6 The Basic Building Blocks

5 Design of a One-Stage FIR Filter
5.1 Introduction
5.2 Coefficient Quantization
5.2.1 Technique to reduce Quantization Noise
5.3 Hardware Implementation

6 Designing a Multistage FIR Filter
6.1 Introduction
6.2 Proposed Structure for this Design
6.2.1 Specifications
6.2.2 Two-Stage Decimation
6.2.3 Three-Stage Decimation
6.3 The Comb - FIR Filter Cascade
6.3.1 Realization of the First Stage
6.3.1.1 Comb Filter 5th order
6.3.1.2 Comb Filter 6th order
6.3.1.3 Comb Filter 7th order
6.3.1.4 Comb Filter 8th order
6.4 The Comb - FIR Filter Cascade Design Example
6.4.1 Filter Properties
6.4.2 Hardware Requirements
6.4.3 An Modification of the Comb - FIR Filter Cascade
6.4.3.1 Filter Properties
6.4.3.2 Hardware Requirements
6.4.4 Sharpened Comb Filter
6.5 The Sharpened Comb Filter - FIR Compensator Cascade Design Example
6.5.1 Filter Properties
6.5.1.1 Quatizated Coefficients
6.5.2 Hardware Requirements
6.6 The Half-Band Filter
6.6.1 Determination of the Coefficients
6.7 The Comb - Half-Band Filter Cascade Design Example
6.7.1 The Half-Band Filter Section
6.7.2 The Front-End realized as Sharpened Comb Filter
6.8 The FIR Filter realized as a PTV Filter Structure
6.8.1 Description of the Topology
6.8.2 Hardware Requirements
6.9 Summary

7 Conclusions

A Filter Coefficients and Matlab Files
A.1 Filter Coefficients for Design Example
A.2 m-file coeff trunc.m
A.3 m-file coeff round.m
A.4 m-file design sinc2sh.m
A.5 m-file design sinc2shcomp.m
A.6 m-file design sinc3.m
A.7 m-file design hbfir.m
A.8 m-file dec2radix.m

Bibliography

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 ΣΔ-modulator with 90dB dynamic range (SNR+THD) will be presented, which has been fabricated 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 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 sincK 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.

Decimation Lowpass Filters for Sigma-Delta Modulators - A Comparative Study Thema der vorliegenden Studienarbeit ist der Vergleich und die Aufwandsabschätzung verschiedener digitaler Dezimationsfilter für den Einsatz bei A/D- Wandlern nach dem Sigma-Delta Prinzip.

Den Anfang macht eine Einführung in die Grundlagen der digitalen Filtertech- nik sowie der Sigma-Delta Modulation. Anschlieend werden die Möglichkeiten der Harwareimplementierung prinzipiell vorgestellt. Im Hinblick auf eine VLSI Implementierung, ist jeweils der Hardwareumfang abgeschätzt worden. Als Referenz dient das unkaskadierte FIR Dezimationsfilter. Die hohen An- forderungen, ein schmales Übergangsband (0.5; 0.55) und eine Sperrdämpfung von 100dB, machen ein Filter der Ordnung 4096 nötig. Das Frequenzverhalten wurde mit Routinen aus den Matlab Toolboxen bestimmt. Es ist der Ein- fluß einer Koeffizientenquantisierung mittels Simulation gezeigt worden. Eine minimale Koeffizientenwortlänge 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 für drei Kaskaden ermittelt. Die erste Stufe ist in jedem Fall ein sincK (Comb) Filter. Ferner wurde eine modifizierte Comb-Filter Struktur untersucht, mit der eine Frequenzgangformung möglich ist. Für beide Struk- turen wurde der Implementierungsaufwand abgeschätzt. Das nachfolgende FIR lowpass Filter kompensiert den “passband droop” im Signalband. Der zu er- wartende Substratbedarf lät sich an Hand der Filterlänge abschätzen. Ferner wurden die Vorteile eines Halbband-Filters bei der Dezimation für diese An- wendung aufgezeigt. Die Realisierung des FIR Filters ist mit konventionellen MAC Bausteinen möglich. Eine alternative Realisierungsform ist die FIR-PTV Struktur (periodical time-varying coefficients), welche abschlieend beschrieben wurde.

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. Davis¹ for his assistance in preparation of this thesis.

I would also like to thank Professor Leland B. Jackson² for helpful discussions regarding the stability of recursive filters.

Finally, I wish to express appreciation to my family and] friends for their support and encouragement.

List of Symbols

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List of Tables

1.1 Filter Specifications, with Filter Order N
1.2 Alias Attenuation for the Comb Cascade
1.3 Parameters and Bit-Precision for the PTV Decimator
1.4 Hardware Requirements for a Radix-2/-3 encoded FIR-PTV Fil- ter, q is the word length of the representation and win is the input signal word length
1.5 Quantization Step for Radix-r represented Coefficients
1.6 Encoding for the Coefficients in Radix-3 Representation
1.7 Encoding for the Coefficients in Radix-4 Representation

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
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
4.3 Number of Levels for O(i) Operands
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)
4.5 Hardware Requirements for the Bit-Serial Architecture in Trans- posed Form, where m is the coefficent lenght and N the filter order
4.6 Basic Logic Blocks

5.1 Filter Specifications
5.2 Quantization Step
5.3 Required Hardware for the Multirate Decimation Filter without Multiplexing, m=22, N=4096, p=128
5.4 Required Hardware for the Multirate Decimation Filter using Barrel Shifter and Time Devision Multiplexing, m=22, N=4096

6.1 Parameters for the Two-Stage Filter Cascade fsa = 5MHz
6.2 Parameters for the Three-Stage Filter Cascade fsa = 5MHz
6.3 Maximum Alias Attenuation and Passband Droop of the conven- tional Comb Filter (fp = 78.125kHz, fsa = 5MHz)
6.4 Maximum Alias Attenuation and Passband Droop of the modified Comb Filter (fp = 78.125kHz, fsa = 5MHz)
6.5 Hardware Requirements for the length-8, length-4 standard Comb Filter K1=6, K2=5
6.6 Maximum Alias Attenuation and Passband Droop of the sharp- ened Comb Filter fp = 78.125kHz, fsa = 5MHz
6.7 Summary of the Filter Parameters for this Example
6.8 Hardware Requirements of the Sharpened Comb Filter
6.9 Possible Constellations using Comb and Half-Band Filters for an overall Decimation of
6.10 Requirements in Logic Blocks for the FIR-PTV Filter
6.11 Summarization of the Filter Topologies considered in this Chapter

List of Figures

1.1 Block Diagram of a Digital System
1.2 FIR Filter in Direct Form
1.3 Single Stage Digital Filter and D to 1 Decimator
1.4 Direct Form of a FIR Filter with Decimation
1.5 Magnitude Spectrum in the Decimation Process
1.6 Direct Form of a FIR Filter with Symmetric Coefficients
1.7 Cascaded Decimation Filter
1.8 Frequency Decimation in a Cascaded Filter Structure
1.9 Magnitude Response of a Comb Filter with D=16
1.10 Comb Filter with Decimation
1.11 Passband Droop
1.12 Cascaded Comb Filter with Decimation
1.13 Magnitude Response of a Cascaded Comb Filter with Decimation
1.14 Block Diagram of the two Stage Comb Filter
1.15 Block Diagram of the length-(D+1) Comb Filter Kth order . .
1.16 Frequency Response of the length-(D+1) Comb Filter 6th order
1.17 Sharpened Comb Filter K = 4, D = 8
1.18 Block Diagram of the Sharpened Comb Filter
1.19 Redrawn Block Diagram of the Sharpened Comb Filter
1.20 Alias Prevention in Oversampled A/D Converters, sampled signal
without (a) and with (b) band-limitation
1.21 Alias Rejection with Comb Filters
1.22 Truncating and Rounding
1.23 Block Diagram of the FIR-PTV Filter
1.24 Scaling to exploit the digit range
1.25 Overall FIR-PTV Filter
1.26 Single Add/Subtract Cell for a Ternary Coefficient Set

2.1 Block Diagram of the Oversampling A/D Converter
2.2 Digital Filter Design Flow
2.3 Block Diagram of the proposed Configuration

3.1 Spectral Noise Density of the fifth-order modulator, Quantization Noise Level
3.2 Integrator Realization in SC-Technique
3.3 Feedback Amplifier
3.4 Ideal Integrator and Integrator with finite Amplifier Gain
3.5 Circuit Diagram of the Integrator
3.6 Desired Amplifier Response versus real Step Response
3.7 Block Diagram of the IFLF5 ΣΔ-Modulator
3.8 Power Spectrum of the IFLF5 ΣΔ-Modulator
3.9 Modulator Output Spectrum (Overall Frequency Range)
3.10 Modulator Output Spectrum
3.11 Modulator Output Spectrum after Lowpass Filtering
3.12 Circuit Diagram of the IFLF5 ΣΔ-Modulator

4.1 LUT-Based Serial Distributed Arithmetic
4.2 Architecture of the Multirate Decimation Filter
4.3 The Registered Adder/Subtractor
4.4 The Bit-Serial Architecture with Symmetric Coefficients
4.5 Bit-Serial two’s Complement Multiplier
4.6 Multilevel Carry-Save Adder
4.7 FIR Filter in Transposed Form
4.8 Bit-Serial Adder/Subtractor
4.9 FIR Filter Realization using Bit-Serial Arithmetic based on the Transposed Form
4.10 Modified Direct Form of a FIR Filter with Decimation and Sym- metric Coefficients

5.1 Block Diagram of the Structure
5.2 One-Stage Decimator for this Example
5.3 Specifications for this Application
5.4 Desired Frequency Response using Chebyshev Window, N = 4096
5.5 Passband Ripple using Chebyshev Window
5.6 Frequency Response with 16 Bit Quantization,fsa = 5MHz, OSR=32
5.7 Frequency Response with 16 Bit Quantization, Transition Band
5.8 Frequency Response with 18 Bit Quantization, fsa = 5MHz, OSR=32
5.9 Frequency Response with 18 Bit Quantization, Transition Band
5.10 Frequency Response with 20 Bit Quantization, fsa = 5MHz, OSR=32
5.11 Frequency Response with 20 Bit Quantization, Transition Band
5.12 Frequency Response with 22 Bit Quantization, fsa = 5MHz, OSR=32
5.13 Frequency Response with 22 Bit Quantization, Transition Band
5.14 Frequency Response with 24 Bit Quantization, fsa = 5MHz, OSR=32
5.15 Frequency Response with 24 Bit Quantization, Transition Band
5.16 Block Scaling of the target Coefficients
5.17 Filter Coefficients after 16 Bit Quantization
5.18 Filter Coefficients in Floating Point Arithmetic
5.19 FIR Filter with Quantized Coefficients, N = 198, q = 16bit
5.20 Architecture of a Multirate Decimation Filter (brute force ap- proach)
5.21 Time Multiplexing of an Add and Accumulate Unit
5.22 Multirate Filter Realization using Multiplexed Add/Accumulators and a Barrel Shifter
5.23 4×4 Barrel Shifter

6.1 Two-Stage Decimation
6.2 Three-Stage Decimation
6.3 Block Diagram of the Comb - FIR Cascade
6.4 5th order Comb Filter with Decimation D = 8
6.5 Cascade of four length-8 and one length-10 comb Filter
6.6 6th order Comb Filter with Decimation D = 8
6.7 Cascade of five length-8 and one length-9 comb Filter
6.8 Cascade of five length-8 and one length-10 comb Filter
6.9 7th order Comb Filter with Decimation D = 16
6.10 Cascade of six length-16 and one length-22 comb Filter
6.11 8th order Comb Filter with Decimation D = 16
6.12 Cascade of seven length-16 and one length-22 comb Filter
6.13 Block Diagram of the Comb Filter Cascade followed by the FIR Compensator
6.14 Composite Frequency Response at the output of the second Comb Filter (1 refers to f1/2=312.5kHz)
6.15 The desired FIR Compensator Frequency Response for the two stage Comb Cascade
6.16 Frequency Response of the Comb Stages and Compensation Filter
6.17 Overall Frequency Response normalized to f1/2=312.5kHz
6.18 Realization of the Comb Filter
6.19 (a) Block Diagram of the recursive Comb Filter (b) Block Dia- gram of the Moving Average Filter
6.20 Block Diagram of the Add/Subtract Unit
6.21 Overall Structure of the Comb Filter, Decimation is not shown
6.22 Block Diagram of the two-stage Comb Cascade with FIR Com- pensator
6.23 Frequency Response at the output of the first Comb Filter nor- malized to fsa = 5MHz
6.24 Composite Frequency Response at the output of the second Comb Filter (1 refers to f2=312.5kHz)
6.25 Frequency Response of the FIR Compensator versus desired Fre- quency Response
6.26 Frequency Response of the Comb Stages and Compensation Fil- ter
6.27 Overall Frequency Response of the Filter Cascade for this Exam- ple, NFIR = 645
6.28 Block Diagram for the modified Comb Cascade
6.29 Sharpened Comb Filter length-8 K = 4
6.30 Sharpened Comb Filter length-16, K = 6
6.31 Block Diagram of the Sharpened Comb-FIR Filter Cascade
6.32 Desired Frequency Response of the Second Filter Stage
6.33 Frequency Response of H21(z), N = 11
6.34 Frequency Response of H22(z), N = 136
6.35 Composite Frequency Response H2(z)=H21(z)· H22(z)
6.36 Used 4th order Sharpened Comb Filter with a following Decima- tion D = 8
6.37 Frequency Response of the Sharpened Comb Filter versus FIR Compensator
6.38 Frequency Response of H31(z), N31=46
6.39 The Passband Region of H31(z)
6.40 Frequency Response of FIR Filter H32(z)
6.41 Composite Frequency Response of H3(z) = H31·H32(z)
6.42 Frequency Response of H2(z) with 12 Bit Quantizated Coefficients113
6.43 Frequency Response of H3(z) with 12 Bit Quantizated Coefficients113
6.44 Passband Ripple of H3(z) with 12 Bit Quantizated Coefficients
6.45 Block Diagram of the Sharpened Comb Filter
6.46 Redrawn Block Diagram of the Sharpened Comb Filter for this Example
6.47 Pipelined Adder for the Comb Filter Implementation
6.48 Half-Band Filter
6.49 Comb - Half-Band Filter Cascade
6.50 Determination of the impulse Response of the Half-Band Filter
6.51 Frequency Response of the Half-Band Filter Example
6.52 Block Diagram of the Sharpened Comb Filter with Half-Band Filters
6.53 Overall Frequency Response of the first Half-Band Filter
6.54 Passband Section of the first Half-Band Filter
6.55 Plotted Coefficients of the first Half-Band Filter, NHB1 = 28
6.56 Overall Frequency Response of the second Half-Band Filter
6.57 Passband Section of the second Half-Band Filter
6.58 Plotted Coefficients of the second Half-Band Filter, NHB2 = 44
6.59 Overall Frequency Response of the third Half-Band Filter
6.60 Passband Section of the third Half-Band Filter
6.61 Plotted Coefficients of the third Half-Band Filter, NHB3 = 600
6.62 Passband Ripple of the overall Half-Band Filter
6.63 Frequency Response of the Sharpened Comb Filter, K=3, D=4
6.64 Passband Droop of the Sharpened Comb Filter, K=3, D=4
6.65 FIR-PTV Filter, Periadically-Time Varying Coefficients c0, · · ·, cL−1
6.66 Overall FIR-PTV Filter
6.67 Desired Frequency Response of the Target Filter, N = 478
6.68 Coefficients of the Target Filter, N = 478
6.69 Radix-3 Encoded Coefficients for a Target Filter with 100dB Stopband Attenuation, N = 478, D = 8
6.70 Radix-3 Encoded Coefficients for a Target Filter with 100dB Stopband Attenuation, N = 478, D = 16
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

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 continoustime 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

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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 R1 if the sequence x(n) is causal, what means x(n)=0 for 0≤ N1 ≤ n ≤ ∞. X(z) converges inside a circle of radius R2, if the sequence x(n) is noncausal, or in a more formal expression for -∞ ≤ n ≤ N2 < 0 is x(n)=0. Finally, if x(n) is defined over -∞ ≤ N1 ≤ n ≤N2 ≤ ∞, then X(z) converges between these circles.

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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

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and the NTF is determined by

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Figure 1.1 shows the block diagram of a digital system with feedback.

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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 difference equation describes the relationship between input and output.

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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

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The transfer function of the nonrecursive FIR filter is

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A FIR filter is characterized by the impulse response written as a finite convolution sum

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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 timedomain, the system is also called moving-average filter. Obviously, the transfer function of a FIR filter in the frequency-domain is given by

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Due to its nonrecursive structure, FIR filters are always stable and provide lin- 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 processing or acoustics in general.

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Figure 1.2: FIR Filter in Direct Form

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 (δs, δp) and the width of the transition band Δf . The peak-to-peak passband ripple in decibels is given by

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and the stopband attenuation is determined by

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where δp is the passband ripple and δs the stopband ripple.

The order of a digital lowpass filter is determined by the empirical Kaiser rela- tion 1.

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with

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The transition band is normalized relative to the input sampling frequency and given by

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where fp is the passband frequency, fs the stopband frequency and fsa the sampling frequency. A more useful equivalent equation to determine the filter length is

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As equation (1.16) demonstrates, the filter length highly depends on the width of the transition band Δf . Table 1.1 shows this trade-off for a FIR lowpass filter with 100dB stopband attenuation. The higher the filter specifications are,

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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

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for the ith filter stage. The transition band becomes

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where fs,i is the stopband frequency, fp,i the passband frequency and fsa,i the input sampling rate for stage i, respectively.

A digital filter is furthermore characterized by the required computation in multiplications per second [2]

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where fsa denotes the sampling frequency, N the filter length and D the decimation ratio. The above presented one-stage FIR filter (N=3814, fsa=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 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 fs/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 Ω = π/D. Figure 1.5 shows the magnitude spectrum of the band-limeted signal. The original spectrum is periodic in Ω = 2π. The downsampled signal is described by

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The resulting downsampled spectrum is periodical in Ω = 2π/D. This can be regarded as a new sampling rate of Ω′=D·Ω. In Figure 1.5 is a new axis with Ω′ depicted.

The designed decimation filter must have two major properties. First, it must

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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.

The downsampling process is described with the following equations. The sequence at the output of the filter is given by (ref. Figure 1.3)

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The final decimated signal is hence

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with n = m·D. The output, depending on the input signal, can be written as

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Figure 1.4: Direct Form of a FIR Filter with Decimation

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Figure 1.5: Magnitude Spectrum in the Decimation Process

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Figure 1.6: Direct Form of a FIR Filter with Symmetric Coefficients

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; fsa]. 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, Δfi 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:

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and transition band:

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where i is the stage index.

At the output of the ith stage, the sampling frequency becomes

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Due to an overall decimation of

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the final output sampling frequency will be

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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]

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Figure 1.7: Cascaded Decimation Filter

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Figure 1.8: Frequency Decimation in a Cascaded Filter Structure

- 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 sincK 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 [illustration not visible in this excerpt]. The comb filter is a recursive filter whose coefficients bj are equal to one. One obtains a conditionally stable liner phase filter with length D. The transfer function is

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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 consider the moving average process

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and rewrite it as

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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 [illustration not visible in this excerpt] seconds. The conventional moving average filter consisting of D delay units needs a computation time of [illustration not visible in this excerpt] seconds for every new output sample, where fsa 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 2π/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

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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

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and in the frequency domain

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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.

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Figure 1.10: Comb Filter with Decimation

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Figure 1.11: Passband Droop

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.

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The transfer function in the frequency domain is, respectively

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with

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where fsa 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 [illustration not visible in this excerpt] is reached.Figure 1.13 shows the magnitude

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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 [illustration not visible in this excerpt]. Hence, the worst alias attenuation is given by

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where fp is the passband frequency.

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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

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Table 1.2: Alias Attenuation for the Comb Cascade

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Figure 1.13: Magnitude Response of a Cascaded Comb Filter with Decimation

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Figure 1.14: Block Diagram of the two Stage Comb Filter

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

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The transfer function for a Kth-order cascade is given by

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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 [illustration not visible in this excerpt] 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 [illustration not visible in this excerpt] 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.

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Figure 1.15: Block Diagram of the length-(D+1) Comb Filter Kth order

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