On Filter Bank Based MIMO Frequency Multiplexing and Demultiplexing

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On Filter Bank Based MIMO Frequency Multiplexing and Demultiplexing

Master thesis performed in Electronics Systems division by

Amir Eghbali

Report number:LiTH-ISY-EX--06/3911--SE September 2006

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Title

On Filter bank Based MIMO Frequency Multiplexing and

Demultiplexing

Master thesis in Electronics Systems

at Linköping Institute of Technology

by

Amir Eghbali LiTH-ISY-EX--06/3911--SE

Supervisor: Prof. Håkan Johansson Examiner: Prof. Håkan Johansson

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Presentation Date 2006-09-26

Publishing Date (Electronic version) 2006-10-02 Division of Electronics Systems Department of Electrical Engineering Abstract

The next generation satellite communication networks will provide multimedia services supporting high bit rate, mobility, ATM, and TCP/IP. In these cases, the satellite technology will act as the internetwork infrastructure of future global systems and assuming a global wireless system, no distinctions will exist between terrestrial and satellite communications systems, as well as between fixed and 3G mobile networks. In order for satellites to be successful, they must handle bursty traffic from users and provide services compatible with existing ISDN infrastructure, narrowcasting/multicasting services not offered by terrestrial ISDN, TCP/IP-compatible services for data applications, and point-to-point or point-to-multipoint on-demand compressed video services. This calls for onboard processing payloads capable of frequency multiplexing and demultiplexing and interference suppression.

This thesis introduces a new class of oversampled complex modulated filter banks capable of providing frequency multiplexing and demultiplexing. Under certain system constraints, the system can handle all possible shifts of different user signals and provide variable bandwidths to users. Furthermore, the aliasing signals are attenuated by the stopband attenuation of the channel filter thus ensuring the approximation of the perfect reconstruction property as close as desired. Study of the system efficient implementation and its mathematical representation shows that the proposed system has superiority over the existing approaches for Bentpipe payloads from the flexibility, complexity, and perfect reconstruction points of view. The system is analyzed in both SISO and MIMO cases. For the MIMO case, two different scenarios for frequency multiplexing and demultiplexing are discussed.

To verify the results of the mathematical analysis, simulation results for SISO, two scenarios of MIMO, and effects of the finite word length on the system performance are illustrated. Simulation results show that the system can perform frequency multiplexing and demultiplexing and the stopband attenuation of the prototype filter controls the aliasing signals since the filter coefficients resolution plays the major role on the system performance. Hence, the system can approximate perfect reconstruction property by proper choice of resolution.

ISBN (Licentiate thesis) ISRN: LiTH-ISY-EX--06/3911—SE Title of series (Licentiate thesis) Series number/ISSN (Licentiate thesis) Language English Number of Pages 95 Type of Publication Licentiate thesis ● Degree thesis Thesis C-level Thesis D-level Report

Other (specify below)

Publication Title

On Filter Bank Based MIMO Frequency Multiplexing and Demultiplexing Author

Amir Eghbali

URL, Electronic Version

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Abstract

The next generation satellite communication networks will provide multimedia services supporting high bit rate, mobility, ATM, and TCP/IP. In these cases, the satellite technology will act as the inter-network infrastructure of future global systems and assuming a global wireless system, no distinctions will exist between terrestrial and satellite communications systems, as well as between fixed and 3G mobile networks. In order for satellites to be successful, they must handle bursty traffic from users and provide services compatible with existing ISDN infrastructure, narrowcasting/multicasting services not offered by terrestrial ISDN, TCP/IP-compatible services for data applications, and point-to-point or point-to-multipoint on-demand compressed video services. This calls for onboard processing payloads capable of frequency multiplexing and demultiplexing and interference suppression.

This thesis introduces a new class of oversampled complex modulated filter banks capable of providing frequency multiplexing and demultiplexing. Under certain system constraints, the system can handle all possible shifts of different user signals and provide variable bandwidths to users. Furthermore, the aliasing signals are attenuated by the stopband attenuation of the channel filter thus ensuring the approximation of the perfect reconstruction property as close as desired. Study of the system efficient implementation and its mathematical representation shows that the proposed system has superiority over the existing approaches for bentpipe payloads from the flexibility, complexity, and perfect reconstruction points of view. The system is analyzed in both Single Input single Output (SISO) and Multiple Input Multiple Output (MIMO) cases. For the MIMO case, two different scenarios for frequency multiplexing and demultiplexing are discussed.

To verify the results of the mathematical analysis, simulation results for SISO, two scenarios of MIMO, and effects of the finite word length on the system performance are illustrated. Simulation results show that the system can perform frequency multiplexing and demultiplexing and the stopband attenuation of the prototype filter controls the aliasing signals since the filter coefficients resolution plays the major role on the system performance. Hence, the system can approximate perfect reconstruction property by proper choice of resolution.

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Acknowledgments

First, I would like to thank my supervisor Prof. Håkan Johansson for the invaluable guidance and incredible patience in answering my questions. I could ask any questions at any time.

Special thanks go to my family for all the support they provided. I will never forget their kindness.

I would also like to thank all the apples and bananas that kept me alive during the time I was working on my thesis!

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Foreword

The next generation information society will include telecommunications, computing, video, TV, videoconferencing, and consumer electronics in every building and requires wideband services to provide multi-application networks at rates around 2 Mbps accessible to everybody everywhere [1]. The terrestrial networks, even with the large bandwidth available due to optical fiber technology, cannot meet these requirements. However, satellites play an important role since if a satellite is in orbit, the subscriber only has to install a satellite terminal and subscribe to the service. To solve the problem of the next generation networks, network technicians suggest asynchronous transfer mode (ATM) comprised of a multiplexer with a high-rate output having every possible lower rate at the input side. On the other hand, telecommunications managers try to provide temporary solutions such as asynchronous digital subscriber line (ADSL) and high-rate DSL (HDSL) [1].

One of the disadvantages of geostationary communications satellites, is the large delay for one up- and downlink, which is disturbing for voice. However, the terrestrial copper, optical fiber, and the cellular radio networks carry most voice traffic. Thus, the satellites can be a suitable choice for interactive data services and delivery of a large amount of data on request. In addition, low earth orbit (LEO) systems such as GLOBALSTAR and ICO are competing with the terrestrial networks for voice applications. Therefore, for wideband multimedia applications, geostationary satellites with several high-gain spot-beam antennas, OnBoard Processing (OBP), and switching seem to be a logical step in migration from pure TV broadcast to interactive multimedia services. The functionality that the OBP system offers is suited to provide the services required by the information society. The elements making up the OBP system are [1]:

• The User Station (UTS): The UTS consists of an outdoor unit and an indoor unit with a capability of being equipped with Integrated Services Digital Network (ISDN), Electronic Network Systems

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for the compatibility of the new networks with the existing protocols and algorithms.

• The switching payload: The payload consists of DSP functions such as digital beamforming, frequency multiplexing and demultiplexing, interference suppression, signal level control and, in a regenerative system, modems [2].

This thesis focuses on digital signal processing of satellite payloads which has two major categories as [2]:

1. Onboard regeneration and baseband processing: Examples for this type are data buffering and multiple access reformatting, data rate conversion, coding, and encryption. These systems decouple noise and interference on the uplink and downlink and are able to optimize access, modulation, and coding techniques for the uplink and downlink.

2. Onboard non-regenerative processing: Here, signals are sampled with appropriate precision and sampling rate. Subsequent processing is performed as arithmetic operations on the signal samples. In particular, such techniques allow the digital demultiplexing of narrowband channels and processing of individual channels to include level control and beamforming. Hence, we need a transparent payload architecture where signals are not regenerated onboard. The system level advantages of this system are power efficiency, frequency reuse, flexibility in response to changing traffic, reproducibility, and lack of sensitivity to temperature changes [2].

This thesis proposes a bentpipe payload architecture that handles all possible frequency shifts and all possible user data rates, has low complexity, achieves high level of parallelism, and is easy to analyze and design. The system uses a new class of oversampled complex modulated Filter Banks (FB), which brings superiority over previously proposed architectures. In particular, it outperforms the regular modulated FB based networks from the flexibility point of view and has better performance over the tree-structured FB based networks in terms of flexibility and complexity. Furthermore, the proposed system outperforms the overlap/save DFT/IDFT based networks if perfect reconstruction property is important.

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• Use of oversampled filter banks: This choice makes the suppression of aliasing easier and allows the combination of smaller subbands into wider subbands without introducing large aliasing distortion. This property brings full flexibility to the system.

• More FB channels than granularity bands: This feature brings the ability to generate all possible frequency shifts and reduces the complexity of the system.

• Complex modulated filter banks: These filter banks result in very low complexity and simplicity in terms of analysis, design, and implementation.

The report is organized in three chapters. In the first chapter, basic building blocks of multirate systems i.e. interpolators, decimators, and polyphase decomposition are introduced. Since the proposed structure uses filter banks, building blocks of filter banks and their mathematical representations are derived. Based on the structure and parameters, the maximally decimated and oversampled filter banks are discussed. Next, the concept of paraunitariness followed by DFT and cosine modulated filter banks in maximally decimated systems is covered. The distinction between uniform and non-uniform filter banks is treated mathematically but the focus is on the uniform filter banks. The oversampled filter bank analysis starts with the definition of frame theory followed by the example on oversampled DFT modulated filter banks. Having discussed the time invariant systems, basics and properties of the time varying filter banks are investigated. The chapter ends with common issues in design of filter banks from a system point of view described as constraints in a hierarchical manner.

The second chapter discusses the basics of transmultiplexers, as duals of filter banks, and derives their mathematical representation. Next, perfect reconstruction, cancelling of multiuser and interblock interference, and channel equalization are discussed. As special cases of transmultiplexers, the multiple access schemes such as Code Division Multiple Access (CDMA), Time Division Multiple Access (TDMA), and Frequency Division Multiple Access (FDMA) are introduced. Having discussed the transmultiplexers, different architectures of payloads i.e.,

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multiplexing and demultiplexing is formulated followed by the introduction to a new class of online variable oversampled complex modulated filter banks. Based on the problem formulation and the filter bank definition, the constraints of the architecture are derived. Next, the characteristics of the filter bank blocks namely analysis/synthesis banks and channel switch are defined. In order to decrease the implementation complexity, the polyphase decomposition is applied to derive the new system architecture. In reality, there are several users in the uplink which must be multiplexed to different downlink spot beams. This calls for a MIMO system capable of performing the multiplexing and demultiplexing and satisfying the defined Perfect Reconstruction (PR) properties. The extension of the proposed system to a MIMO case is covered in two scenarios.

The last part of the chapter illustrates simulation results of the proposed architecture from the functionality and performance points of view. To do so, the system test setup and the error measurement algorithm which is Mean Square Error (MSE) are described. Next, examples on SISO and MIMO cases verifying the system functionality to multiplex and demultiplex signals are presented. To evaluate the system performance, the finite word length effects are introduced and examples for a 64-QAM signal with different resolutions are illustrated. The chapter ends with conclusion and topics for future research.

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Outline of Tasks

The tasks assigned in this thesis work were as follows: 1. Study of the multirate signal processing basics.

2. Literature review on different filter bank architectures. 3. Literature review on different satellite payload systems.

4. Implementation of a MIMO polyphase Frequency Band Reallocation network in MATLAB including the finite word length effects.

5. Evaluation of the MIMO polyphase network from the BER point of view.

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Figure 28: FFBR system with Fixed Analysis and Adjustable Synthesis Bank. ... 59 Figure 29: FFBR system with Fixed Analysis/Synthesis Banks and Channel Switch. ... 63 Figure 30: Polyphase Implementation of the FFBR Network. ... 64 Figure 31: K-Input K-Output MIMO FFBR with Fixed Analysis and Synthesis FBs... 65 Figure 32: S-Input K-Output MIMO FFBR with Fixed Analysis and Synthesis FBs... 66 Figure 33: Transmit and Receive Filter Characteristics to Evaluate the FFBR Network... 68 Figure 34: Test Setup for FFBR Network Evaluation. ... 69 Figure 35: Example Channel Switch for SISO Case. ... 70 Figure 36: Input, Output, and Analysis Filters for SISO Polyphase FFBR Network... 71 Figure 37: Example Channel Switch for Two-Input Two-Output MIMO FFBR Network... 71 Figure 38: Inputs and Outputs for MIMO FFBR Network with two Inputs and two Outputs. ... 72 Figure 39: Input and Outputs of the FFBR Network without Channel Switch. ... 73 Figure 40: Example One-Input/Two-Output Channel Switch for MIMO FFBR Network... 73 Figure 41: Input and Outputs of the FFBR Network with Channel Switch of Figure 40... 74 Figure 42: Quantization in the Polyphase FFBR Network. ... 75 Figure 43: Multiplexed 64-QAM Data Constellation for Three Filter Coefficient Lengths... 76 Figure 44: FFBR Network Noise Variance for Channels in Figure 38. . 77

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List of Abbreviations

Abbreviation Comments AFB Analysis Filter Bank ATM Asynchronous Transfer Mode BER Bit Error Rate

CDMA Code Division Multiple Access DCT Discrete Cosine Transform

DFT Discrete Fourier Transform DSL Digital Subscriber Line DSP Digital Signal Processing

ENS Electronic Network Systems ESA European Space Agency

FB Filter Bank

FDM Frequency Division Multiplexing FDMA Frequency Division Multiple Access FFBR Flexible Frequency Band Reallocation FIR Finite Impulse Response

GDFT Generalized Discrete Fourier Transform HDSL High bit rate Digital Subscriber Line IDFT Inverse Discrete Fourier Transform IIR Infinite Impulse Response ISDN Integrated Services Digital Network ISI Inter Symbol Interference ISP Internet Service Provider LEO Low Earth Orbit

LP Low Pass

LTI Linear Time Invariant LTV Linear Time Variant MCC Master Control Center

MIMO Multiple Input Multiple Output

MSE Mean Square Error

MUI Multi User Interference

OBP OnBoard Processing

PFBR Perfect Frequency Band Reallocation PR Perfect Reconstruction PU ParaUnitary

QAM Quadrature Amplitude Modulation SFB Synthesis Filter Bank

SISO Single Input Single Output SNR Signal to Noise Ratio

SS/TDMA Satellite-Switched Time Division Multiple Access TCP/IP Transmission Control Protocol/Internet Protocol TDMA Time Division Multiple Access

TM TransMultiplexer TVFB Time Varying Filter Banks

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Chapter One: Overview of Multirate Systems

and Filter Banks

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1. Introduction

Multirate digital filters and filter banks find wide application in areas such as speech processing, communications, analog voice privacy systems, image compression, antenna systems, and digital audio industry. This applicability has excited immense amount of research leading to a substantial progress in multirate systems including decimation and interpolation filters, polyphase structures, and several types of analysis/synthesis filter banks with specific properties that suit some applications. To analyze different systems mathematically, it is useful to have some blocks that are common among the systems and furthermore, ease the analysis process. In the analysis of the multirate systems and filter banks, which is the subject of this chapter, the basic building blocks are the interpolators and decimators, which used along with the concept of the polyphase decomposition, reduce the implementation complexity.

In this chapter, we start with the definition of these building blocks, and then we proceed to define the basics of filter bank theory. In this context, different types of maximally decimated and oversampled filter banks are discussed. Furthermore, a brief introduction to time varying filter banks is provided.

1.1. Basic Building Blocks of Multirate Systems

In the area of the multirate signal processing, interpolators and decimators are the basic blocks that alter the sampling frequency at different parts of the system leading to name “Multirate”. An interpolator is a combination of an upsampler and a lowpass filter where the upsampler inserts M −1 zeros between consecutive samples of the

original signal. Doing so, the output signal spectrum is a compressed version of the input signal spectrum. In the mathematical representation of an upsampler, we have [3]

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) ( ) ( ) ( ) ( M j j M e X e Y or z X z Y = ω = ω , 1.1

where y(n) and x(n) are the output and input sequences, respectively. If

) (_ejω

X is periodic with 2π , then ( jω) e

Y will be periodic with

M

π

2 [3]. On the other hand, a decimator is the combination of a lowpass filter and a downsampler where the downsampler retains only the Mth

samples of the input signal. In the mathematical representation, assuming the notations on interpolator, we have

M j M k M k j j M k k M _X _e _W _e M e Y or W z X M z Y π π ω ω 1 2 0 ) 2 ( 1 0 1 , ) ( 1 ) ( ) ( 1 ) ( − − = − − = = = =

∑

. 1.2 Hence, )( jω e

Y is a sum of M uniformly shifted versions of an M −fold stretched version of ( jω)

e

X [3]. An important issue in the analysis of

these blocks is imaging and aliasing.

Looking at Equation (1.2), one can conclude that if x(n) is band limited to M M ω π π _< _< − (more generally M π α ω α_< _< ₊2 [3]), the original signal can be recovered from y(n) by the use of a lowpass filter. Otherwise, the problem of aliasing can occur damaging the information. So, an interpolator can cause imaging due to compression of the input signal spectrum, which must be removed by a lowpass filter following the upsampler.

Similarly, a decimator can cause aliasing due to the stretching of the input signal spectrum. To deal with this problem, a lowpass filter must remove unnecessary signals before the downsampler. The imaging and aliasing effects and the characteristics of the lowpass filters for a system with a decimation and interpolation ratio of three are shown in

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Figure 1: Effect of Aliasing and Imaging in Upsamplers and Downsamplers.

It must be added that in reality, the brick wall filters can not be realized, so the filters should have transition bands. This can be solved by considering the fact that data signals are not also strictly band limited which allows for filters to have transition bands. To reduce the complexity of the interpolator and decimator implementation, the idea of polyphase decomposition is used and will be discussed in the next section.

1.1.1. Polyphase Decomposition

Polyphase decomposition realizes any lowpass filter as the sum of polyphase components [3]. Any finite or infinite length sequence

{ }

h(n) with a z-transform H(z) can be written as [4]

[

]

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = = = − − − − − = − ∞ −∞ = −

∑

) ( . . . ) ( ) ( ... 1 ) ( ) ( ) ( 1 1 0 ) 1 ( 1 1 0 M M M M M M k M k k n n z H z H z H z z z H z z n h z H . 1.3 Interpolation Filter 2 3π 2π 2 3π − π 2 −

Aliasing in the absence of the filter

π 2 π 2 − +π₂ 2 π − Decimation Filter ) ( jω e X ) ( jω e X π 2 π + π − π 2 − 3 2π 3 π 3 π − Images to be removed

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decompositions. As the first type, any lowpass filter with cutoff frequency at M π can be written as

∑

− = − = 1 0 ) ( ) ( M i M i i z H z z H , 1.4

where H_i(z) are the polyphase components. In the time domain, the

impulse responses of the polyphase components can be derived as

1 0 ), ( ) (n =h i+Mn ≤i≤M −

hi . It must be noted that the polyphase

components can have different lengths. As an example, the 2-fold and 3-fold polyphase components of a 6th order filter with transfer function

) (z H can be derived as . 3 , ]) 5 [ ] 2 [ ( ]) 4 [ ] 1 [ ( ]) 6 [ ] 3 [ ] 0 [ ( 2 , ]) 5 [ ] 3 [ ] 1 [ ( ]) 6 [ ] 4 [ ] 2 [ ] 0 [ ( ] 6 [ ] 5 [ ] 4 [ ] 3 [ ] 2 [ ] 1 [ ] 0 [ ) ( ) ( 3 2 ) ( 3 1 ) ( 6 3 ) ( 4 2 1 ) ( 6 4 2 6 5 4 3 2 1 3 2 3 1 3 0 2 1 2 0 fold h z h z h z h z h z h z h fold h z h z h z h z h z h z h h z h z h z h z h z h z h z H z E z E z E z E z E − + + + + + + = − + + + + + + = + + + + + + = − − − − − − − − − − − − − − − − − − 4 4 3 4 4 2 1 4 4 3 4 4 2 1 4 4 4 4 3 4 4 4 4 2 1 4 4 4 4 3 4 4 4 4 2 1 4 4 4 4 4 4 3 4 4 4 4 4 4 2 1 1.5

The second type of the polyphase decomposition can be derived as

∑

− = − − − = 1 0 ) 1 ( ₍ ₎ ) ( M i M i i M _R _z z z

H and is useful in the analysis of synthesis bank filters

[3]. The relationship between these types is Ri(z)=EM−1−i(z)[5]. The advantage of polyphase components can be better understood by the use of two noble identities shown in Figure 2 whose properties are proved in [5]. It must be added that these noble identities are different in the case of time varying systems and are defined in [6].

Figure 2: Noble Identities in Multirate Systems.

Having these tools, we can derive the efficient decimation and interpolation filter implementations as shown in Figure 3.

≡ ≡

[ ]

n x

[ ]

n x y

[ ]

m

[ ]

m y

[ ]

m v₁

[ ]

m v₁ L M H(z) ) ( L z H

[ ]

m y

[ ]

n x

[ ]

m y

[ ]

n x

[ ]

n v2 L ) (z H ) ( M z H v2

[ ]

n M

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Figure 3: Efficient Polyphase Decimator and Interpolator Implementation.

In these structures, the filters run at lower sampling rates compared to the input signal sampling rate i.e. 1T. Since the samples across the

adders are phased by Tseconds and hence they do not interact in the

adder, some commutator models as described in [7] can be used to avoid the adders for easier implementation. A generalization of the polyphase decomposition is called the structural subband decomposition given by [4]

[

]

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = − − − − ) ( . . . ) ( ) ( ... 1 ) ( 1 1 0 ) 1 ( 1 M M M M M z V z V z V T z z z H , 1.6

where T =

[ ]

t_ij is an M×M non-singular matrix. A non-singular square

matrix is one that has a matrix inverse. In other words, a square matrix is nonsingular if and only if its determinant is nonzero. The relationship between the polyphase components and the generalized polyphase components is as Polyphase Interpolator . . . L L ) ( 0 z H ) ( 2 z HL− ) ( 1 z H_L− ₋₁ z 1 − z [ ]m y + + L [ ]n x Polyphase Decimator 1 − z 1 − z . . . [ ]n x _{[ ]} m y M ) ( 1 M M z H − ) ( 1 zM H ) ( 0zM H +

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⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − − − ( ) . . . ) ( ) ( ) ( . . . ) ( ) ( 1 1 0 1 1 1 0 M M M M M M M M z X z X z X T z H z H z H . 1.7

As with the case for the polyphase decomposition, the structural decomposition can be used to realize an FIR filter. Suppose H(z) is an

FIR filter with an impulse response of length N =P×M, where P and M are positive integers. One can apply the structural subband

decomposition and write the filter as [4]

[

]

. 1 ,..., 1 , 0 , ) ( , ) ( ) ( ) ( . . . ) ( ) ( ... 1 ) ( 1 0 1 , 1 1 0 1 1 0 ) 1 ( 1 − = = = ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ =

∑

− = − + + − = − − − − M k z t z I z V z I z V z V z V T z z z H M j j j k k M k M k k M M M M M 1.8

Finally, the filter can be realized as shown in Figure 4.

Figure 4: Filer Realization Using Subband Decomposition. where F(z)=I (z)V(zM), i=0,1,...,M−1

i i

i . It must be mentioned that by

choosing simple invertible transform matrices T , the complexity can further be reduced. Polyphase decomposition reduces the complexity of the filter realization and hence finds extensive use in the analysis and implementation of filter banks, as discussed in the next sections.

) ( 1 z F . . . + ) (n x ) (n y ) ( 0 z F ) ( 1 z FM−

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1.2. Digital Filter Banks

The idea of filter banks is to split the input signal x(n) into subband

signals x_k(n) through the use of analysis filters H_k(z). The subband

signals can then be processed which is usually called subband processing. The last stage is the reconstruction to approximate the output signal x^k(n)

by the use of synthesis filters F_k(z) to combine the subband signals [3].

The typical system diagram is shown in Figure 5.

Figure 5: Typical Analysis and Synthesis Banks.

In this section, we will introduce the main blocks of the filter banks and their properties for specific types of filter banks namely maximally decimated, oversampled, and time varying filter banks which will be discussed in the later subsections. Generally, a filter bank has five main blocks namely analysis bank, downsampler, subband processing, upsampler, and synthesis bank. These blocks will be discussed in the next subsections.

1.2.1. Analysis Filter Bank

This block is a collection of M so called analysis or decimation filters with a common input signal. The typical frequency responses of these filters can be overlapping, marginally overlapping, and non-overlapping as shown in Figure 6 .

) (n x ) ( 1z H ) ( 0 z H ) ( 1 z HM− ) ( 0 n x ) ( 1 n x_M− ) ( 1 n x . . . Analysis Bank . . . ) ( 0 z F ) ( 1z F_M− ) ( 1z F + + ) ( ^ n x ) ( 1 n y ) ( 1 n yM− ) ( 0 n y Synthesis Bank

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Figure 6: Typical Frequency Responses of Analysis Filters.

1.2.2. Downsamplers

In order to increase the subband processing efficiency, the sampling rate can be reduced. The choice of down sampling ratio leads to two types of systems as:

• Maximally decimated filter banks: In this case, the number of the subband channels is equal to the down sampling ratio leading to equal number of samples in the subband and full band signals. Although this seems to bring maximum efficiency, but it causes aliasing.

• Oversampled filter banks: Contrary to the maximally decimated case, one can choose the decimation ratio to be less than the number of subband channels. The draw back here is that the number of subband samples is larger than the number of full band samples. This has some advantages though and will be discussed later.

1.2.3. Subband Processing

In this block, the subband signals are processed according to the requirements. Examples of the processing can be coding, decoding, etc. In the design of filter banks, this part is usually ignored and the prefect reconstruction properties are defined for the filter bank only. Throughout this document, we will assume the frequency response of the subband processing block to be unity for all frequencies.

Overlapping Marginally Overlapping Non-Overlapping T ω T ω T ω

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1.2.4. Upsamplers

In order to have the data at the original sampling rate, upsampling which simply inserts a number of zeros in between every two samples is used.

1.2.5. Synthesis Filter Bank

As discussed in Section 1.1, upsampling causes imaging and must be removed by an interpolation filter. The synthesis bank is a collection of M so called synthesis or interpolation filters with a summed output which is simply a combination of the subband signals. In order to have perfect reconstruction, the frequency responses of the synthesis filters must be matched to frequency responses of the analysis filters. The waveform h(t) is said to be matched to the waveform s(t) if [8]

∆ − ∆ − ₌ − = − ∆ =_ks _or _H _j _f _kS _j _f _e j f _kS _j _f _e j f t h() ( τ) ( 2π ) ( 2π ) 2π *( 2π ) 2π , 1.9

where k and s are arbitrary constants.

In other words, ignoring the delay and amplitude factors, the transfer function of a matched filter is the complex conjugate of the spectrum of the filter to which it is matched. The use of a matched filter gives the maximum Signal to Noise Ratio (SNR). However, in most cases, the synthesis and filters are exactly the same as the analysis filters.

1.3. General Filter Bank Architecture

As a conclusion of the previous discussion, the filter bank architecture can be drawn as shown in Figure 7.

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Figure 7: General Filter Bank Architecture.

In general, the decimation and interpolation ratios R can be m

different resulting in the aliased channel outputs as [9]

) ( ) ( 1 ) ( ) 2 ( 1 0 ) 2 ( m m m m m R k R j m R k R k R j m j m X e H e R e Y π ω π ω ω − − = −

∑

= . 1.10 The set

{

ym(n)

}

forms a critically sampled time-frequency

representation of the original signal. To construct the input signal and assuming there is no processing, the signals

{

ym(n)

}

must be upsampled

and filtered through the synthesis filters F_m(z). The reconstructed signal

can be written as ) ( ) ( ) ( 1 ) ( ) ( ) ( 1 0 1 0 ) 2 ( 2 ( 1 0 ^ ^ ω π ω π ω ω ω ω j m M m R k R k j m R k j m M m j m R j m j e F e H e X R e F e Y e X m m m m

∑ ∑

∑

− = − = − − − = = = . 1.11

The drawback of this system is that the information about the aliased signals in one channel is available in the other channel signals. However, it is possible to design exactly reconstructing analysis and synthesis systems despite existence of aliasing in every individual channel [9]. A special case can be derived letting Ri =M,0≤i≤M −1 and

is called a maximally decimated filter bank where the number of samples in the set of

{

ym(n)

}

and x(n) is equal. This type of filter bank will be

discussed in the next section.

0 Y 1 Y 1 − M Y . . . ) (z X ) ( 1z HM− ) ( 1z H ) ( 0z H R0 1 R 1 − M R Processing Processing Processing . . . 1 ^ − M Y 0 ^ Y 1 ^ Y 1 − M R 0 R 1 R ) ( 1z FM− ) ( 1z F ) ( 0 z F + ) ( ^ z X

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1.4. Maximally Decimated Filter Banks

As stated before, a simplification by setting R_i =M,0≤i≤M −1 in

the general filter bank system of Figure 7 leads to maximally decimated case. In this system, the number of samples for full band and subband signals is equal. To analyze this system, the input-output relationship can be written as [10] ) ( )} ( ) ( { 1 ) ( } ) ( ) ( { 1 ) ( 1 1 1 0 1 0 l M l M k k l k M k k k H zW F z X zW M z X z F z H M z Y

∑

∑ ∑

− = − = − = + = . 1.12

The output signal has two parts as follows:

• The first term represents the amplitude and phase distortion and its distortion function is as _{ 1 ₍ ₎ ₍ ₎_}

0

∑

− = M k k k z F z

H . For PR, the distortion

function should be a pure delay.

• The second term represents the aliased signal and its transfer function is as

_∑

− = 1 0 ) ( ) . ( M k k l k zW F z

H which in the ideal case, must be

zero.

The system can be analyzed by the use of the polyphase representation. To do this, the architecture is redrawn according to the polyphase matrices as shown in Figure 8.

Figure 8: Realization of the Analysis and Synthesis Banks Based on Polyphase Matrices. In this architecture, the matrices ( M)

z

E and R(zM) represent the

polyphase components of the analysis and synthesis filters in the sense that, the ith row of E(zM) and the ith column of R(zM) have the

) 1 (n− M+ y ) (n x ._. . . . . 1 ) 1 ( − − M z ) 2 ( − − M z 1 1 − z ) 1 ( − − M z M M M ) (_zM E _R(_zM) M M M +

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⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ = ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ = ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ − − − − − − − − − − − − − − ) ( ... ) ( : : ) ( ... ) ( . 1 : ) ( : ) ( : 1 . ) ( ... ) ( : : ) ( ... ) ( ) ( : ) ( 1 , 1 0 , 1 1 , 0 0 , 0 ) 1 ( 1 0 ) 1 ( 1 , 1 0 , 1 1 , 0 0 , 0 1 0 N N N N N N N N T N T N N N N N N N N N N N z R z R z R z R z z F z F z z E z E z E z E z H z H . 1.13

Using the noble identities, this system can further be simplified to ease the extraction of perfect reconstruction conditions as shown in Figure 9.

Figure 9: Simplified Realization of Filter Banks Using Noble Identities.

In this system, the only part affecting the PR is the product )

( ) (z R z

E since the rest can be proved to be a PR system. It can be

shown that the system is a PR system if this product is a pseudo circulant matrix. A pseudo circulant matrix is a circulant matrix i.e., a matrix whose rows are cyclically shifted versions of a sequence, but the elements below the main diagonal are multiplied by −1

z . So, the matrix is

of the form [10] ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − − − − − − − − − − ) ( ... ) ( ) ( ) ( ) ( ) ( ) ( ... ) ( ) ( ) ( ... ) ( ) ( 0 2 1 1 1 0 1 1 2 1 2 0 1 1 1 1 0 z p z p z z p z z p z p z z p z z p z p z p z z p z p z p M M M M M . 1.14

In this case, the first row is comprised of the polyphase components of distortion function ( ) ( ) ... 1( ) ) 1 ( 1 1 0 M M M M M z p z z p z z p + − + + − − − which must be a

pure delay for a PR system. As a conclusion, the condition for PR can be derived as [10] ) (n x M M M M M M + ) (z E R(z) 1 1 − z ) 1 ( − − M z 1 ) 1 ( − − M z ) 2 ( − − M z _y₍_n_{− M}₊₁₎ . . . . . .

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1 0 , 0 0 ) ( ) ( ₁ _⎥ ≤ ≤ − ⎦ ⎤ ⎢ ⎣ ⎡ = ₋ ₋ − − N r I z I z z E z R r r N δ δ 1.15

where I is the identity matrix with r being a constant. The PR _N

condition can be stated in another way. If we have the set of power complementary analysis filters, by a proper choice of the synthesis filters [3], the subband signals can be combined in a way to produce the original input signal at the output. In general, a set of filters H_k(z) is said to be complementary of order p if we have [11]

. 1 ) ( 1 0 =

∑

− = p M k j k e H ω 1.16

Here p is a positive integer. In special cases, the magnitude and power complementary filters are the set which satisfy the general equation for values of p=1,2as

. 1 ) ( , 1 ) ( 2 1 0 1 0 = =

∑

− = − = M k j k M k j k e H e H ω ω 1.17

It can be shown [11] that the higher order complementary filters can generate ordinary magnitude and power complementary filters while maintaining superior cut-off characteristics.The procedure to design a maximally decimated filter bank has the following steps [10]:

• An appropriate method should be chosen to design all the analysis filters.

• Having designed the analysis filters, polyphase matrix E(z) can be determined.

• The polyphase matrix of the synthesis filters R(z) can be determined by inverting E(z).

In general, we prefer the Finite Impulse Response (FIR) solutions which are guaranteed to be stable despite having larger delays compared to their Infinite Impulse Response (IIR) counterparts. However, the

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also. Usage of paraunitary matrices, leads to paraunitary PR filter banks which will be discussed in the next section.

1.5. Paraunitary Filter Banks

As stated before, paraunitary filter banks constitute a special class of the maximally decimated filter banks where the polyphase matrices are paraunitary. The definition of paraunitariness needs the concept of paraconjugation to be defined. This property can be defined for two types of transfer matrices as follows [10].

1. In the case of a scalar transfer function H(z), the paraconjugate is

defined as ~( ) ( −1)

∗

=H z z

H . Thus, to obtain the paracojugate, one

has to replace z by −1

z and also replace each coefficient by its

complex conjugate. On the unit circle, paraconjugation is equivalent to complex conjugation since we have

* 1 * ~ } ) ( { ) ( ) ( _ω ω ω j j j z e z e e z z H z H z H ₌ = − = = = . 1.18

2. In the case of a matrix transfer function H(z), paraconjugate is defined as ( ) ( 1) * ~ − =H z z

H T . To obtain the paracojugate, one has to

transpose the matrix, replace z by −1

z , and replace each

coefficient by its complex conjugate. On the unit circle, paraconjugation is equivalent to transpose conjugation since we have T e z e z T e z j j j z H z H z H( ) _*( 1) { ( ) } ~ ω ω ω = = − = = = . 1.19

Having these definitions, a matrix transfer function H(z), is defined to be paraunitary if H(z)H(z)=I

~

. In the case of a square matrix function, and using the concept of the inverse matrix, we have

{

}

1 ~ ) ( ) (z = H z −

H . So the paraconjugate can be derived from the inverse

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Another property of the paraunitary matrices is that if two matrices ) ( ), ( ₂ 1 z P z

P are paraunitary, then the product P₁(z)P₂(z) will also be

paraunitary. This fact can be used to make a conclusion about the system in Figure 9 . If the polyphase matrices satisfy the relationship

N I z E z

R( ) ( )= , and assuming E(z) to be paraunitary i.e. E(z)E(z)=I ~

, the PR system can be obtained choosing R(z)=E~(z). In this case, If E(z) is

FIR, then R(z) will also be FIR and there is no concern about the stability. In the next section, some properties of the paraunitary filter banks will be introduced.

1.5.1. Properties of Paraunitary PR Filter banks

The choice of matrices being paraunitary brings some useful properties as follows:

1. If the polyphase matrix E(z) is paraunitary, then E(zN) is

paraunitary also. Hence, assuming

⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ = ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ − − − − − − − 1,0 1, 1 ( 1) 1 , 0 0 , 0 1 0 : 1 ) ( ... ) ( : : ) ( ... ) ( ) ( : ) ( N N N N N N N N N N E z E z z z E z E z H z H , 1.20

it can be shown that the vector transfer function H(z) composed of all the analysis filters is paraunitary.

2. If the vector transfer function H(z) is paraunitary and assuming that its components are power complementary as

. ) ( 1 0 2 const e H N k j k =

∑

− = ω 1.21

then, we have a lossless system with one input and N outputs.

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) (z

Hk while its phase is negative. In this sense, they satisfy the

definition of the matched filters [13].

4. As a straightforward result, if the analysis filters are power complementary, then the synthesis filters are power complementary also.

A uniform DFT filter bank is a system where a cascade of DFT and IDFT matrices replaces the polyphase matrices and will be discussed in the next section.

1.6. DFT Modulated Filter Banks

Before moving to the discussion of DFT modulated filter banks, we will define the concept of uniform and non-uniform filter banks.

1.6.1. Uniform and Non-uniform Filter Bank

Based on the characteristics of the data signals, one can choose to shift the analysis and synthesis filters uniformly or non-uniformly along the frequency axis. This leads to new classes of filter banks whose sample filter characteristics are shown in Figure 10.

Figure 10: Filter Characteristics for Uniform and Non-Uniform Filter Banks.

In the uniform case, the channel filters are derived from a real linear-phase LowPass (LP) prototype filter g(n) of length L by modulation as [14] , 1 ,..., 0 , ) ( ) ( ) ( 2 ) ) 1 ( )( 5 . 0 ( * ) 2 ) 1 ( )( 5 . 0 ( − = + =ag ne− + − − a g n e + − − i M n h L n i M j i L n i M j i i π π 1.22 Uniform Non-uniform T ω T ω

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where subscript * denotes the complex conjugation. In this system, since the LP prototype has real coefficients, the channel filters are obtained by cosine modulation and will be discussed in Section 1.7. The complex multiplying factors a define the modulation phase. The synthesis filters i

are similar to the analysis filters but with a different modulation phase usually chosen so that the resulting filters are the time-reverse of the analysis filters. In this case, the overall filter bank response will have linear phase. By appropriate design of the prototype filter, the overall frequency response can be made flat also.

On the other hand, for the case of non-uniform filter banks, the analysis filter for channel i is generated by modulation of a possibly complex lowpass prototype g_i(n) of length L , as [14] _i

. 1 ,..., 0 , ) ( ) ( ) ( 2 ) ) 1 ( )( 5 . 0 ( * * ) 2 ) 1 ( )( 5 . 0 ( − = + = − − + − − + − M i e n g a e n g a n h i i i i i i L n k M j i i L n k M j i i i π π 1.23 Hence, the synthesis filters are given as

. 1 ,..., 0 , ) ( ) ( ) ( 2 ) ) 1 ( )( 5 . 0 ( * * ) 2 ) 1 ( )( 5 . 0 ( − = + = − − + − − + − M i e n g b e n g b n f i i i i i i L n k M j i i L n k M j i i i π π 1.24

Here, the term ± ( _i +0.5)

i k M

π _{defines the}

ith channel center frequency, k is i

an integer, and M_i is the decimation factor. The coefficients a ,i bi are

complex and define the modulation phase. The choice of different decimation factors M_i gives the possibility of having narrow channels at

low frequencies and wider channels at high frequencies or vice versa.

1.6.2. Uniform DFT Modulated Filter Banks

DFT filter banks can realize linear-phase analysis and synthesis filters using a proper complex modulation of a real-valued lowpass prototype filter. In an N-channel uniform filter bank, the prototype filter

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) ( ) ( , ] [ ] [ 2 2 N k j k N n k j k n p ne H z P ze h π π ₋ = = . 1.25

Assuming a non-causal prototype filter and in order to obtain causal analysis and synthesis filters, the impulse responses are delayed by

2 1 −

N

samples. Therefore, the time-domain representation of the analysis filters will be [16] 1 ,..., 0 , 1 ,..., 1 , 0 , ] 2 1 [ ] [ 2 ) 1 ( 2 − = − = − − = pn N e − − n N k M n h N n N k j k π . 1.26

The synthesis filters are identical to the analysis filters. It can be shown [16] that if the prototype filter has the zero phase property, then all the analysis and synthesis filters will be linear-phase. In the implementation phase, the polyphase decomposition can be used to reduce the implementation complexity. The polyphase components of the prototype filter can be written as

∑

− = − = 1 0 ) ( ) ( N l N l l z E z z P . 1.27

So, the analysis filters can be written as

) ( ) ( ) ( ) ( 1 0 1 0 / 2 / 2 / 2

∑

− = − − − = − − − ₌ ₌ = N l N l kl l N l N kN j N l N kl j l N k j k z P ze z e E z e z W E z H π π π , 1.28

and can be arranged in a matrix formulation as

), ( ) ( . : ) ( . ) ( . ) ( ... : : : : ... ... ... ) ( ) ( : ) ( ) ( ) ( 1 1 2 2 1 1 0 ) 1 ( ) 1 ( 2 ) 1 ( 0 ) 1 ( 2 4 2 0 ) 1 ( 2 1 0 0 0 0 0 1 2 1 0 2 z X z E z z E z z E z z E IDFT W W W W W W W W W W W W W W W W z X z H z H z H z H N N N N N N N N N N N N ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − + − − − − − − − − − − − − − − − − − − 4 4 4 4 4 4 4 8 4 4 4 4 4 4 4 7 6 1.29 where j N e W = − 2π/ .

As a conclusion, the whole analysis bank can be implemented at the cost of one filter plus an IDFT as shown in Figure 11. At the design

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phase, we only need to design the prototype filter since the other filters are shifted versions of the prototype filter.

Figure 11: Analysis Bank Polyphase Realization of DFT Modulated Filter Banks.

To analyze the system in another way, we can use the definition of a transfer function. The relationship between signals xi(n) and yk(n) can

be written as [3]

∑

− = − = 1 0 ) ( 1 ) ( N i ik i k x nW M n y . 1.30

By defining the transfer function

) ( ) ( ) (z Y z _X _z H k

k = , it can be verified that

) (z

Hk is a shifted version of a prototype response P(z) through Equation

1.26 namely ( ) ( k)

k z P zW

H = [3].

To analyze the synthesis side, we assume the synthesis filters to be

) ( ),..., ( ), ( 1 1 0 z F z F z

F _N− that satisfy the time reverse property as

) ( ) ( j2k/N j2k/N k z e P ze F = π − π . 1.31

Again, using the polyphase representation, we have

) ( ) ( 1 0

∑

− = − =N l N l l z R z z P . 1.32

In general, for all of the synthesis filters, we have

) (n x 1 − z 1 − z N N N . . . . . . IDFT ) ( 0 n y ) ( 1 n y ) ( 1 n y_N− ) ( 0 n x ) ( 1 n x_N− ) ( 1 n x ) ( 1 z E ) ( 0 z E ) ( 1 z EN−

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[ ] ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = − − ) ( : ) ( ) ( ) ( ) ( ... ) ( ) ( ) ( ) ( 1 2 1 0 1 2 1 0 z Y z Y z Y z Y z F z F z F z F z Y N N . 1.34

which using Equation 1.34 can be rewritten as

[

]

. ) ( : ) ( ) ( ) ( ... : : : : ... ... ... ) ( ) ( ) ( ... ) ( 1 2 1 0 ) 1 ( ) 1 ( 2 ) 1 ( 0 ) 1 ( 2 4 2 0 ) 1 ( 2 1 0 0 0 0 0 0 1 1 2 2 1 1 2 ⎥⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − − − − − − − − − + − z Y z Y z Y z Y DFT W W W W W W W W W W W W W W W W z R z R z z R z z R z N N N N N N N N N N N N 4 4 4 4 4 4 4 8 4 4 4 4 4 4 4 7 6 1.35 Finally, the architecture of the synthesis bank can be drawn as Figure 12.

Figure 12: Analysis Bank Polyphase Realization of DFT Modulated Filter Banks.

where ( ) 1 ( )

1 z E z

R_l = _N−₋₋_l and is the second type of the polyphase

representation [3]. In this scheme, the first filter should be centered at

0 =

ω but using a Generalized Discrete Fourier Transform (GDFT) [7], this constraint can be removed. It must be mentioned that by appropriate choices of GDFT matrix, one can obtain a filter bank with

2

M filters

having real coefficients for even M .

In the GDFT case, the analysis filters hk(n) are derived from a

real-valued LP prototype FIR filter p(n) that has even length L as [17]

N n k e n p n h jM k k n n k = ∈ + + , , ) ( ) ( ( )( ) 2 0 0 π . 1.36 DFT ) ( 0n y ) ( 1n yN− ) ( 1n y . . . ) ( 1 z RN− ) ( 2 z RN− ) ( 0 z R + + y(n) 1 − z 1 − z N N N

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Here, the offsets k₀and n0 are introduced leading to the name GDFT. Choosing a linear-phase prototype filter and setting n0 in a way to have a transform symmetric to L−12, the modulated filters will have the

linear-phase property also. If we choose k0=0.5, the frequency range(0,2π)will

be covered by M 2 subbands for even M. In this case, the remaining

subbands are complex conjugate versions and can be ignored in the processing reducing the complexity. So, we have a filter bank with M 2

filters. The synthesis filters can be obtained by time reversion of the analysis filter as ( )₌ *( ₋ ₊1)

n L h n

f_k _k . Thus, all filters can be derived from

one single prototype. The procedure to design these filter banks can be summarized in the following steps [10]:

• The prototype filter must be designed according to the system requirements.

• Having the prototype filter, the polyphase components Ek(z) can

be achieved.

• Assuming that E_k(z) can be inverted, the synthesis filters can be

chosen as ( ) 1 ( )

1 z

E z

R_k = _N−−−_k .

It can be shown that the maximally decimated DFT filter banks at the same time satisfy perfect reconstruction, have FIR analysis and synthesis filters, and are paraunitary.

As a simple case, assume a prototype filter of the form

1 2 1 _... 1 ) ( ₌ ₊ − ₊ − ₊ ₊ −N+ z z z z P , 1.37

which leads in polyphase components as delays reducing the filter bank structure to Figure 13. ) (n x ) 1 (n− M+ y M M M M M M

+

IDFT DFT ) 2 ( − − M z 1 ) 1 ( − − M z 1 1 − z ) 1 ( − − M z . . . . . .

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system response to be still a delay but doing so, the filtering order must be changed. The analysis to derive equations for the analysis and synthesis filters is similar to the general case. It can be shown that the analysis filters can be modeled as

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ + − − − − − − − − − 1 2 1 ) 1 ( ) 1 ( 2 1 0 ) 1 ( 2 4 2 0 1 2 1 0 0 0 0 0 1 2 1 0 : 1 ... : : : : ... ... ... ) ( : ) ( ) ( ) ( 2 _N DFT N N N N N N z z z W W W W W W W W W W W W W W W W z H z H z H z H 4 4 4 4 4 4 8 4 4 4 4 4 4 7 6 . 1.38

In other words, we can assume the analysis filters as ) ( ) ( j j( 2 k/N) k e P e H ω = ω−π , 1.39

which are obviously the uniformly shifted versions of the prototype filter.

1.7. Cosine

Modulated

Filter

Banks

In uniform DFT modulated filter banks, assuming P(z) to be the

prototype lowpass filter with a cutoff frequency at ±π/N, the analysis

filters can be derived as ( ) ( )

2 N k j k z P ze H π −

= . Cosine modulated filter banks

are defined by the use of Discrete Cosine Transform (DCT) which has four types as [18]

On Filter Bank Based MIMO Frequency Multiplexing and Demultiplexing

Title

On Filter bank Based MIMO Frequency Multiplexing and

Demultiplexing

Master thesis in Electronics Systems

at Linköping Institute of Technology

by

Abstract

Acknowledgments

Foreword

Outline of Tasks

Table of Contents

List of Abbreviations

Chapter One: Overview of Multirate Systems

and Filter Banks

1. Introduction

1.1.

Basic Building Blocks of Multirate Systems

∑

∑

{ }

[

]

∑

∑

∑

∑

[ ]

[ ]

[ ]

[ ]

[ ]

[ ]

[ ]

[ ]

[ ]

[ ]

[ ]

[ ]

[

]

[ ]

[

]

∑

∑

1.2.

Digital Filter Banks

1.3.

General Filter Bank Architecture

∑

{

}

{

}

∑ ∑

∑

{

}

1.4.

Maximally Decimated Filter Banks

∑

∑ ∑

∑

∑

∑

∑

∑

1.5.

Paraunitary Filter Banks

{

}

∑

1.6.

DFT Modulated Filter Banks

∑

∑

∑

∑

∑

_∑