The Roles of Majorization and Generalized Triangular Decomposition in Communication and Signal Processing

Signal processing is an art that deals with the representation, transformation and manipulation of the signals and the information they contain based on their specific characteristics. The second part of the thesis focuses on signal processing algorithms for data compression and filter bank designs.

MIMO Transceiver Optimization

MIMO Channel Models

Two of the most common techniques for this are the zero-padding and cyclic prefix precoding techniques [89]. The zero-padding precoding produces the effective frequency flat channel matrix H with Toeplitz structure, and the cyclic-prefix precoding produces circular matrix H.

Figure 1.1: (a) Frequency flat MIMO channel model. (b) Frequency selective MIMO channel model.

Transceiver Optimization and History

Many of the existing works were shown as special cases within this framework. Most of the results described above assume that the channel, precoder, and equalizer are constant matrices.

Figure 1.2: (a) The general form of linear transceivers with channel H(e jω ) , precoder F(e jω ) , and equalizer G(e jω )

Transform Coder and Signal Adapted Filter Bank Optimization

For the special case where the polyphase matrix E(ejω) is paraunitary (ie, unitary for all) and R(ejω) =E†(ejω), the filter bank is called an anorthonormal filter bank. A filterbank whose filters depend on knowledge of the input statistics is called a signal-matched filterbank.

Figure 1.3: The M -channel maximally decimated filter bank with uniform decimation ratio M .

Outline and Scope of the Thesis

• Review of Majorization, Matrix Theory, and Generalized Triangular Decom-
• Transceiver Designs for MIMO Frequency Flat Channels (Chapter 3)
• Transceiver Designs for MIMO Frequency Selective Channels (Chapter 4)
• The Role of GTD in Transform Coding (Chapter 5)
• The Role of GTD in Filter Bank Optimization (Chapter 6)

Other special cases of GTD-TC are GMD (geometric mean decomposition) and BID (bidiagonal transformation). However, the performance of the GMD transform encoder degrades significantly in the low-rate case.

Notations

Additive Majorization and Schur Convexity

Additionally, φ is said to be Schur concave if and only if−φ is Schur convex. Note that the sets of Schur-concave and Schur-convex functions do not constitute a partition of the set of all functions.

Multiplicative Majorization

There are a number of simple but useful facts regarding compositions involving Schur-convex and Schur-concave functions (p.61 of [65]). The following theorem is a direct consequence of the composition rule of Schur convex functions with increasing convex functions.

Figure 2.1: The illustration of the relations between sets of functions [41].

Relation to Matrix Theory

Hermitian Matrices

Complex-Valued Square Matrices

Generalized Triangular Decomposition

Block-Diagonal Geometric Mean Decomposition

In BD-GMD, one of the identity matrices in (2.4) is restricted to the block diagonal. Third, for the case of linear receivers and transmitters, the joint optimization of the precoder, (linear) equalizer and bit assignment was studied in [48] (under ZF constraint) and in [74] (without ZF constraint).

MIMO Transceivers with Decision Feedback and Bit Loading

Problem Formulation

In the following sections, we first discuss the problem of minimizing the transmitted power subject to a specified total bit rate and a specified error probability at each substream. In this section we will consider the error probability Pe(k) as a quality of service (QoS) specification.

Figure 3.1: The MIMO transceiver with linear precoder and DFE.

Minimum Power Achieved by DFE Systems

Equality can be achieved in the AM-GM inequality if and only if the terms are identical for allk, i.e. 1 In general (3.15) can give an incomplete or even negative numberbk. However, in the case of high bitrate (largeb), it is large enough to be replaced by integer values ​​without compromising the optimization too much.

GTD-Based Transceivers

With the above choice of transmitter matrices, the error variance (3.5) in the subcurrent becomes 3.8) the transmitted power required to meet the specified QoS constraints and the bit rate can be expressed as. The QR decomposition of the channel matrix can be written as H = QR, where Q has orthonormal columns and is upper triangular.

Figure 3.2: The proposed form of optimal solution for the DFE transceiver.

Other Transceiver Problems Solved by GTD-Based Transceiver

The bit rate maximization problem subject to a transmitted power constraint is the counterpart of the problem described in Eq. Thus, the bit rate maximization problem is reduced to maximizing (3.42) subject to zero forcing.

Simulation Results with Perfect CSI

This means that there exists a GTD for the Hi channel such that both (3.36) and the full bit distribution (3.46) hold simultaneously. Example 1: High bit rate case: In this example we consider GTD transmitters with approximate bit allocation (3.26).

Figure 3.5: Example 1. BER versus Tx-Power for P

Simulation Results with Limited Feedback

From the plots, we see that the proposed “QR-limited-FB” scheme significantly outperforms the state-of-the-art limited feedback schemes [56, 91] and comes close to the optimal “GMD-perfect-CSI” scheme. Even with such limited feedback, the proposed "QR-limited-FB" scheme works very well.

Figure 3.9: BER versus Tx-Power with limited feedback (8 feedback bits per block, and 32 bits transmitted per block).

Concluding Remarks

MIMO Transceivers with Linear Constraints on Transmit Covariance Matrix

• Signal Model and Problem Formulation
• Linear Transceivers
• DFE Transceivers
• Numerical Simulations
• Concluding Remarks

In the first step we will minimize the AM-MSE (arithmetic mean of the mean square error) of the system. In the first step we will minimize the GM-MSE (geometric mean mean square error) of the system.

Figure 3.11: The system with linear precoding and DFE.

Conclusions

If the precoder matrix Fi is not square, for example, when the channel matrix is ​​HisN×P and P > M, the rank of U must not be greater than M. When P > M, suppose we first relax the rank constraint, then the rank-relaxing covariance matrixU is solved by the SDP solver as before.

Appendix

Proofs of Lemma 3.2.1

We have shown that the semi-definite programming (SDP) technique provides a nice framework for unifying the design of linear and DFE receivers.

Proofs of Theorem 3.2.4

Therefore, ZP-BD-GMD systems have a much smaller implementation cost than optimal systems. In particular, ZP-BD-GMD systems are average BER minimizers within the family of systems using unitary block diagonal precoders.

Signal Model

In the following sections, we will discuss some important properties of the proposed ZP-BD-GMD receivers. In addition, the receiver structure of the ZP-BD-GMD systems is computationally simple.

Transceivers with Zero-Forcing DFEs

It is therefore important to characterize the performance of the ZF-BD-GMD system for finite block sizes. The following theorem represents the optimality of the ZF-BD-GMD system for any finite block size.

Figure 4.1: The ZP-BD-GMD transceiver. The signal vector s i is first linear precoded by the unitary matrix P i

Transceivers with MMSE DFEs

Similar to the ZF case, in the following we derive the BD-GMD system for the MMSE counterpart. We can prove that within this family, the MMSE-BD-GMD system is one of the minimizers for the average BER.

Trade-Off between BW Efficiency and Performance

Since the precoder matrix in the MMSE-BD-GMD system is diagonal block, the transmitter implementation cost is the same as the ZF-BD-GMD case. For the receiver part, we can prove that the implementation cost of the receiver is the same as that of the ZF-BD-GMD system.

ZP for SISO Frequency Selective Channel

Thus, for the example of a frequency-selective SISO channel, systems with a lazy precoder and ZF-DFE are asymptotically optimal in the class of systems with linear precoders and ZF-DFE receivers. In particular, systems with a lazy precoder and MMSE-DFE are asymptotically optimal in the class of systems with a linear unity precoder and an MMSE-DFE receiver.

Numerical Simulations

We can see that the ZF-BD-GMD system indeed has similar performance to the ZF-Optimal system. ZFBDG” represents the ZF-BD-GMD system; "ZFOPT" represents the ZF-Optimal system; and "ZF-Lazy" represents the lazy precoder with zero-forcing DFE.

Fig. 4.3 shows the subchannel gains. For the ZF-BD-GMD systems, the first 10 subchannel gains when K = 5 are the same as that when K = 10

Concluding Remarks

Appendix

Proof of Lemma 4.3.1

Proof of Theorem 4.3.5

Therefore, to prove this theorem, what we need to prove is that the vector consisting of the absolute values ​​of the subchannel gains of the P0-BD system, the vector consisting of the subchannel gains of the ZF-BD-GMD- system exists, multiplicatively majorizes. Let ap be defined as the product of the largest pabsolute values ​​of the subchannel gains of the P0-BD system, and let bp be defined as the product of the largest psubchannel gains of the ZF-BD-GMD system.

Proof of Theorem 4.3.6

The above advantage of the GMD encoder is shown to be true in the high bit rate case. However, the performance of the GMD transform encoder is degraded in the low-speed case.

GTD Transform Coder for Optimizing Coding Gain

Preliminaries and Reviews

Under the assumption of a high bit rate (5.1), the optimal bit allocation is given by the bit loading formula [35, 107]. At each step of both the Minlab encoder and decoder, a prediction is made based on the quantized data, while in the structure in Fig.

Figure 5.1: Schematic of a transform coder with scalar quantizers.

Generalized Triangular Decomposition Transform Coder

Note that this result is true because of the minimum noise structure for the PLT (which has unity noise gain). The first group is the significant group where the K1 data streams contain a rough approximation of the signal.

Figure 5.4: The GTD transform coder implemented using MINLAB(I) structure.

Simulations

It can be seen from the figure that at the optimal bit load, all GTD-TCs perform approximately the same. It can be seen from the figure that without applied bit load, GMD performs much better than other methods, since GMD without bit allocation is theoretically as good as other methods with optimal bit allocation.

Figure 5.7: Performance of different transform coders with optimal bit allocation. Input covariance matrix has a high condition number ( 10 7 ).

Dithered GMD Transform Coder for Low Rate Applications

Dithered GMD Quantizer

In the GMD-NSD encoder, the quantized signal is directly multiplied by the predictive coefficients for use of subsequent substream quantizers, without first being subtracted from the dither. The complexity of the successive decompositions in the above algorithm is in the same order as that in the GMD transform coder described in [122].

Figure 5.12: Subtractive dithered GMD transform coder.

Numerical Example

5.15, we see that in the low-rate regime, the two proposed split transform encoders perform better than all other transform encoders. In the high-rate regime, the two proposed GMD split transform encoders perform comparably to the three encoders ("KLTwBL", "PLTwBL", and "GMD") which are designed under the high-rate assumption.

Figure 5.15: Performance of different transform coders.

Concluding Remarks

We will first show that there are two fundamental properties in both the optimal orthonormal GTD FB and the optimal biorthogonal GTD FB, namely, total correlation and spectrum equalization. We will show that the optimal systems are related to the frequency-dependent GTD of the channel response matrix.

Subband Coder Signal Model

The psd matrix of the errore(n) is therefore See(ejω) =R(ejω)Sqq(ejω)R†(ejω)The average mean square error of the coderεcoderis3. 4The optimal bit loading formula for conventional filter banks with perfect reconstruction usually has the granularity problem, that is, the number of bits in the formula must be rounded to the nearest integer when used in practice.

Figure 6.1: The biorthogonal GTD subband coders for M = 4.

Optimal Orthonormal GTD Filter Banks

If this filter pair also results in the spectrum equalizing property, then the product of the subband variances will be. 6.3.4 we know that the optimal GTD coder will produce the product of the subband variances.

Biorthogonal GTD Filter Banks

For the case where the optimal orthonormal GTD filter banks are designed as GMD filter banks, the diagonal matrix D(ejω) can be written as D(ejω) = Mp. To summarize from Theorem 6.4.1, Corollary 6.4.4 and Corollary 6.4.5, the optimal biorthogonal GTD filter bank also has total decorrelation and spectrum equalization as the two necessary conditions.

Figure 6.2: A restricted case of the biorthogonal GTD subband coders for M = 4.

Performance Comparison of Optimal Filter Banks Designs

In each case, we also state the necessary and sufficient conditions for optimal solutions. In the following, we compare the coding performance of the optimal subband encoders in these four cases.

Table 6.1: Features of Optimal Filter Banks Used in Subband Coders φ (Coding Gain = σ x2 /φ 1/M ) Nec

The Role of Frequency Dependent GTD in Transceivers for the QoS Problem

Transceivers with Orthonormal Precoder Constraint

We assume at frequencyω, H(ejω) has rankKω, andKω≥M for allω due to the zero-forcing assumption, i.e., there is no channel zero. We are now ready to design our transceiver based on this particular GTD shape of the channel matrix.

Transceivers with Arbitrary Precoder

We have derived the optimal transceiver designs for the case of orthonormal precoder and unconstrained precoder. The results of Theorems 6.6.3 and 6.6.5 provide elegant design methods for both cases – the optimal orthonormal precoder transceiver design can be obtained from a frequency-dependent GTD form of the channel matrix; the optimal unconstrained ZF transceivers can be obtained by first designing the optimal orthonormal transceiver and then cascading with the filter λ(ejω).

Concluding Remarks

This is very similar to the subband coder case, where the optimal biorthogonal GTD subband coders can be obtained by using the optimal orthonormal GTD subband coders and a series of scalar filters.

Appendix

• Proof of Theorem 6.3.1
• Proof of Theorem 6.3.3
• Proof of Lemma 6.6.1
• Proof of Lemma 6.6.2

Vaidyanathan, “Filterbank Optimization with Convex Objectives and Optimality of Principal Component Shapes,” IEEE Trans. Ottersten, “Joint bit allocation and precoding for MIMO systems with decision feedback sensing,” IEEE Trans.

The M -channel maximally decimated filter bank with uniform decimation ratio M

The polyphase representation of the M -channel maximally decimated filter bank

The illustration of the relations between sets of functions [41]

The MIMO transceiver with linear precoder and DFE

The proposed form of optimal solution for the DFE transceiver

The SVD system, which represents a linear transceiver

The QR transceiver, which has the lazy precoder. This is identical to the ZF-VBLAST

Example 1. BER versus Tx-Power for P

Example 2. BER versus Tx-Power for P

BER versus Tx-Power with limited feedback (8 feedback bits per block, and 32 bits

BER versus Tx-Power with limited feedback (8 feedback bits per block, and 24 bits

The system with linear precoding and DFE

A direct implementation of the PLT

The PLT implemented using MINLAB(I) structure

The GTD transform coder implemented using MINLAB(I) structure

The BID Transform coder implemented using MINLAB(I) structure

Use of GTD-TC in the progressive transmission context

Performance of different transform coders with optimal bit allocation. Input covari-

Performance of different transform coders with optimal bit allocation. Input covari-

Comparison of coding gain of different transform coders with optimal bit allocation

Performance of different transform coders with uniform bit allocation. Input covari-

Performance of different transform coders with uniform bit allocation. Input covari-

Subtractive dithered GMD transform coder

Nonsubtractive dithered GMD transform coder

The equivalent model of dithered GMD transform coder

Performance of different transform coders

The biorthogonal GTD subband coders for M = 4

A restricted case of the biorthogonal GTD subband coders for M = 4

Coding gain of subband coders with M = 3 for the AR(1) process with ρ from 0.85 to

Coding gain of subband coders with M = 4 for the AR(2) process with ρ from 0.95 to

Monotone behavior of the coding gain as a function of the number of channels for the

Nonmonotone behavior of the coding gain as a function of the number of channels for

Schematic of a frequency selective transceiver with linear precoder and zero-forcing