Transceiver designs and analysis for LTI, LTV and broadcast channels - new matrix decompositions and majorization theory

Signal processing techniques play important roles in the design of digital communication systems. These include information manipulation, transmitter signal processing, channel estimation, channel equalization and receiver signal processing. By interacting with communication theory and system implementing technologies, signal processing specialists develop efficient schemes for various communication problems by wisely exploiting various mathematical tools such as analysis, probability theory, matrix theory, optimization theory, and many others. In recent years, researchers realized that multiple-input multiple-output (MIMO) channel models are applicable to a wide range of different physical communications channels. Using the elegant matrix-vector notations, many MIMO transceiver (including the precoder and equalizer) design problems can be solved by matrix and optimization theory. Furthermore, the researchers showed that the majorization theory and matrix decompositions, such as singular value decomposition (SVD), geometric mean decomposition (GMD) and generalized triangular decomposition (GTD), provide unified frameworks for solving many of the point-to-point MIMO transceiver design problems.

In this thesis, we consider the transceiver design problems for linear time invariant (LTI) flat MIMO channels, linear time-varying narrowband MIMO channels, flat MIMO broadcast channels, and doubly selective scalar channels. Additionally, the channel estimation problem is also considered. The main contributions of this dissertation are the development of new matrix decompositions, and the uses of the matrix decompositions and majorization theory toward the practical transmit-receive scheme designs for transceiver optimization problems. Elegant solutions are obtained, novel transceiver structures are developed, ingenious algorithms are proposed, and performance analyses are derived.

The first part of the thesis focuses on transceiver design with LTI flat MIMO channels. We propose a novel matrix decomposition which decomposes a complex matrix as a product of several sets of semi-unitary matrices and upper triangular matrices in an iterative manner. The complexity of the new decomposition, generalized geometric mean decomposition (GGMD), is always less than or equal to that of geometric mean decomposition (GMD). The optimal GGMD parameters which yield the minimal complexity are derived. Based on the channel state information (CSI) at both the transmitter (CSIT) and receiver (CSIR), GGMD is used to design a butterfly structured decision feedback equalizer (DFE) MIMO transceiver which achieves the minimum average mean square error (MSE) under the total transmit power constraint. A novel iterative receiving detection algorithm for the specific receiver is also proposed. For the application to cyclic prefix (CP) systems in which the SVD of the equivalent channel matrix can be easily computed, the proposed GGMD transceiver has K/log_2(K) times complexity advantage over the GMD transceiver, where K is the number of data symbols per data block and is a power of 2. The performance analysis shows that the GGMD DFE transceiver can convert a MIMO channel into a set of parallel subchannels with the same bias and signal to interference plus noise ratios (SINRs). Hence, the average bit rate error (BER) is automatically minimized without the need for bit allocation. Moreover, the proposed transceiver can achieve the channel capacity simply by applying independent scalar Gaussian codes of the same rate at subchannels.

In the second part of the thesis, we focus on MIMO transceiver design for slowly time-varying MIMO channels with zero-forcing or MMSE criterion. Even though the GGMD/GMD DFE transceivers work for slowly time-varying MIMO channels by exploiting the instantaneous CSI at both ends, their performance is by no means optimal since the temporal diversity of the time-varying channels is not exploited. Based on the GTD, we develop space-time GTD (ST-GTD) for the decomposition of linear time-varying flat MIMO channels. Under the assumption that CSIT, CSIR and channel prediction are available, by using the proposed ST-GTD, we develop space-time geometric mean decomposition (ST-GMD) DFE transceivers under the zero-forcing or MMSE criterion. Under perfect channel prediction, the new system minimizes both the average MSE at the detector in each space-time (ST) block (which consists of several coherence blocks), and the average per ST-block BER in the moderate high SNR region. Moreover, the ST-GMD DFE transceiver designed under an MMSE criterion maximizes Gaussian mutual information over the equivalent channel seen by each ST-block. In general, the newly proposed transceivers perform better than the GGMD-based systems since the super-imposed temporal precoder is able to exploit the temporal diversity of time-varying channels. For practical applications, a novel ST-GTD based system which does not require channel prediction but shares the same asymptotic BER performance with the ST-GMD DFE transceiver is also proposed.

The third part of the thesis considers two quality of service (QoS) transceiver design problems for flat MIMO broadcast channels. The first one is the power minimization problem (min-power) with a total bitrate constraint and per-stream BER constraints. The second problem is the rate maximization problem (max-rate) with a total transmit power constraint and per-stream BER constraints. Exploiting a particular class of joint triangularization (JT), we are able to jointly optimize the bit allocation and the broadcast DFE transceiver for the min-power and max-rate problems. The resulting optimal designs are called the minimum power JT broadcast DFE transceiver (MPJT) and maximum rate JT broadcast DFE transceiver (MRJT), respectively. In addition to the optimal designs, two suboptimal designs based on QR decomposition are proposed. They are realizable for arbitrary number of users.

Finally, we investigate the design of a discrete Fourier transform (DFT) modulated filterbank transceiver (DFT-FBT) with LTV scalar channels. For both cases with known LTV channels and unknown wide sense stationary uncorrelated scattering (WSSUS) statistical channels, we show how to optimize the transmitting and receiving prototypes of a DFT-FBT such that the SINR at the receiver is maximized. Also, a novel pilot-aided subspace channel estimation algorithm is proposed for the orthogonal frequency division multiplexing (OFDM) systems with quasi-stationary multi-path Rayleigh fading channels. Using the concept of a difference co-array, the new technique can construct M^2 co-pilots from M physical pilot tones with alternating pilot placement. Subspace methods, such as MUSIC and ESPRIT, can be used to estimate the multipath delays and the number of identifiable paths is up to O(M^2), theoretically. With the delay information, a MMSE estimator for frequency response is derived. It is shown through simulations that the proposed method outperforms the conventional subspace channel estimator when the number of multipaths is greater than or equal to the number of physical pilots minus one.

Generalized modal identification of linear and nonlinear dynamic systems

This dissertation is concerned with the problem of determining the dynamic characteristics of complicated engineering systems and structures from the measurements made during dynamic tests or natural excitations. Particular attention is given to the identification and modeling of the behavior of structural dynamic systems in the nonlinear hysteretic response regime. Once a model for the system has been identified, it is intended to use this model to assess the condition of the system and to predict the response to future excitations.

A new identification methodology based upon a generalization of the method of modal identification for multi-degree-of-freedom dynaimcal systems subjected to base motion is developed. The situation considered herein is that in which only the base input and the response of a small number of degrees-of-freedom of the system are measured. In this method, called the generalized modal identification method, the response is separated into "modes" which are analogous to those of a linear system. Both parametric and nonparametric models can be employed to extract the unknown nature, hysteretic or nonhysteretic, of the generalized restoring force for each mode.

In this study, a simple four-term nonparametric model is used first to provide a nonhysteretic estimate of the nonlinear stiffness and energy dissipation behavior. To extract the hysteretic nature of nonlinear systems, a two-parameter distributed element model is then employed. This model exploits the results of the nonparametric identification as an initial estimate for the model parameters. This approach greatly improves the convergence of the subsequent optimization process.

The capability of the new method is verified using simulated response data from a three-degree-of-freedom system. The new method is also applied to the analysis of response data obtained from the U.S.-Japan cooperative pseudo-dynamic test of a full-scale six-story steel-frame structure.

The new system identification method described has been found to be both accurate and computationally efficient. It is believed that it will provide a useful tool for the analysis of structural response data.

Relativistic mean field theory: methods and applications

We develop a method for performing one-loop calculations in finite systems that is based on using the WKB approximation for the high energy states. This approximation allows us to absorb all the counterterms analytically and thereby avoids the need for extreme numerical precision that was required by previous methods. In addition, the local approximation makes this method well suited for self-consistent calculations. We then discuss the application of relativistic mean field methods to the atomic nucleus. Self-consistent, one loop calculations in the Walecka model are performed and the role of the vacuum in this model is analyzed. This model predicts that vacuum polarization effects are responsible for up to five percent of the local nucleon density. Within this framework the possible role of strangeness degrees of freedom is studied. We find that strangeness polarization can increase the kaon-nucleus scattering cross section by ten percent. By introducing a cutoff into the model, the dependence of the model on short-distance physics, where its validity is doubtful, is calculated. The model is very sensitive to cutoffs around one GeV.

Mean-speed measurements in two-dimensional, incompressible, fully-developed turbulent channel flow

Mean velocity profiles were measured in the 5” x 60” wind channel of the turbulence laboratory at the GALCIT, by the use of a hot-wire anemometer. The repeatability of results was established, and the accuracy of the instrumentation estimated. Scatter of experimental results is a little, if any, beyond this limit, although some effects might be expected to arise from variations in atmospheric humidity, no account of this factor having been taken in the present work. Also, slight unsteadiness in flow conditions will be responsible for some scatter.

Irregularities of a hot-wire in close proximity to a solid boundary at low speeds were observed, as have already been found by others.

That Kármán’s logarithmic law holds reasonably well over the main part of a fully developed turbulent flow was checked, the equation u/u_t = 6.0 + 6.25 log₁₀ y^ut/v being obtained, and, as has been previously the case, the experimental points do not quite form one straight line in the region where viscosity effects are small. The values of the constants for this law for the best over-all agreement were determined and compared with those obtained by others.

The range of Reynolds numbers used (based on half-width of channel) was from 20,000 to 60,000.

A generalized treatment of the order-disorder transformation in alloys and its effects on their magnetic properties

A theory of the order-disorder transformation is developed in complete generality. The general theory is used to calculate long range order parameters, short range order parameters, energy, and phase diagrams for a face centered cubic binary alloy. The theoretical results are compared to the experimental determination of the copper-gold system, Values for the two adjustable parameters are obtained.

An explanation for the behavior of magnetic alloys is developed, Curie temperatures and magnetic moments of the first transition series elements and their alloys in both the ordered and disordered states are predicted. Experimental agreement is excellent in most cases. It is predicted that the state of order can effect the magnetic properties of an alloy to a considerable extent in alloys such as Ni₃Mn. The values of the adjustable parameter used to fix the level of the Curie temperature, and the adjustable parameter that expresses the effect of ordering on the Curie temperature are obtained.

On an embedding property of generalized Carter subgroups

If E and F are saturated formations, we say that E is strongly contained in F if for any solvable group G with E-subgroup, E, and F-subgroup, F, some conjugate of E is contained in F. In this paper, we investigate the problem of finding the formations which strongly contain a fixed saturated formation E.

Our main results are restricted to formations, E, such that E = {G|G/F(G) ϵT}, where T is a non-empty formation of solvable groups, and F(G) is the Fitting subgroup of G. If T consists only of the identity, then E=N, the class of nilpotent groups, and for any solvable group, G, the N-subgroups of G are the Carter subgroups of G.

We give a characterization of strong containment which depends only on the formations E, and F. From this characterization, we prove:

If T is a non-empty formation of solvable groups, E = {G|G/F(G) ϵT}, and E is strongly contained in F, then

(1) there is a formation V such that F = {G|G/F(G) ϵV}.

(2) If for each prime p, we assume that T does not contain the class, S_p’, of all solvable p’-groups, then either E = F, or F contains all solvable groups.

This solves the problem for the Carter subgroups.

We prove the following result to show that the hypothesis of (2) is not redundant:

If R = {G|G/F(G) ϵS_r’}, then there are infinitely many formations which strongly contain R.

Extension theorems for functions of vanishing mean oscillation

A locally integrable function is said to be of vanishing mean oscillation (VMO) if its mean oscillation over cubes in R^d converges to zero with the volume of the cubes. We establish necessary and sufficient conditions for a locally integrable function defined on a bounded measurable set of positive measure to be the restriction to that set of a VMO function.

We consider the similar extension problem pertaining to BMO(ρ) functions; that is, those VMO functions whose mean oscillation over any cube is O(ρ(l(Q))) where l(Q) is the length of Q and ρ is a positive, non-decreasing function with ρ(0⁺) = 0.

We apply these results to obtain sufficient conditions for a Blaschke sequence to be the zeros of an analytic BMO(ρ) function on the unit disc.

A generalized Hausdorff dimension for functions and sets

Let E be a compact subset of the n-dimensional unit cube, 1_n, and let C be a collection of convex bodies, all of positive n-dimensional Lebesgue measure, such that C contains bodies with arbitrarily small measure. The dimension of E with respect to the covering class C is defined to be the number

d_C(E) = sup(β:H_{β, C}(E) > 0),

where H_{β, C} is the outer measure

inf(Ʃm(C_i)^β:UC_i Ↄ E, C_i ϵ C) .

Only the one and two-dimensional cases are studied. Moreover, the covering classes considered are those consisting of intervals and rectangles, parallel to the coordinate axes, and those closed under translations. A covering class is identified with a set of points in the left-open portion, 1’_n, of 1_n, whose closure intersects 1_n - 1’_n. For n = 2, the outer measure H_{β, C} is adopted in place of the usual:

Inf(Ʃ(diam. (C_i))^β: UC_i Ↄ E, C_i ϵ C),

for the purpose of studying the influence of the shape of the covering sets on the dimension d_C(E).

If E is a closed set in 1₁, let M(E) be the class of all non-decreasing functions μ(x), supported on E with μ(x) = 0, x ≤ 0 and μ(x) = 1, x ≥ 1. Define for each μ ϵ M(E),

d_C(μ) = lim/c → inf/0 log ∆μ(c)/log c , (c ϵ C)

where ∆μ(c) = v/x (μ(x+c) – μ(x)). It is shown that

d_C(E) = sup (d_C(μ):μ ϵ M(E)).

This notion of dimension is extended to a certain class Ӻ of sub-additive functions, and the problem of studying the behavior of d_C(E) as a function of the covering class C is reduced to the study of d_C(f) where f ϵ Ӻ. Specifically, the set of points in 1₁,

(*) {d_B(F), d_C(f)): f ϵ Ӻ}

is characterized by a comparison of the relative positions of the points of B and C. A region of the form (*) is always closed and doubly-starred with respect to the points (0, 0) and (1, 1). Conversely, given any closed region in 1₂, doubly-starred with respect to (0, 0) and (1, 1), there are covering classes B and C such that (*) is exactly that region. All of the results are shown to apply to the dimension of closed sets E. Similar results can be obtained when a finite number of covering classes are considered.

In two dimensions, the notion of dimension is extended to the class M, of functions f(x, y), non-decreasing in x and y, supported on 1₂ with f(x, y) = 0 for x · y = 0 and f(1, 1) = 1, by the formula

d_C(f) = lim/s · t → inf/0 log ∆f(s, t)/log s · t , (s, t) ϵ C

where

∆f(s, t) = V/x, y (f(x+s, y+t) – f(x+s, y) – f(x, y+t) + f(x, t)).

A characterization of the equivalence d_C1(f) = d_C2(f) for all f ϵ M, is given by comparison of the gaps in the sets of products s · t and quotients s/t, (s, t) ϵ C_i (I = 1, 2).