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In this paper, we consider the design of statistical multiple-input-multiple-output orthogonal frequency division multiplexing (MIMO-OFDM) beamformers for millimeter wave (mmWave) channels. The transmitter designs the subcarrier beamformers based on the statistics of the channel, without instantaneous channel information. To overcome the radio frequency (RF) limitation in mmWave application, the subcarrier beamformers are implemented in a hybrid structure, which imposes constraints on the design of subcarrier beamformers. We analyze the unconstrained statistical subcarrier beamformers using spectral analysis of the subcarrier channels. The analysis shows that, for each subcarrier channel, the optimal statistical beamformer is approximately a linear combination of optimal statistical beamformers for some appropriately defined narrowband single-cluster subchannels. The result suggests a design of the subcarrier beamformers that can be readily implemented in a hybrid structure. Furthermore, a hybrid design for the receiver is proposed based on the concept of vector quantization. Simulations are given to show that the use of a hybrid beamforming structure incurs a minor degradation in transmission rate. With three RF chains, the performance is close to that of all digital statistical beamforming.

For a time-correlated channel, we consider the differential feedback of the geometrical mean decomposition (GMD) precoder, which is known to be optimal for a number of criteria. When the channel varies slowly, we can expect the optimal GMD precoders of consecutive channel uses to be close. We consider the feedback of the so-called differential precoder and show that it lies in a neighborhood of the identity matrix using matrix perturbation theory. Furthermore, the radius of the neighborhood is shown to be proportional to a time-correlation parameter. Such a characterization allows for subsequent efficient quantization of the differential precoder. We show how to design codewords for the prescribed radius by perturbing the identity matrix and applying QR decomposition on the perturbed matrix. Simulations are given to demonstrate that, with a small feedback rate, the performance of the proposed differential GMD comes close to the case when perfect channel state information is available to the transmitter for slowly time-varying channels like those in indoor and microcellular environments.

Hybrid precoding that combines analog RF processing and digital baseband processing has been shown to be a promising technique for transceivers with limited number of RF chains. The RF processing, due to power and complexity issue, is typically done by using phase shifters. A recent study shows that hybrid precoding is no loss of generality if each coefficient of the analog precoder is implemented by using two phase shifters. In this paper, we consider two possible implementation structures that use Two pHase shIfters for each Coefficient (THIC). These two structures, having the same performance when the phases can take on continuum values, behave differently upon quantization, as the analysis shows. For a small number of quantization bits, the precoder implemented using THIC is a good approximation of the unquantized one. With 3-bit phase shifters, the performance is close to that of the unquantized system in multiuser communications, where the performance is more sensitive to quantization errors. While the analysis in this paper is done assuming high-resolution quantization, simulations show that the result is accurate even for coarse quantization.

In this paper, we consider variable-rate transmission over a slowly varying multiple-input multiple-output (MIMO) channel with a decision feedback receiver. The transmission rate is adapted to the channel by dynamically assigning bits to the subchannels of the MIMO system. Predictive quantization is used for the feedback of bit loading to take advantage of the time correlation inherited from the temporally correlated channel. Due to the use of decision feedback at the receiver, the bit loading is related to the Cholesky decomposition of the channel Gram matrix. Assuming the channel is modeled by a slowly varying Gauss–Markov process, we show that the nested submatrices generated during the process of Cholesky decomposition can be updated as time evolves. Based on the update, we derive the optimal predictor of the next bit loading for predictive quantization. Furthermore, we derive the statistics of the prediction error, which are then exploited to design the quantizer to achieve a smaller quantization error. Simulations are given to demonstrate that the proposed predictive quantization gives a good approximation of the desired transmission rate with a low feedback rate.

In this paper we consider variable-rate transmission for time-correlated MIMO (multi-input multi-output) channels with limited feedback. The number of bits loaded on each sub- channel of the MIMO system is dynamically assigned according to the current channel condition and fed back to the transmitter. As the channel is time-correlated, bit loading is a vector signal that is also time-correlated. We propose to feedback bit loading using predictive coding, which is known to be a powerful quanti- zation technique when the underlying signal is correlated in time. Assuming the channel is a first-order Gauss-Markov random process, we derive the predictor for the bit loading to be coded and analyze the corresponding prediction error variance when the channel is varying slowly. By exploiting the prediction error variance, we adapt the quantizer of the prediction error to have a smaller quantization error. Furthermore we show that the pre- diction error variance is proportional to a term that depends only on the time-correlation coefficient. This leads to the conclusion that, a codebook that is designed for a particular time correlation coefficient can be easily modified to a codebook for a different correlation coefficient without redesign. Simulations are presented to demonstrate that the proposed predictive coding can achieve a very good approximation of the desired transmission rate with a very low feedback rate.

This paper jointly considers limited feedback of bit loading and precoder. In the past when both precoder and bit loading are fed back to the transmitter, the feedback rate is often allocated between the two using an ad hoc approach and codebooks are designed separately rather than jointly. In this paper we allocate feedback rate in a systematic manner by analyzing the effect of quantization on transmission power. The analysis allows us to obtain the rate allocation that minimizes the power penalty due to limited feedback. As both precoder and bit loading are fed back to the transmitter, the information embedded in one can be exploited for the design of the other. To take advantage of bit loading feedback, which carries valuable information on the importance of individual subchannels, we employ multiple precoder codebooks, each tailored to a bit loading vector in the bit loading codebook. The multi-codebook scheme enjoys significant gain over the single-codebook case that does not take bit loading feedback into consideration. Simulations are given to demonstrate that the proposed system can achieve a very good performance due to carefully designed feedback rate allocation and joint codebook designs.

Equal gain transmission (EGT) requires only the feedback of phase information and is known to enjoy several implementational advantages compared to maximum ratio transmission (MRT). This paper investigates the use of lattice quantization for quantizing phases in EGT systems. In practice, the feedback rate is finite and the lattices need to be truncated to have finite code rate. The truncation of lattices results in boundary effect; hence, earlier results developed for untruncated lattices, e.g., fast algorithm and mean-square error (MSE) analysis, cannot be applied directly. Exploiting the fact that the quantized signals are phases, we show how to quantize using truncated lattices so that there is no boundary effect. Moreover, we propose efficient methods to convert between lattice codewords and binary digits so that the lattice quantization can be used in practical systems. Furthermore, the mean-square quantization error is analyzed for several useful truncated lattices. We also show how to analyze the signal-to-noise ratio (SNR) loss when the proposed quantization is applied to an EGT system.

Bit rate and power are two commonly used optimality criteria for MIMO transceiver design. In the literature, bit rate maximization and power minimization problems are viewed as different problems and solved independently. In this paper, we derive the duality between these two problems for both the cases with and without integer constraint on bit allocation. We will show that if a transceiver is optimal for the power-minimizing problem, it is also optimal for the rate maximizing problem, and the converse is true. Such a duality has not been stated and proved in the literature to the best of our knowledge. The derivation does not involve any existing optimal solution and we can establish duality result even for the rate maximization problem with integer bit constraint, which the optimal solution is not known. The duality also allows us to develop an algorithm for finding the rate-maximizing transceiver with integer bit allocation using the solution of power-minimizing system. We will also consider some possible generalizations of the problem, for example, when there is a constraint on the maximal constellation size and when the subchannel bit error rates (BERs) are constrained. For each of these cases, we will see that the duality between the two problems continued to hold. In the simulations, we will compute the optimal solutions for these two problems and demonstrate the duality between these two.

In this correspondence, we propose a new iterative algorithm for finding the minimum sampling frequency of a signal that consists of multiple bandpass signals. This finds important application in software radio where it is desirable to downconvert multiple bandpass signals simultaneously. We will derive a new set of conditions for alias-free sampling. The minimum sampling frequency can be found by iteratively increasing the sampling frequency to meet the alias-free conditions. We will show how the algorithm can be generalized to find alias-free sampling frequency intervals. Simulations will be given to demonstrate the usefulness of the proposed method.

It is known that multicarrier code-division multiple-access (MC-CDMA) systems suffer from multiaccess interference (MAI) when the channel is frequency-selective fading. In this paper, we propose a Hadamard-Walsh code-based MC-CDMA system that achieves zero MAI over a frequency-selective fading channel. In particular, we will use appropriately chosen subsets of Hadamard-Walsh code as codewords. For a multipath channel of length L, we partition a Hadamard-Walsh code of size N into G subsets, where G is a power of two with GgesL. We will show that the N/G codewords in any of the G subsets yields an MAI-free system. That is, the number of MAI-free users for each codeword subset is N/G. Furthermore, the system has the additional advantage that it is robust to carrier frequency offset (CFO) in a multipath environment. It is also shown that the MAI-free property allows us to estimate the channel of each user separately and the system can perform channel estimation much more easily. Owing to the MAI-free property, every user can enjoy a channel diversity gain of order L to improve the bit error performance. Finally, we discuss a code priority scheme for a heavily loaded system. Simulation results are given to demonstrate the advantages of the proposed code and code priority schemes

The carrier frequency offset (CFO) effect of a new multiaccess orthogonal frequency division multiplexing (OFDM) transceiver is analyzed. We show that multiaccess interference (MAI) due to CFO can be mitigated by choosing a proper set of orthogonal codes. That is, if we use M/2 symmetric or antisymmetric codewords of the M Hadamard-Walsh codewords, MAI can be greatly reduced to a negligible amount. The new OFDM system has been shown to be approximately MAI-free when there is no CFO. The new result developed here shows that the system continues to be approximately MAI-free even in the presence of CFO. Furthermore, we derive a close form for the reduced MAI, which reveals an interesting property of the system. That is, when the number of users increases, the reduced MAI and the inter carrier interference (ICI) due to self-CFO will decrease. Finally, it is demonstrated by simulation that the proposed OFDM scheme with proper code selection outperforms the OFDMA and the multicarrier code division multiplexing access (MC-CDMA) systems in a CFO environment.

We consider window designs for discrete Fourier transform (DFT) based multicarrier transceivers without using extra cyclic prefix. As in previous works of window designs for DFT-based transceivers, a postprocessing matrix that is generally channel dependent, is needed to have a zero-forcing receiver. We show that postprocessing is channel independent if and only if the window itself has the cyclic-prefixed property. We design optimal windows with minimum spectral leakage subject to the cyclic-prefixed condition. Moreover, we analyze how postprocessing affects the signal-to-noise ratio (SNR) at the receiver, which is an aspect that is not considered in most of the earlier works. The resulting SNR can be given in a closed form. Join optimization of spectral leakage and SNR are also considered. Furthermore, examples demonstrate that we can have a significant reduction in spectral leakage at the cost of a small SNR loss. In addition to cyclic-prefixed systems, window designs for zero-padded DFT-based transceivers are considered. For the zero-padded transceivers, windows that minimize spectral leakage can also be designed.

The implementation of orthogonal frequency division multiplexing (OFDM) transmitters typically consists of a discrete DFT matrix and a digital-to-analog (DAC) converter. Many existing results on the analysis of OFDM systems, e.g., spectral roll-off, are based on a convenient analog representation. We show that the analog representation and the DFT-based OFDM transmitters are equivalent only in special cases. Using the analog system to analyze the DFT-based OFDM system may not be valid if there is no equivalent analog representation.

We consider the minimization of uncoded bit error rate (BER) for the orthogonal frequency division multiplexing (OFDM) system with an orthogonal precoder. We analyze the BER performance of precoded OFDM systems with zero forcing and minimum mean squared error (MMSE) receivers. In the case of MMSE receivers, we show that for quadrature phase shift keying (QPSK), there exists a class of optimal precoders that are channel independent. Examples of this class include the discrete Fourier transform (DFT) matrix and the Hadamard matrix. When the precoder is the DFT matrix, the resulting optimal transceiver becomes the single carrier system with cyclic prefix (SC-CP) system. We also show that the worst solution corresponds to the conventional OFDM system; the conventional OFDM system has the largest BER. In the case of zero forcing receivers, the design of optimal transceiver depends on the signal-to-noise ratio (SNR). For higher SNR, solutions of optimal precoders are the same as those of MMSE receivers.

Two types of lapped transforms have been studied in detail in the literature, namely, the lapped orthogonal transform (LOT) and its extension, the biorthogonal lapped transform (BOLT). In this paper, we will study the lapped unimodular transform (LUT). All three transforms are first-order matrices with finite impulse response (FIR) inverses. We will show that like LOT and BOLT, all LUTs can be factorized into degree-one unimodular matrices. The factorization is both minimal and complete. We will also show that all first-order systems with FIR inverses can be minimally factorized as a cascade of degree-one LOT, BOLT, and LUT building blocks. Two examples will be given to demonstrate that despite having a very small system delay, the LUTs have a satisfactory performance in comparison with LOT and BOLT.

There has been great interest in the design of filterbank transceivers. Usually, with proper time domain equalization, the channel is modeled as an FIR filter. It is known that for FIR channels, the introduction of certain redundancy allows the receiver to cancel intersymbol interference (ISI) completely, and channel equalization is performed implicitly using FIR transceivers. This scheme allows us to trade bandwidth for ISI cancellation. In this paper, we derive the minimum redundancy required for the existence of FIR transceivers for a given channel. We see that the minimum redundancy is directly related to the zeros of the channel and to the Smith form of an appropriately defined channel matrix.

The design of optimal DMT transceivers for distorted channel with colored noise has been of great interest. Of particular interest is the class of block based DMT, where the transmitter and the receiver consist of constant matrices. Two types of block- based DMT transceivers are considered: the DMT system with zero padding (ZP-DMT) and the DMT system with general prefix (GP-DMT). We derive the bit allocation formula. For a given channel and channel noise spectrum, we design the ISI-free optimal transceiver that minimizes the transmission power for a given transmission rate and probability of error. For both ZP-DMT and GP-DMT systems, the optimal ISI-free transceiver can be given in closed from. We will see that for both classes, the optimal transceiver has an orthogonal transmitter. Simulation shows that the optimal DMT system can achieve the same transmission rate and the same probability of error with a much lower transmission power compared with other existing DMT systems.

Discrete multitone modulation transceivers (DMTs) have been shown to be very useful for data transmission over frequency-selective channels. The DMT scheme is realized by a transceiver that divides the channel into subbands. The efficiency of the scheme depends on the frequency selectivity of the transmitting and receiving filters. The receiving filters with good stopband attenuation are also desired for combating narrowband noise. The filterbank transceiver or discrete wavelet multitone (DWMT) system has been proposed as an implementation of the DMT transceiver that has better frequency band separation, but usually, intersymbol interference (ISI) cannot be completely cancelled in these filterbank transceivers, and additional equalization is required. We show how to use over interpolated filterbanks to design ISI-free FIR transceivers. A finite impulse response (FIR) transceiver with good frequency selectivity can be designed, as demonstrated by the design examples.

A new nonunitary transform called the prediction-based lower triangular transform (PLT) is introduced for signal compression. The new transform has the same decorrelation property as the Kahurnen-Loeve transform (KLT), but its implementational cost is less than one half of KLT. Compared with the KLT, the design cost of an M/spl times/M PLT is much lower and is only of the order of O(M/sup 2/). Moreover, the PLT can be factorized into simple building blocks. Using two different factorizations, we introduce two minimum noise structures that have roughly the same complexity as the direct implementation of PLT. These minimum noise structures have the following properties: (1) its noise gain is unity even though the transform is nonunitary; (2) perfect reconstruction is structurally guaranteed; (3) it can be used for both lossy/lossless compression. We show that the coding gain of PLT implemented using the minimum noise structure is the same as that of KLT. Furthermore, universal transform coders using PLT are derived. For AR(1) process, the M/spl times/M PLT has a closed form and needs only (M-1) multiplications and additions.

Previously, discrete Fourier transform (DFT)-based discrete multitone modulation (DMT) systems have been widely applied to various applications. In this paper, we study a broader class of DMT systems using more general unitary matrices instead of DFT matrices. For this class, we show how to design optimal DMT systems over frequency-selective channels with colored noise. In addition, asymptotical performance of DFT-based and optimal DMT systems are studied and shown to be equivalent. However, for a moderate number of bands, the optimal DMT system offers significant gain over the DFT-based DMT system, as is demonstrated by examples.

We introduce a minimum noise structure for ladder-based biorthogonal (MINLAB) filter bank. The minimum noise structure ensures that the quantization error has a unity noise gain, even though the filter bank is biorthogonal. The coder has a very low design and implementation cost. The perfect reconstruction property is structurally preserved. Optimal bit allocation and coding gain formulas are derived. We show that the coding gain of the optimal MINLAB coder is always greater than or equal to unity. For both AR(1) process and MA(1) process, the MINLAB coder with two taps has a higher coding gain than the optimal orthonormal coder with an infinite number of taps. In addition to its superior decorrelation ability, it has many other desired features that make it a potentially valuable and attractive alternative to the orthonormal coder, especially for a high-fidelity compression.

It is well known that the analysis and synthesis filters of orthonormal DFT filter banks can not have good frequency selectivity. The reason for this is that each of the analysis and synthesis filters have only one passband. Such frequency stacking (or configuration) in general does not allow alias cancellation when the individual filters have good stopband attenuation. A frequency stacking of this nature is called nonpermissible and should be avoided if good filters are desired. In a usual M-channel filter bank with real-coefficient filters, the analysis and synthesis filters have two passbands. It can be shown that the configuration is permissible in this case. Many designs proposed in the past demonstrate that filter banks with such configurations can have perfect reconstruction and be good filters at the same time. We develop the two-parallelogram filter banks, which is the class of 2-D filter banks in which the supports of the analysis and synthesis filters consist of two parallelograms. The two-parallelogram filter banks are analyzed from a pictorial viewpoint by exploiting the concept of permissibility. Based on this analysis, we construct and design a special type of two-parallelogram filter banks, namely, cosine-modulated filter banks (CMFB). In two-parallelogram CMFB, the analysis and synthesis filters are cosine-modulated versions of a prototype that has a parallelogram support. Necessary and sufficient conditions for perfect reconstruction of two-parallelogram CMFB are derived.

We propose a novel way to design maximally decimated FIR cosine modulated filter banks, in which each analysis and synthesis filter has a linear phase. The system can be designed to have either the approximate reconstruction property (pseudo-QMF system) or perfect reconstruction property (PR system). In the PR case, the system is a paraunitary filter bank. As in earlier work on cosine modulated systems, all the analysis filters come from an FIR prototype filter. However, unlike in any of the previous designs, all but two of the analysis filters have a total bandwidth of 2/spl pi//M rather than /spl pi//M (where 2M is the number of channels in our notation). A simple interpretation is possible in terms of the complex (hypothetical) analytic signal corresponding to each bandpass subband. The coding gain of the new system is comparable with that of a traditional M-channel system (rather than a 2M-channel system). This is primarily because there are typically two bandpass filters with the same passband support. Correspondingly, the cost of the system (in terms of complexity of implementation) is also comparable with that of an M-channel system. We also demonstrate that very good attenuation characteristics can be obtained with the new system.