Finite Number Of Bits Research Articles

Asymptotic stabilization of dynamically quantized nonlinear systems in feedforward form

This paper studies the stabilizability of an n-dimensional quantized feedforward nonlinear system. The state of that system is first quantized into a finite number of bits, and then sent through a digital network to the controller. We want to minimize the number of transmitted bits subject to maintaining asymptotic stability. In the prior literature, n bits are used to stabilize the n-dimensional system by assigning one bit to each state variable (dimension). Under the stronger assumption of global Lipschitz continuity, this paper extends that result by stabilizing the system with a single bit. Its key contribution is a dynamic quantization policy which dynamically assigns the single bit to the most “important” state variable. Under this policy, the quantization error exponentially converges to 0 and the stability of the system can, therefore, be guaranteed. Because 1 is the minimum number of quantization bits (per sampling step), the proposed dynamic quantization policy achieves the minimum stabilizable bit number for that n-dimensional feedforward nonlinear system.

Achievable rates with quantised channel state information in a multiple-access channel with one cognitive transmitter

The authors investigate the achievable rates of a multiple-access channel with one cognitive transmitter in a fading channel. The cognitive transmitter uses dirty-paper coding to mitigate the known interference. They assume partial transmit channel state information which is obtained through a quantised limited feedback. In this model, the receiver sends as feedback only a finite number of bits describing the fading coefficients. Numerical results show that different quantisation schemes must be used for different transmitters (cognitive and non-cognitive) in order to maximise their rates. Also, the number of quantisation bits allocated to each channel coefficient (i.e. the channels from the non-cognitive and cognitive transmitters to the receiver), for a fixed total feedback bits, can affect the achievable rates of the transmitters, and even for each channel coefficient, how to dedicate the considered quantisation bits between the coefficient magnitude and phase can have substantial effects on the rate.

Achievable Rates with Imperfect Transmitter Side Information Using a Broadcast Transmission Strategy

rdquoWe investigate the performance of the broadcast approach for various fading distributions, which correspond to different models of partial transmit channel state information (CSI). The first model considered is the quantized limited feedback. In this model, the receiver can send as feedback only a finite number of bits describing the fading gain. We derive the optimal power allocation for the broadcast approach for the quantized feedback model. For a Rayleigh fading channel, numerical results here show that if the feedback word can be longer than one bit, the broadcasting gain becomes negligible, due to diminished channel uncertainty. The second partial transmit CSI model is a stochastic Gaussian model with mean and variance information, which is commonly used for modeling the channel estimation error. In a single-input single-output (SISO) channel, this model also corresponds to the Ricean fading distribution, for which we derive maximal achievable broadcasting rates. We further consider a multiple-input single-output (MISO) channel, and derive the optimal power allocation strategy in a broadcast approach. Numerical results here show that uniform power allocation is preferable over beamforming power allocation in the region where broadcasting gain over single level coding is non-negligible.

Performance analysis of b-bit digital receivers for TR-UWB systems with inter-pulse interference

An ultra-wideband (UWB) transmitted reference (TR) system transmits an un-modulated pulse and a delayed modulated pulse pair. Then, a correlation receiver uses the former to demodulate the latter. Because of the long spread of a typical UWB channel, time delay between the two pulses is preferable to be larger than the channel delay spread for reduced noise at the receiver. However, for bandwidth efficiency, that delay should be made small, resulting in inter-pulse interference at the receiver. In this paper, digital receivers are constructed for TR-UWB systems including inter-pulse interference. A typical mean matching technique, appropriate for both PPM and PAM schemes, is implemented digitally to obtain a good template for symbol detection. Joint estimation and detection performance of this family of digital receivers, using finite number of bits in analog-to-digital conversion and finite noisy observations, is analyzed. Closed form results are derived and verified by computer simulations. In addition, the effect of time offset between the reference pulse and information carrying pulse is studied. Overlap of the two pulses does not incur noticeable performance degradation. The proposed analytical framework can be applied to study detection performance of other related digital receivers not covered in this paper

Efficient feedback methods for MIMO channels based on parameterization

In this paper, we propose two efficient low-complexity quantization methods for multiple-input multiple-output (MIMO) systems with finite-rate feedback based on proper parameterization of the information to be fed back followed by quantization in the new parameter domain. For a MIMO channel which has multiple orthonormal vectors as channel spatial information, we exploit the geometrical structure of orthonormality while quantizing the spatial information matrix. The parameterization is of two types: one is in terms of a set of unit-norm vectors with different lengths, and the other is in terms of a minimal number of scalar parameters. These parameters are shown to be independent for the i.i.d. flat-fading Rayleigh channel, facilitating efficient quantization. In the first scheme, each of the unit-norm vectors is independently quantized with a finite number of bits using an optimal vector quantization (VQ) technique. Bit allocation is needed between the vectors, and the optimum bit allocation depends on the operating SNR of the system. In the second scheme, the scalar parameters are quantized. In slowly time-varying channels, the scalar parameters are also found to be smoothly changing over time, leading to the development of a simple quantization and feedback method using adaptive delta modulation. The results show that the proposed feedback scheme has a channel tracking feature and achieves a capacity very close to perfect feedback with a reasonable feedback rate

On the encoding rate and discrete modulation adaptation design for MIMO links

It is well-known that MIMO links in slow fading channels can benefit significantly through rate and power adaptations when there is knowledge of channel state information at the transmitter (CSIT). However, practical MIMO transmitter can only support finite combinations of the channel encoding rates and modulation constellation levels due to the finite number of bits allocated in the control overhead (embedded in the packet transmission). In this paper, we shall address various practical MIMO adaptation design issues associated with discrete transmission modes. We propose a systematic design framework to address the MIMO adaptation design based on information theoretical framework. The design problem is cast into an optimization problem similar to the classical vector quantization problem with a modified distortion measure. As an illustration, we apply the design framework to a 2 times 2 MIMO link and compare the goodput performance with the optimal performance predicted by the theory. We found that the goodput performance of the MIMO link predicted by the theoretical framework matches the actual performance closely

Recursive and Trellis-Based Feedback Reduction for MIMO-OFDM with Rate-Limited Feedback

We investigate an adaptive MIMO-OFDM system with a feedback link that can only convey a finite number of bits. We consider three different transmitter configurations: i) beamforming applied per OFDM subcarrier, ii) precoded spatial multiplexing applied per subcarrier, and iii) precoded orthogonal space time block coding applied per subcarrier. Depending on the channel realization, the receiver selects the optimal beamforming vector or precoding matrix from a finite-size codebook on each subcarrier, and informs the transmitter through finite-rate feedback. Exploiting the fact that the channel responses across OFDM subcarriers are correlated, we propose two methods to reduce the amount of feedback. One is recursive feedback encoding that selects the optimal beamforming/precoding choices sequentially across the subcarriers, and adopts a smaller-size time-varying codebook per subcarrier depending on prior decisions. The other is trellis-based feedback encoding that selects the optimal decisions for all subcarriers at once along a trellis structure via the Viterbi algorithm. Our methods are applicable to different transmitter configurations in a unified fashion. Simulation results demonstrate that the trellis-based approach outperforms the recursive method as well as an existing interpolation-based alternative at high signal-to-noise-ratio, as the latter suffers from diversity loss

Bounds and observations on the transmission of data packets with minimum energy

Transmitting bursty data energy-efficiently is of interest in emerging wireless communication systems. We observe that in the AWGN channel a finite number of bits B cannot be transmitted with finite energy per bit epsivb as the error probability goes to zero. We show, however, that any sub-linear increase of B with the blocklength suffices. We derive structural properties of the behavior of epsivb with rate for fixed B and apply classical random coding error probability bounds to obtain numerical bounds. Implications are discussed for the design of practical systems where transmission energy is not the only component of energy consumption.

Computing machine-efficient polynomial approximations

Polynomial approximations are almost always used when implementing functions on a computing system. In most cases, the polynomial that best approximates (for a given distance and in a given interval) a function has coefficients that are not exactly representable with a finite number of bits. And yet, the polynomial approximations that are actually implemented do have coefficients that are represented with a finite---and sometimes small---number of bits. This is due to the finiteness of the floating-point representations (for software implementations), and to the need to have small, hence fast and/or inexpensive, multipliers (for hardware implementations). We then have to consider polynomial approximations for which the degree- i coefficient has at most m i fractional bits; in other words, it is a rational number with denominator 2 m i . We provide a general and efficient method for finding the best polynomial approximation under this constraint. Moreover, our method also applies if some other constraints (such as requiring some coefficients to be equal to some predefined constants or minimizing relative error instead of absolute error) are required.

Minimum Energy Decentralized Estimation in a Wireless Sensor Network with Correlated Sensor Noises

Consider the problem of estimating an unknown parameter by a sensor network with a fusion center (FC). Sensor observations are corrupted by additive noises with an arbitrary spatial correlation. Due to bandwidth and energy limitation, each sensor is only able to transmit a finite number of bits to the FC, while the latter must combine the received bits to estimate the unknown parameter. We require the decentralized estimator to have a mean-squared error () that is within a constant factor to that of the best linear unbiased estimator (BLUE). We minimize the total sensor transmitted energy by selecting sensor quantization levels using the knowledge of noise covariance matrix while meeting the target requirement. Computer simulations show that our designs can achieve energy savings up to when compared to the uniform quantization strategy whereby each sensor generates the same number of bits, irrespective of the quality of its observation and the condition of its channel to the FC.

Quantifying the power loss when transmit beamforming relies on finite-rate feedback

Transmit beamforming achieves optimal performance in systems with multiple transmit antennas and a single receive antenna from both the capacity and the received signal-to-noise ratio (SNR) perspectives but ideally requires perfect channel knowledge at the transmitter. In practical systems where the feedback link can only convey a finite number of bits, transmit beamformer designs have been extensively investigated using either the outage probability or the average SNR as the figure of merit. In this paper, we study the symbol error rate (SER) for transmit beamforming with finite-rate feedback in a multi-input single-output setting. We derive an SER lower bound that is tight for good beamformer designs. Comparing this bound with the SER corresponding to the ideal case, we quantify the power loss due to the finite-rate constraint across the entire SNR range.

Performance limit of finite wordlength FIR digital filters

In many practical situations, it is necessary to represent the coefficients of a finite impulse response (FIR) digital filter by a finite number of bits. This not only degrades the filter frequency response but also introduces a theoretical limit on the performance of the filter. Derivation of a lower bound on filter degradation is the purpose of this paper. We consider a general case of a length N filter with a discrete set of allowable coefficients. A theorem that gives the lower bound on the increase in minimax approximation error that is caused by the finite wordlength restriction is presented. Its extension and application to filter design cases is demonstrated. The importance of this bound is not only theoretical. Its practical effectiveness is shown in the algorithm for optimal finite wordlength FIR filter design where it significantly reduces the amount of computation.

Opportunistic File Transfer Over a Fading Channel Under Energy and Delay Constraints

We consider transmission control (rate and power) strategies for transferring a fixed-size file (finite number of bits) over fading channels under constraints on both transmit energy and transmission delay. The goal is to maximize the probability of successfully transferring the entire file over a time-varying wireless channel modeled as a finite-state Markov process. We study two implementations regarding the delay constraints: an average delay constraint and a strict delay constraint. We also investigate the performance degradation caused by the imperfect (delayed or erroneous) channel knowledge. The resulting optimal policies are shown to be a function of the channel-state information (CSI), the residual battery energy, and the number of residual information bits in the transmit buffer. It is observed that the probability of successful file transfer increases significantly when the CSI is exploited opportunistically. When the perfect instantaneous CSI is available at the transmitter, the faster channel variations increase the success probability under delay constraints. In addition, when considering the power expenditure in the pilot for channel estimation, the optimal policy shows that the transmitter should use the pilot only if there is sufficient energy left for packet transfer; otherwise, a channel-independent policy should be used.

The Access-Control Problem on Capacitated FIFO Networks With Unique O-D Paths is Hard

This paper is concerned with the performance of multicommodity capacitated networks in a deterministic but time-dependent environment. For a given time-dependent origin-destination table, this paper asks if it is easy to find a way of regulating the input flows into the network to avoid queues from growing internally, i.e., to avoid capacity violations. Problems of this type are conventionally approached in the traffic/transportation field with variational methods such as control theory (if time is continuous) and with mathematical programming (if time is discrete). However, these approaches can only be expected to work well ifthe set off easible solutions is convex. Unfortunately, it is found in this paper that this is not the case. It is shown that continuous-time versions of the problem satisfying the smoothness conditions of control theory can have a finite but very large number off easible solutions. The same happens for the discrete time case. These difficulties arise even with the simplest versions of the problem (with unique origin-destination paths, perfect information, and deterministic travel times). The paper also shows that the continuous-time feasibility problem is NP-hard, and that if we restrict our attention to (practical) problems whose data can be described with a finite number ofbits (e.g., in discrete time), then the problem is NP-complete. These results are established by showing that the problem instances of interest can be related to the Directed Hamiltonian Path problem by a polynomial transformation.

Classical simulation of quantum entanglement without local hidden variables

Recent work has extended Bell's theorem by quantifying the amount of communication required to simulate entangled quantum systems with classical information. The general scenario is that a bipartite measurement is given from a set of possibilities and the goal is to find a classical scheme that reproduces exactly the correlations that arise when an actual quantum system is measured. Previous results have shown that, using local hidden variables, a finite amount of communication suffices to simulate the correlations for a Bell state. We extend this in a number of ways. First, we show that, when the communication is merely required to be finite {\em on average}, Bell states can be simulated {\em without} any local hidden variables. More generally, we show that arbitrary positive operator valued measurements on systems of $n$ Bell states can be simulated with $O(n 2^n)$ bits of communication on average (again, without local hidden variables). On the other hand, when the communication is required to be {\em absolutely bounded}, we show that a finite number of bits of local hidden variables is insufficent to simulate a Bell state. This latter result is based on an analysis of the non-deterministic communication complexity of the NOT-EQUAL function, which is constant in the quantum model and logarithmic in the classical model.

Optimal transform coding in the presence of quantization noise

The optimal linear Karhunen-Loeve transform (KLT) attains the minimum reconstruction error for a fixed number of transform coefficients assuming that these coefficients do not contain noise. In any real coding system, however, the representation of the coefficients using a finite number of bits requires the presence of quantizers. We formulate the optimal linear transform using a data model that incorporates the quantization noise. Our solution does not correspond to an orthogonal transform and in fact, it achieves a smaller mean squared error (MSE) compared to the KLT, in the noisy case. Like the KLT, our solution depends on the statistics of the input signal, but it also depends on the bit-rate used for each coefficient. Especially for images, based on our optimality theory, we propose a simple modification of the discrete cosine transform (DCT). Our coding experiments show a peak signal-to noise ratio (SNR) performance improvement over JPEG of the order of 0.2 dB with an overhead less than 0.01 b/pixel.

Error analysis and reduction for angle calculation using the CORDIC algorithm

In this paper, we consider the errors appearing in angle computations with the CORDIC algorithm (circular and hyperbolic coordinate systems) using fixed-point arithmetic. We include errors arising not only from the finite number of iterations and the finite width of the data path, but also from the finite number of bits of the input. We show that this last contribution is significant when both operands are small and that the error is acceptable only if an input normalization stage is included, making unsatisfactory other previous proposals to reduce the error. We propose a method based on the prescaling of the input operands and a modified CORDIC recurrence and show that it is a suitable alternative to the input normalization with a smaller hardware cost. This solution can also be used in pipelined architectures with redundant carry-save arithmetic.

Capacity of autocorrelation associative memory with quantized synaptic weight

AbstractThe neural chips to speed up the process of a neural network have recently been developed actively. Since a conventional neural chip requires quantized synaptic weights, it is important to know the properties of the network with such weights. Devices for a 2‐layer network with the same number of input and output elements have been well developed, and such a device can be used as an autocorrelation associative memory by feeding its output to its input.This paper investigates theoretically the capacity of an autocorrelation associative memory when its synaptic weights are quantized with a finite number of bits. It also proposes an optimum quantization function. A system with a finite number of elements is computer simulated. The proposed method can be applied to the design of a neural chip for associative memory.

On the precision constraints of threshold elements

The authors address the issue of the precision required by a threshold element in order to implement a linearly separable mapping of N binary inputs. In contrast to previous works they require only the ability to correctly implement, with high probability, the mapping of P randomly chosen training examples, as opposed to the requirement of perfect mapping for every possible set of inputs. Their results are obtained within the statistical mechanics approach and are thus average case results as opposed to the extreme case analyses in the computational learning theory literature. They show that as long as the fraction P/N is finite, then with probability close to 1 as N→∞, a finite number of bits suffice to implement the mapping. This should be compared with the extreme case analysis which requires at least N bits. They also calculate the ability of the constrained network to predict novel examples and compare their predictions with those of an unconstrained network, showing that as long as the training set is not too large, the learning and generalization abilities of the constrained network are not very different from the unconstrained one.

On the precision constraints of threshold elements

The authors address the issue of the precision required by a threshold element in order to implement a linearly separable mapping of N binary inputs. In contrast to previous works they require only the ability to correctly implement, with high probability, the mapping of P randomly chosen training examples, as opposed to the requirement of perfect mapping for every possible set of inputs. Their results are obtained within the statistical mechanics approach and are thus average case results as opposed to the extreme case analyses in the computational learning theory literature. They show that as long as the fraction P/N is finite, then with probability close to 1 as N→∞, a finite number of bits suffice to implement the mapping. This should be compared with the extreme case analysis which requires at least N bits. They also calculate the ability of the constrained network to predict novel examples and compare their predictions with those of an unconstrained network, showing that as long as the training set is not too large, the learning and generalization abilities of the constrained network are not very different from the unconstrained one.