This paper is about a new efficient method for the implementation of convolvers and correlators using the Fermat Number Transform (FNT) and the inverse (IFNT). The latter present advantages compared to Inverse Fast Fourier Transform (IFFT). An efficient state space method for implementing the Inverse FNT (IFNT) over rectangular windows is proposed for the cases where there is a large overlap between the consecutive input signals. This is called Inverse Generalized Sliding Fermat Number Transform (IGSFNT) and is useful for reducing the computational complexity of finite ring convolvers and correlators. This algorithm uses the technique of Generalized Sliding associated to matricial calculation in the Galois Field. The computational complexity of this method is compared with that of standard IFNT.

This paper is about a new efficient method for the implementation of a Block Proportionate Normalized Least Mean Square (BPNLMS++) adaptive filter using the Fermat Number Transform (FNT) and its inverse (IFNT). These transforms present advantages compared to Fast Fourier Transform (FFT) and the inverse (IFFT). An efficient state space method for implementing the FNT over rectangular windows is used in the cases where there is a large overlap between the consecutive input signals. This is called Generalized Sliding Fermat Number Transform (GSFNT) and is useful for reducing the computational complexity of finite ring convolvers and correlators. In this contribution, we propose, as a first objective, an efficient state algorithm with the purpose of reducing the complexity of IFNT. This algorithm, called Inverse Generalized Sliding Fermat Number Transform (IGSFNT), uses the technique of Generalized Sliding associated to matricial calculation in the Galois Field. The second objective is to realize an implementation of the BPNLMS++ adaptive filter using GSFNT and IGSFNT, which can significantly reduce the computation complexity of the filter implantation on digital signal processors.

This study deals with two linear precoders: the maximization of the minimum Euclidean distance between received symbol-vectors, called here max-dmin, and the maximization of the post-processing signal-to-noise ratio termed max-SNR or beamforming. Both have been designed for reliable MIMO transmissions operating over uncorrelated Rayleigh fading channels. Here, we will explain why performances in terms of bit error rates show a significant enhancement of the max-dmin over the max-SNR whenever the number of antennas is increased. Then, from theoretical developments, we will demonstrate that, like the max-SNR precoder, the max-dmin precoder achieves the maximum diversity order, which is warrant of reliable transmissions. The current theoretical knowledge will be applied to the case-study of a system with two transmit- or two receive-antennas to calculate the probability density functions of two channel parameters directly linked to precoder performances for uncorrelated Rayleigh fading channels. At last, this calculation will allow us to quickly get the BER of the max-dmin precoder further to the derivation of a tight semi-theoretical approximation.

Multiple-input multiple-output (MIMO) maximum-SNR (max-SNR) system employs the maximum ratio combiner (MRC) at the receiver side and the maximum ratio transmitter (MRT) at the transmitter side. Its performances highly depend on MIMO channel characteristics, which vary according to both the number of antennas and their distribution between the transmitter and receiver sides. By using the decomposition of the ordered Wishart distribution in the uncorrelated Rayleigh case, we derived a closed-form expression of the largest eigenvalue probability density function (PDF). The final result yields to an expression form of the PDF where polynomials are multiplied by exponentials; it is worth underlining that, though this form had been previously observed for given couples of antennas, to date no formally-written closed-form was available in the literature for an arbitrary couple. Then, this new expression permits one to quickly and easily get the well known largest eigenvalue PDF and use it to determine the binary error probability (BEP) of the max-SNR.

This paper is about an efficient implementation of adaptive filtering for echo cancelers. The first objective of this paper is to propose a simplified method of the flexible block Multi-Delay Filter (MDF) algorithm in the time-domain. Then, we will derive a new method for the step-size adaptation coefficient. The second objective is about the realization of a Block Proportionate Normalized Least Mean Squares (BPNLMS++) with the simplified MDF (SMDF) implementation. Using the new step-size method and the smaller block dimension proposed by SMDF, we achieve a faster convergence of the adaptive process with a limited computational cost. Then, an efficient implementation of the new procedure (SMDF-BPNLMS++) block filtering is proposed using Fermat Number Transform, which can significantly reduce the computation complexity of filter implantation on Digital Signal Processor.