# Robust communication in a time-varying noisy environment

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## Abstract

Matched filter detectors are used to detect known signal waveforms transmitted under noisy conditions. Moving-average matched filters (MAMF's) are a class of digital filters whose performance is measured in terms of Signal to Noise Ratio (SNR). The overall performance of a MAMF is described by the SNR Improvement (SNRI) which is the ratio of Output SNR (OSNR) to Input SNR (ISNR). The OSNR and ISNR are the SNR at the output and input of the MAMF respectively. SNRI is maximized by maximizing OSNR since ISNR is fixed for a received signal and noise. The OSNR of a MAMF is a function of the noise autocorrelation sequence and the transmitted signal vector. The maximum OSNR of a MAMF is produced when the signal vector is the eigenvector associated with the smallest eigenvalue of the Toeplitz matrix formed from the noise autocorrelation sequence. If the noise autocorrelation is not known in advance of transmission, or not stationary, then it must be estimated at the receiver. Since autocorrelation estimators derive their estimates from noise samples, i.e. a random process, the estimates are probabilistic. In a practical implementation wherein the signal vector is fixed, the noise is stationary over short periods of time, and the noise autocorrelation sequence is estimated, the SNRI or performance of the MAMF varies and can even become less than unity if either the estimates are poor or the noise characteristics differ from those expected when the signal vectors were selected. A SNRI less than unity is highly undesirable as processing, which is done with the objective of obtaining higher OSNR than ISNR, i.e. a SNRI greater than unity, has become counterproductive.

This thesis proposes a variation to the classical MAMF communication system and investigates the performance of the resulting MAMF. In the classical MAMF communication system the N-dimensional signal vector is treated as a single vector. In the proposed MAMF communication system, the N-dimensional signal vector is composed of two or more linearly independent basis vectors spanning a signal vector subspace of dimension M. By combining the linearly independent basis vectors in the receiver, one can effectively change the transmitted signal vector to any signal vector in the signal vector subspace to maximize OSNR. The OSNR of a MAMF is a function of the autocorrelation of the noise as well as the signal vector. The autocorrelation of the noise is estimated in both the classical and proposed systems. For relatively few noise samples, the estimated autocorrelation of the noise deviates from the actual autocorrelation. The proposed system is formed from the classical system by proceeding the MAMF with a processor that extracts the received linearly independent basis vectors with additive colored Gaussian noise from the received transmission and combines them to yield maximum OSNR assuming the estimated autocorrelation of the noise is exact. Since the autocorrelation of the noise is estimated from the random noise process, the autocorrelations themselves are probabilistic and hence the maximum OSNR is too. As the estimated noise autocorrelation approaches the actual noise autocorrelation, the OSNR approaches the absolute maximum OSNR for the M-dimensional system. The theoretical aspects of both the classical and proposed MAMF communication systems are developed in this thesis.

The performance of the proposed MAMF communication system is investigated for a practical implementation wherein the signal vector is composed of two linearly independent basis vectors and the noise characteristics vary over time. The performance of the proposed system is first compared to that of the classical system with both systems using various signal vectors, over various noise colors, and with the exact noise autocorrelation given. The performance comparison between the classical and proposed systems is then repeated with the noise autocorrelation, as in a practical implementation, estimated using either the classical biased or Burg estimator. The performance is measured by SNRI and the results are tabulated and graphed.

Finally, the proposed system is implemented and its performance measured by bit error rates as a function of ISNR. This will show whether SNRI performance is a good prediction of bit error rate performance. The color of the stationary Gaussian noise is kept constant during transmission of a particular bit. The color of the stationary Gaussian noise is changed between bit transmissions to observe the robustness of the system under different colored noise conditions while maintaining the same signal vectors, or signal subspace. The results are again tabulated and graphed.