# Non-Linear Estimation of Convolutionally Encoded Sequences

## TL;DR

This paper introduces a non-linear estimation method for convolutionally encoded sequences transmitted over AWGN channels, utilizing a novel probability measure to facilitate MAP decoding.

## Contribution

It constructs a new probability measure under which observations become independent Gaussian vectors, enabling recursive calculation of conditional probabilities for MAP decoding.

## Key findings

- Derived a recursive formula for conditional probabilities
- Connected the measure change to Girsanov theorem
- Enabled MAP decoding via the new probabilistic framework

## Abstract

Suppose that a convolutionally encoded sequence is transmitted symbol by symbol over an AWGN channel using BPSK modulation. In this case, pairs of the signal (i.e., code symbol) and observation are not jointly Gaussian and therefore, a linear estimation method cannot be applied. Hence, in this paper, non-linear estimation of convolutionally encoded sequences is discussed. First a probability measure (denoted Q), whose Radon-Nikodym derivative with respect to the underlying probability measure P is an exponential martingale, is constructed. It is shown that with respect to Q, the observations are mutually independent Gaussian random vectors with zero mean and identity covariance matrix. We see that the relationship between observation noises (with respect to P) and observations (with respect to Q) has a close relation to the Girsanov theorem in continuous case. Next, using the probability measure Q, we calculate the conditional probability of an event related to any encoded symbol conditioned by the observations. Moreover, we transform it into a recursive form. In the process of derivation, the metric associated with an encoded sequence comes out in a natural way. Finally, it is shown that maximum a posteriori probability (MAP) decoding of convolutional codes is realized using the derived conditional probability.