Wavelet Codes: Detection and Correction Using Kalman Estimation

Wavelet codes encoded by uniform oversampled filter banks where the wavelet coefficients represent code symbols are decoded employing Kalman fixed-lag estimation procedures. The code symbol error estimates are based on the mean-squared error (MSE) criterion and are first used to detect the presence of large numerical errors. Then the correction of symbols uses these Kalman estimators directly. The wavelet code symbols are corrupted by low levels of processing noise continuously and occasionally by large disruptive errors, e.g., impulsive noise. The modeling variables have time-varying statistics allowing both types of errors to be handled naturally. Kalman time-varying tracking filters, which use wavelet syndromes as their measurement inputs, develop the fixed-lag smoothed estimates for the larger errors. The syndromes are also modeled as corrupted by processing noise modeling computational failure errors, large and small. Hypothesis detection methods are used to locate large errors and their statistics must include Kalman mismatch characteristics to be valid. Estimators from several adjacent lag positions assist in the detection procedures. Simulation results show correction accuracy in the time domain as well as on a MSE basis. Large classes of wavelet codes are designed using binary convolutional codes as starting points from which the proper filter weights for encoding and syndrome evaluating are determined.