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Signals transmitted over certain randomly time-varying linear channels undergo spreading both in time and in frequency. This paper considers a particular class of doubly spread channels: those with an impulse response that can be represented as the output of a distributed linear system, the dynamics of which are governed by partial differential state equations driven by a white noise process. The model is shown to include channels characterized by wide-sense stationary scattering (WSS), uncorrelated scattering (US), or both (WSSUS). When reception takes place in white Gaussian noise, the optimum, quadratic detector can be realized as an estimator-correlator. For the doubly spread channels encompassed by the model, the causal least-squares estimator of the channel response is obtained in the form of a linear filter with a distributed-parameter state-variable description. Certain error probability bounds and approximations can be evaluated upon solution of the receiver equations. Numerical results are given for an example: binary frequency-shift-keying (FSK) signaling over a particular WSSUS channel. The effects of the channel's frequency-time spread factor (BL) and the signaling interval on the optimum receiver performance, as measured by the Bhattacharyya distance, are investigated.
Date of Publication: Sep 1971