A linear coding for transmitting a set of correlated signals

A coding scheme is described for the transmission of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> continuous correlated signals over <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</tex> channels, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</tex> being equal to or less than <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> . Each of the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</tex> signals is a linear combination of the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> original signals. The coefficients of this linear transformation, which constitute an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m × n</tex> matrix, are constants of the coding scheme. For the purpose of decoding, the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</tex> signals are once more combined linearly into <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> output signals which approximate the input signals. The coefficients of the coding matrix which minimize the sum of the mean square differences between the original signals and the reconstructed ones are shown to be the components of the eigenvectors of the matrix of the correlation coefficients of the original signals. The decoding matrix is the transpose of the coding matrix. As an example, the coding scheme is applied to a channel vocoder in which speech is transmitted by means of a set of signals proportional to the speech energy in the various frequency bands. These signals are strongly correlated, and the coding results in a substantial reduction in the number of signals necessary to transmit highly articulate speech. The coding theory can be extended to include the minimization of the expectation of any positive definite quadratic function of the differences between the original and reconstructed signals. In addition, if the signals are Gaussian, the sum of the channel capacities necessary to transmit the transformed signals is shown to be equal to or less than that necessary to transmit the original signals.