A unified approach to signal processing using matrices

The matrix representational forms of linear systems commonly used in digital signal processing applications, are unified. The systems discussed include both impulse response and difference equation matrix representations, of linear shift invariant and linear shift varying, FIR and IIR systems.Two types of matrix systems are presented. Those that involve non-square coefficient matrices, and those that involve square coefficient matrices.The non-square matrix systems result from strict adherence to linear system definitions, throughout system development. The matrix representation of convolution and the concept of the window of observation are used to develop infinite size, and then finite size matrix forms, that accurately represent the original system.Square matrix systems are commonly used matrix systems, that benefit from their square coefficient matrix forms. These systems are carefully analyzed in terms of the various linear system definitions.Both square and non-square matrix systems are used to synthesize systems that solve specific digital signal processing problems such as estimation, detection and correlation. Square and non-square matrix system, minimum mean square error (MSE) derivations are presented. Estimation, detection and correlation systems are unified under the general class of minimum mean square error systems, by careful choice of the system desired output. An alternate detection system derivation based on a generalized derivative maximization, is presented