The notion of observability for higher order discrete-time systems of algebraic and difference equations is studied. Such systems are also known as polynomial matrix descriptions . Attention is first given to a special form of descriptor systems with a state lead in the output. This system is transformed into its causal and noncausal subsystems and observability criteria are given in terms of the subsystem's matrices, and the fundamental matrix sequence of the matrix pencil (sigma E - A). Afterwards, the higher order system is studied. By transforming it into a first-order descriptor system of the above form, an observability criterion is provided for the higher order system in terms of the Laurent expansion at infinity of the system's polynomial matrix. In addition, observability is connected with the coprimeness of the polynomial matrices of the higher order system and the coprimeness of the matrix pencils of the descriptor system.