A modification to conventional block pulse functions (BPF) is proposed by describing a set of linearly pulse-width modulated block pulse functions (LPWM-BPF) that has been utilised to develop a generalised convolution matrix of operational nature. This matrix is used to determine convolution of time-varying functions and is also employed to solve linear feedback system identification problem. Also, a recursive technique for solving the identification problem in the conventional BPF domain has been derived. Numerical examples are treated to establish the validity of the proposal. The representational error analysis has also been carried out for the LPWM-BPF to show that this kind of BPF set introduces less error than the conventional equal-width BPF.