lecture 5 | mathematical model of discrete time system part 1

A mathematical model of a discrete-time system describes how the system behaves over time, with time being divided into distinct intervals. Unlike continuous-time systems where variables change continuously, discrete-time systems have variables that change only at specific points in time. There are two main ways to represent a discrete-time system mathematically: Difference Equations: These equations relate the output of the system at a current time step (n) to its past outputs (n-1, n-2, etc.) and its current and past inputs (if applicable). They are often written in the form: y(n) = ay(n-1) + by(n-2) + ... + mx(n) + nx(n-1) + ... where: y(n) is the output of the system at time step n x(n) is the input of the system at time step n a, b, ..., m, n are constants that determine the system's behavior Transfer Function: This approach uses the z-transform, which converts a discrete-time system from the time domain (where variables are functions of time) to the z-domain (where variables are functions of a complex number z). The transfer function is the ratio of the system's output transform (Y(z)) to its input transform (X(z)). Here are some additional points to consider: Sampling: Discrete-time models often arise from discretizing continuous-time systems, where we sample the continuous signal at specific intervals. Techniques like zero-order hold can be used for this conversion. Advantages: Discrete-time models are often easier to analyze and implement on computers compared to continuous-time models. Applications: They are widely used in various fields like signal processing, control systems, digital filtering, and many others where data is acquired or processed at specific intervals. For a deeper understanding, you can explore resources like textbooks on control systems or online materials such as lectures on YouTube: [YouTube discrete time systems].