Systems in the Time-Domain 2.6 2000/07/21 2003/07/24 11:00:43.826 GMT-5 Don Johnson dhj@rice.edu Prashant Singh prash@ece.rice.edu Richard G. Baraniuk richb@rice.edu Mariyah Poonawala mariyah@rice.edu superposition shift-invariant difference equation Discrete-time systems allow for mathematically specified processes like the difference equation. A discrete-time signal sn is delayed by n0 samples when we write s n n0 , with n0 0 . Choosing n0 to be negative advances the signal along the integers. As opposed to analog delays, discrete-time delays can only be integer valued. In the frequency domain, delaying a signal corresponds to a linear phase shift of the signal's discrete-time Fourier transform: ↔ s n n 0 2 f n 0 S 2 f . Linear discrete-time systems have the superposition property. Superposition S a 1 x 1 n a 2 x 2 n a 1 S x 1 n a 2 S x 2 n A discrete-time system is called shift-invariant (analogous to time-invariant analog systems) if delaying the input delays the corresponding output. Shift-Invariant If   S x n y n , Then   S x n n 0 y n n 0 We use the term shift-invariant to emphasize that delays can only have integer values in discrete-time, while in analog signals, delays can be arbitrarily valued. We want to concentrate on systems that are both linear and shift-invariant. It will be these that allow us the full power of frequency-domain analysis and implementations. Because we have no physical constraints in "constructing" such systems, we need only a mathematical specification. In analog systems, the differential equation specifies the input-output relationship in the time-domain. The corresponding discrete-time specification is the difference equation. The Difference Equation y n a 1 y n 1 … a p y n p b 0 x n b 1 x n 1 … b q x n q Here, the output signal yn is related to its past values y n l , l 1 … p , and to the current and past values of the input signal xn . The system's characteristics are determined by the choices for the number of coefficients p and q and the coefficients' values a1 … ap and b0 b1 … bq . There is an asymmetry in the coefficients: where is a 0 ? This coefficient would multiply the y n term in the difference equation. We have essentially divided the equation by it, which does not change the input-output relationship. We have thus created the convention that a0 is always one. As opposed to differential equations, which only provide an implicit description of a system (we must somehow solve the differential equation), difference equations provide an explicit way of computing the output for any input. We simply express the difference equation by a program that calculates each output from the previous output values, and the current and previous inputs.