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Systems in the Time-Domain

Module by: Don Johnson. E-mail the author

Summary: Discrete-time systems allow for mathematically specified processes like the difference equation.

A discrete-time signal s⁢n sn is delayed by n0 n0 samples when we write s⁢n−n0 s n n0 , with n0>0 n0 0 . Choosing n0 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 ↔e−(j⁢2⁢π⁢f⁢ n 0 )⁢S⁢ej⁢2⁢π⁢f ↔ 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 S a 1 x 1 n a 2 x 2 n a 1 S x 1 n a 2 S x 2 n
(1)
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 If   S x n y n , Then   S x n n 0 y n n 0
(2)
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 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
(3)
Here, the output signal y⁢n yn is related to its past values y⁢n−l y n l , l=1…p l 1 … p , and to the current and past values of the input signal x⁢n xn. The system's characteristics are determined by the choices for the number of coefficients pp and qq and the coefficients' values a1…ap a1 … ap and b0b1…bq b0 b1 … bq .

Aside:

There is an asymmetry in the coefficients: where is a 0 a 0 ? This coefficient would multiply the y⁢n 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 a0a0 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.

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A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

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