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Systems in the Time-Domain

Module by: Don Johnson. E-mail the author

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

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A discrete-time signal sn sn is delayed by n0 n0 samples when we write sn−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: sn− n 0 ↔ⅇ-ⅈ2πf n 0 Sⅇⅈ2πf ↔ 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   Sxn=yn , Then   Sxn− n 0 =yn− 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

yn= a 1 yn−1+…+ a p yn−p+ b 0 xn+ b 1 xn−1+…+ b q xn−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 yn yn is related to its past values yn−l y n l , l=1…p l 1 … p , and to the current and past values of the input signal xn 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 yn 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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