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# Fundamentals of Signal Processing

Science and Technology

## Systems in the Time-Domain

Tác giả: Don Johnson

A discrete-time signal $s(n)$ is delayed by ${n}_{0}$ samples when we write $s(n-{n}_{0})$, with ${n}_{0}> 0$. Choosing ${n}_{0}$ 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: $↔(s(n-{n}_{0}), e^{-(i\times 2\pi f{n}_{0})}S(e^{i\times 2\pi f}))$.

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
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
$y(n)={a}_{1}y(n-1)+\dots +{a}_{p}y(n-p)+{b}_{0}x(n)+{b}_{1}x(n-1)+\dots +{b}_{q}x(n-q)$
Here, the output signal $y(n)$ is related to its past values $y(n-l)$, $l=\{1, \dots , p\}$, and to the current and past values of the input signal $x(n)$. The system's characteristics are determined by the choices for the number of coefficients $p$ and $q$ and the coefficients' values $\{{a}_{1}, \dots , {a}_{p}\}$ and $\{{b}_{0}, {b}_{1}, \dots , {b}_{q}\}$. 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.

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