1.3 Linear time-invariant systems

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 signalcorresponds 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.

We want to concentrate on systems that are both linear and shift-invariant. It will be these that allow us thefull power of frequency-domain analysis and implementations. Because we have no physical constraints in "constructing" suchsystems, we need only a mathematical specification. In analog systems, the differential equation specifies the input-outputrelationship in the time-domain. The corresponding discrete-time specification is the difference equation .

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 thatcalculates each output from the previous output values, and the current and previous inputs.