Inside Collection (Course): ECE 454 and ECE 554 Supplemental reading

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

A discrete-time signal
*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:

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 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.

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
a 0 a 0
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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