Discrete Time Signals

Common Representations

Difference equation e.g. \(y[n] = x[n] - x[n-1]\). These are declarative, in that they describe the rules the system obeys
Block diagram. These are imperative, in that they tell you how to implement the system
Operator algebra e.g. \(Y = (1 - R)X\), where \(R\) is the right shift operator

Operator Algebra

A term like \(x[n]\) describes a sample of a signal, specifically a sampling at the \(n\)th index. Operators manipuilate entire signals rather than samples. Specifically, signals are primitives and operators are the tools for manipulation.

Example: \(Y = RX \Leftrightarrow y[n] = x[n-1]\)
Example: \(Y = (1 -R)X \Leftrightarrow y[n] = x[n] - x[n-1]\)

Operators are useful because they can be manipulated as polynomials. For instance, \(Y = (1 - R)^2 X = (1 - 2R + R^2)X\). This algebra gives rise to equivalence classes for systems. For instance, the three following systems would be implemented differently, but the input-output signal-to-signal map is the same:

\[Y = (1 + 4R + 4 R^2) X \Leftrightarrow Y = (2R + 1)^2 X \Leftrightarrow Y = X + 4 R X + 4 R^2 X\]

Recipes vs Constraints

Some systems are better described as recipes (i.e. instructions for transforming signals) while others are better described as constraints (i.e. requirements that must be met). For instance, \(Y = (1 - R)X\) is a recipe (“To construct Y, subtract a right-shifted version of input from the input”) whereas \((1- R)Y = X\) is a constraint (“Y minus its right-shifted version must equal the input”).