Signals defined only at discrete instants of time are discrete signals, denoted by symbols , , and so on, where is an integer. Systems whose inputs and outputs are discrete signals are discrete systems.

If the samples are uniformly distributed, we further simplify this notation to

Discrete signals naturally occur in scenarios where the input is inherently discrete. They can also result from sampling continuous-time signals. For example, in digital filtering, a continuous signal is first sampled to convert into a discrete signal , which then is processed by a discrete-time system to yield an output , and at the end we reconstruct a continuous system . 1 processing continuous signal as discrete.png

Discrete System Representation

Instead of differential equations, we can use difference equations to represent a discrete system. For example,

An alternative is a block diagram like the following:

Difference equations are mathematically precise and concise. However, since difference equation is declarative, it does not show the signal flow path. On the other hand, the block diagram is imperative, and can directly translate into a hardware implementation or an algorithm. 2

Another alternative representation that combines the strength of difference equations and block diagrams are the operator notation. For example, let be the right-shift operation, and let and represents the whole input and output signals, then we can represent the above system as

Footnotes

  1. Linear Systems and Signals, 3rd Edition, 1.7.5

  2. Lecture 2: Discrete-time systems | Signals and Systems | MIT OpenCourseWare