A discrete-time signal is represented symbolically as

Discrete-Time Cosine Signal |
---|

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

Summary: Signals can be represented by discrete quantities instead of as a function of a continuous variable. These discrete time signals do not necessarily have to take real number values. Many properties of continuous valued signals transfer almost directly to the discrete domain.

So far, we have treated what are known as
analog signals and systems. Mathematically, analog
signals are functions having continuous quantities as their
independent variables, such as space and time. Discrete-time signals are
functions defined on the integers; they are sequences. One of
the fundamental results of
signal theory will detail conditions under which an
analog signal can be converted into a discrete-time one and
retrieved *without error*. This result is
important because discrete-time signals can be manipulated by
systems instantiated as computer programs. Subsequent modules
describe how virtually all analog signal processing can be
performed with software.

As important as such results are, discrete-time
signals are more general, encompassing signals derived from
analog ones *and* signals that aren't. For
example, the characters forming a text file form a sequence,
which is also a discrete-time signal. We must deal with such
symbolic
valued signals and systems as well.

As with analog signals, we seek ways of decomposing real-valued discrete-time signals into simpler components. With this approach leading to a better understanding of signal structure, we can exploit that structure to represent information (create ways of representing information with signals) and to extract information (retrieve the information thus represented). For symbolic-valued signals, the approach is different: We develop a common representation of all symbolic-valued signals so that we can embody the information they contain in a unified way. From an information representation perspective, the most important issue becomes, for both real-valued and symbolic-valued signals, efficiency; What is the most parsimonious and compact way to represent information so that it can be extracted later.

A discrete-time signal is represented symbolically as

Discrete-Time Cosine Signal |
---|

The most important signal is, of course, the complex exponential sequence.

Discrete-time sinusoids have the obvious form
*only* when

The second-most important discrete-time signal is the unit sample, which is defined to be

Unit Sample |
---|

Examination of a discrete-time signal's plot, like that of the
cosine signal shown in Figure 1,
reveals that all signals consist of a sequence of delayed and
scaled unit samples. Because the value of a sequence at each
integer
*any* signal as a sum of
unit samples delayed to the appropriate location and scaled by
the signal value.

Discrete-time systems can act on discrete-time signals in ways similar to those found in analog signals and systems. Because of the role of software in discrete-time systems, many more different systems can be envisioned and “constructed” with programs than can be with analog signals. In fact, a special class of analog signals can be converted into discrete-time signals, processed with software, and converted back into an analog signal, all without the incursion of error. For such signals, systems can be easily produced in software, with equivalent analog realizations difficult, if not impossible, to design.

Another interesting aspect of discrete-time signals is that
their values do not need to be real numbers. We do have
real-valued discrete-time signals like the sinusoid, but we
also have signals that denote the sequence of characters typed
on the keyboard. Such characters certainly aren't real
numbers, and as a collection of possible signal values, they
have little mathematical structure other than that they are
members of a set. More formally, each element of the
symbolic-valued signal
*entirely* of
analog circuit elements. Furthermore, the transmission and
reception of discrete-time signals, like e-mail, is
accomplished with analog signals and systems. Understanding
how discrete-time and analog signals and systems intertwine is
perhaps the main goal of this course.

Download LabVIEW Source

- « Previous module in collection m19 - Wavlet-Based Signal Analysis
- Collection home: ECE 454 and ECE 554 Supplemental reading
- Next module in collection » Difference Equation

Comments:"Electrical Engineering Digital Processing Systems in Braille."