A discrete-time signal is represented symbolically as
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Summary: (Blank Abstract)
Mathematically, analog signals are functions having as their independent variables continuous quantities, such as space and time. Discrete-time signals are functions defined on the integers; they are sequences. As with analog signals, we seek ways of decomposing discrete-time signals into simpler components. Because this approach leads 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
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The most important signal is, of course, the complex exponential sequence.
Discrete-time sinusoids have the obvious form
The second-most important discrete-time signal is the unit sample, which is defined to be
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The unit step in discrete-time is well-defined at the origin, as opposed to the situation with analog signals.
An 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
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.
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This work is licensed by Don Johnson under a Creative Commons Attribution License (CC-BY 3.0), and is an Open Educational Resource.
Last edited by Don Johnson on May 17, 2013 9:45 am -0500.
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