Page 22 - DCAP608_REAL TIME SYSTEMS

Unit 2: Introduction to Real-time Applications

Examples of real-time systems include command and control systems, process control systems, Notes
flight control systems, and so on. Digital signal processing systems analyse, produce, and
transform discrete-time signals.

2.3.1 Discrete-time Signals

A discrete-time signal is a function that is defined only at a particular set of values of time. In the
common case called uniform sampling, a discrete-time signal x(n) is related to a continuous
analog signal xa(t) by
n
x ( ) x a (nT ), n
where T is the sampling period, and f = 1/T is the sampling frequency.
s

Notes Internal and External Signals
Signals can be internal or external, in the latter case; they constitute an interface between
the signal processing system and an external environment. Many signal processing systems
are “offline”, that is, the external signals are not processed and produced simultaneously
with their occurrence in the external environment. In real-time signal processing, however,
they are.
Although discrete-time signals are defined over all time, computer implementations find
it more convenient to consider only non-negative time indices, where time zero is the
nominal time at which the real-time program begins operating. Thus:

x ( ) x (nT n 0
n
),
a
From here on, quantification of time indexes over positive n is implicit. Let t denote the
x
clock of signal x, that is, the sequence of times at which x is defined. Each element of a clock
is called a “tick.” The clock of a uniformly- sampled signal x with period T is
t {nT }|n 0}
x
A non-uniformly-sampled signal is not characterised quite so easily. Let  x be a non-uniformly-

sampled signal with clock t . Then:
x

t
x ( )n x a ( ( )), n 0
n
x
In general, a discrete-time signal does not represent samples of an analog signal. The sequence
of values representing changes of the state of my refrigerator door – open, closed, or almost-
closed is an example. A sequence of messages between processes in a multi-tasking operating
system is another. The term uniformly-clocked is used to mean a signal with clock {nT | n >= 0}
for constant T, and non-uniformly-clocked for any other signal.

2.3.2 Streams and Channels

A signal is represented in a computer by a register or data structure that updates in time to
contain zero, one, or more values of the signal. This register or data structure will call a channel.
It could be implemented in software or hardware; in either case, a program (or hardware) can
write to and read from the channel. Define a stream as the sequence of values that passes through

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