Introduction to signals - Continuous and Discrete Time Signals and Systems - page 52

(1) A time-varying signal is classified as a continuous time (CT) signal if it is
defined for all values of time t . A time-varying discrete time (DT) signal is
defined for certain discrete values of time, t = kT s , where T s is the sampling
interval. In our notation, a CT signal is represented by x ( t ) and a DT signal
is denoted by x [ k ].
(2) An analog signal is a CT signal whose amplitude can take any value. A
digital signal is a DT signal that can only have a discrete set of values.
The process of converting a DT signal into a digital signal is referred to as
quantization.
(3) A periodic signal repeats itself after a known fundamental period, i.e.
x ( t ) = x ( t + T 0 ) for CT signals and x [ k ] = x [ k + K 0 ] for DT signals. Note
that CT complex exponentials and sinusoidal signals are always periodic,
whereas DT complex exponentials and sinusoidal signals are periodic only
if the ratio of their DT fundamental frequency Ω 0 ,to2 π is a rational
number.
(4) A signal is classified as an energy signal if its total energy has a non-zero
finite value. A signal is classified as a power signal if it has non-zero finite
power. An energy signal has zero average power whereas a power signal
has an infinite energy. Periodic signals are generally power signals.
(5) A deterministic signal is known precisely and can be predicted in advance
without any error. A random signal cannot be predicted with 100%
accuracy.
(6) A signal that is symmetric about the vertical axis ( t = 0) is referred to
as an even signal. An odd signal is antisymmetric about the vertical axis
( t = 0). Mathematically, this implies x ( t ) = x ( − t ) for the CT even signals
and x ( t ) =− x ( − t ) for the CT odd signals. Likewise for the DT signals.
In Section 1.2, we introduced a set of 1D elementary signals, including rectan-
gular, sinusoidal, exponential, unit step, and impulse functions, defined both in
the DT and CT domains. We illustrated through examples how the elementary
signals can be used as building blocks for implementing more complicated sig-
nals. In Section 1.3, we presented three fundamental signal operations, namely
time shifting, scaling, and inversion that operate on the independent variable.
The time-shifting operation x ( t − T ) shifts signal x ( t ) with respect to time.
If the value of T in x ( t − T ) is positive, the signal is delayed by T time
units. For negative values of T , the signal is time-advanced by T time units.
The time-scaling, x ( ct ), operation compresses ( c > 0) or expands ( c < 0) sig-
nal x ( t ). The time-inversion operation is a special case of the time-scaling
operation with c =− 1. The waveform for the time-scaled signal x ( − t ) is the
reflection of the waveform of the original signal x ( t ) about the vertical axis
( t = 0). The three transformations play an important role in the analysis of lin-
ear time-invariant (LTI) systems, which will be covered in Chapter 2. Finally,
in Section 1.4, we used M ATLAB to generate and analyze several CT and DT
signals.