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 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
sn
s
n
, where
n=…−101…
n
…
1
0
1
…
.

We usually draw discrete-time signals as stem plots to emphasize the fact
they are functions defined only on the integers. We can delay a
discrete-time signal by an integer just as with analog ones. A delayed
unit sample has the expression
δn−m
δ
n
m
, and equals one when
n=m
n
m
.

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

sn=ej2πfn
s
n
j
2
f
n

(1)
Discrete-time sinusoids have the obvious form
sn=Acos2πfn+φ
s
n
A
2
f
n
φ
.
As opposed to analog complex exponentials and sinusoids that can have their
frequencies be any real value, frequencies of their discrete-time counterparts
yield unique waveforms *only *when
f
f lies in the interval
−12
12
1
2
1
2
.
This property can be easily understood by noting that adding an integer to
the frequency of the discrete-time complex exponential has no effect on the
signal's value.

ej2π(f+m)n=ej2πfnej2πmn=ej2πfn
j
2
f
m
n
j
2
f
n
j
2
m
n
j
2
f
n

(2)
This derivation follows because the complex exponential evaluated at an
integer multiple of

2π
2
equals one.

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

δn={1 if n=00 if otherwise
δ
n
1
n
0
0
otherwise

(3)
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
m
m is denoted by
sm
s
m
and the unit sample delayed to occur at
m
m
is written
δn−m
δ
n
m
, we can decompose *any *signal as a sum of unit samples
delayed to the appropriate location and scaled by the signal value.

sn=∑
m
=−∞∞smδn−m
s
n
m
s
m
δ
n
m

(4)
This kind of decomposition is unique to discrete-time signals, and will prove useful
subsequently.

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
sn
s
n
takes on one of the values
a
1
…
a
K
a
1
…
a
K
which comprise the alphabet
A
A.
This technical terminology does not mean we restrict symbols to being members of
the English or Greek alphabet. They could represent keyboard characters, bytes
(8-bit quantities), integers that convey daily temperature. Whether controlled by
software or not, discrete-time systems are ultimately constructed from digital
circuits, which consist *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.