Chapter 2

Signal Classification and

Representation
EE 370 Communications Engineering I Chapter 2

Classification of Signals
Analog vs. Digital signals:
Analog signals - amplitude may take any real value in a specific range
Digital Signals - amplitude takes only a finite number of values

Continuous-time vs. discrete-time signals:

Continuous-time signals - magnitudes are defined for all values of time (t)
(can be analog or digital signals)
Discrete-time signals - magnitudes are defined at specific instants of time
only and are undefined for other time instants (can be analog or digital)

2
 Classification of Signals
Examples

Figure 2.4 Examples of signals: (a) analog and continuous time; (b) digital and continuous time;
(c) analog and discrete time; (d) digital and discrete time.
EE 370 Communications Engineering I Chapter 2

Classification of Signals
Periodic vs. aperiodic signals:
Periodic signals - those that are constructed from a specific shape that
repeats regularly after a specific amount of time T0

f(t) = f(t+nT0) for all integer values of n

Aperiodic Signals - do not repeat regularly

Deterministic vs. probabilistic signals:

Deterministic signals - that can be computed beforehand at any instant of time

Probabilistic signals - one that is random and cannot be determined beforehand

4
EE 370 Communications Engineering I Chapter 2

Classification of Signals
Energy vs. Power signals:

Figure 2.1 Examples of signals: (a) signal with finite energy;

(b) signal with finite power.
EE 370 Communications Engineering I Chapter 2

Classification of Signals
Energy vs. Power signals:
∞

∫
2
Energy: Ef = | f (t ) | dt
−∞
T /2
1
Power: Pf = lim
T →∞ T ∫
−T / 2
| f (t ) |2 dt

T +t0
1
∫
2
PPeriodic f = | f (t ) | dt
T t0

Energy Signals: an energy signal is a signal with finite energy and zero
average power (0 ≤ Ef < ∞, P = 0),

Power Signals: a power signal is a signal with infinite energy but finite
average power (0 < P < ∞, Ef → ∞).
6
EE 370 Communications Engineering I Chapter 2

Comments
of a power signal is what is usually defined as the
1. P RMS value of that signal.

2. In most of the cases if a signal approaches zero as t approaches ∞

then the signal is an energy signal. But this is not always as you
can verify in part (d) in the following example.

3. All periodic signals are power signals (but not all non–periodic
signals are energy signals).

4. Any signal f(t) that has limited amplitude (| f(t) | < ∞) and is time
limited (f(t) = 0 for | t | > t0 for some t0 > 0) is an energy signal
as in part (g) in the following example.
EE 370 Communications Engineering I Chapter 2

Exercise 2–1: determine if the following signals are Energy signals,

Power signals, or neither, and evaluate E and P for each signal
a) a ( t ) = 3 s in ( 2 π t ), − ∞ < t < ∞

b) b (t ) = 5e −2|t | , −∞ < t < ∞

4e +3t , | t |≤ 5
c) c (t ) = 
 0, | t |> 5
 1 ,
 , t >1
d) d (t ) =  t
 0, t ≤ 1

e) e (t ) = −7t 2 , −∞ <t < ∞

f) f (t ) = 2 cos 2 (2π t ), −∞ <t < ∞

12 cos 2 (2π t ), −8 < t < 31
g) g (t ) = 
 0, elsewhere
EE 370 Communications Engineering I Chapter 2
Basic Signal Operations

A) Time Shifting: given the signal f(t), the signal f(t–t0) is a

time-shifted version of f(t); t0 > 0 => RIGHT shift, t0 < 0 => LEFT shift.

B) Magnitude Shifting: given the signal f(t), the signal c + f(t) where c is a
constant is a magnitude-shifted version of f(t);
c >0 => UP shift, c <0 => DOWN shift.

C) Time Scaling and Time Inversion: given f(t), the signal f(a⋅t) is a
time-scaled version of f(t), where a is a constant;
0 < |a| < 1 => f(a⋅t) is an EXPANDED version of f(t), and
|a| > 1 => f(a⋅t) is a COMPRESSED version of f(t) .
a<0 => f(a⋅t) is also a time-inverted version of f(t).

D) Magnitude Scaling and Magnitude Inversion: given f(t), the signal b⋅f(t)
is a magnitude-scaled version of f(t), where b is a constant,
0 < |b| < 1 => b⋅f(t) is an ATTENUATED version of f(t).
|b| > 1 => b⋅f(t) is an AMPLIFIED version of f(t).
b < 0 => the signal b⋅f(t) is also a magnitude-flipped version of f(t).
EE 370 Communications Engineering I Chapter 2

Basic Signal Operations

Example 1: given the signal f(t) shown below,
sketch the signal 4 – 3f(– 2t – 6)
f(t)

-2 -1 1 2 3 4
-1

Example 2: assume the function shown above is

–3g(–t/2+1), sketch g(t).
 EE 370 Communications Engineering I Chapter 2

Unit Impulse Functions (Dirac delta function)

.
Graphical Definition: Mathematical Definition:
The rectangular pulse shape shown The unit impulse function δ(t) satisfies
Below approaches the unit impulse the following conditions:
function as ε approaches 0
(area under the curve=1). 1. δ(t) = 0 if t ≠ 0, (therefore it is
non-zero only at t = 0).
∞
2. ∫ δ (t )dt = 1concentrated
−∞
(so, all of the area under it is
at t = 0)

t
0
EE 370 Communications Engineering I Chapter 2

Properties of the Unit Impulse Function

a) Multiplication of a function by the unit impulse function:
f (t )δ (t − t 0 ) = f (t 0 )δ (t − t 0 )
b) Sampling of a function using the unit impulse function (sifting property):
∞ . ∞ ∞

∫ f (t ) ⋅δ (t − T )dt = ∫ f (T ) ⋅δ (t − T )dt = f (T ) ∫ δ (t − T )dt = f (T )

−∞ −∞ −∞

c) Obtaining the unit step function from the unit impulse function

t
0, t < 0
∫− ∞δ (τ )dτ = 1, t ≥ 0 du (t )
dt
= δ (t )

= u (t )
EE 370 Communications Engineering I Chapter 2

Trigonometric Fourier Series

A periodic function g(t ) can be expressed by a trigonometric Fourier series
with period T0. => frequency fo =1/To Hz (ωo=2π fo radiance/second)
. ∞
g (t ) = a0 + ∑ an cos nω0t + bn sin nω0t t1 ≤ t ≤ t1 + T0
n =1
t 1 +T0
1
a0 =
T0 ∫ g ( t ) dt
t1
t 1 +T0
where 2
an =
T0 ∫ g ( t ) cos nω tdt
t1
0 n = 1, 2,3,L

t 1 +T0
2
bn =
T0 ∫ g ( t ) sin nω tdt
t1
0 n = 1, 2,3,L
EE 370 Communications Engineering I Chapter 2

Compact Trigonometric Fourier Series

We may write:
an cos nω0t + bn sin n ω0t = Cn cos ( nω0t + θn ) where b2 − 4ac
 b 
Cn = an2 + bn2 and θn = tan −1  − n  . Let C0 = a0 . Then,
 an 
∞
g ( t ) = C0 + ∑ C n cos ( nω0t + θn ) → Amplitude and phase spectra
n=1
Note: Either g(t) should be periodic or the Fourier series is valid for t0 ≤ t ≤t0+T0.

This is known as single sided spectra of the signal

EE 370 Communications Engineering I Chapter 2
EE 370 Communications Engineering I Chapter 2

Exponential Fourier Series

Cn j (nω0t +θ n ) − j (nω0t +θ n )
Using Euler’s formula: Cn cos(nω0t + θ n ) =
2
e [ +e ]
C C
Dn = n e jθ n , D− n = − n e − jθ n
2 2
∞
1
We may write g (t ) = ∑ Dn e jnω0t where Dn = ∫ g ( t ) e − jnω0t dt
n =−∞ T0 T0
Existence of the Fourier Series: Dirichlet Conditions

1. ∫ g (t ) dt < ∞
T0
weak Dirichlet condition (Fourier series exists but the convergence is not guaranteed.

2. g(t) has only a finite number of maxima and minima in one period and it may have
only a finite number of finite discontinuities in one period.

1 and 2 together known as strong Dirichlet conditions.

This is known as double sided spectrum of the signal

EE 370 Communications Engineering I Chapter 2

Figure 2.14 Exponential (double sided) Fourier spectra for the signal in Fig. 2.7.
EE 370 Communications Engineering I Chapter 2

Exponential Fourier Spectra (Fourier Spectra) – real g(t)

Cn
Dn = D−n = even
2
∠ D n = θ n , ∠ D − n = −θ n odd

Parseval’s Theorem
∞
For the trigonometric Fourier Series g ( t ) = C0 + ∑ C n cos ( nω0t + θ n )
n =1
1 ∞ 2
The power of g(t) is given by Pg = C02 + ∑ Cn
2 n =1
∞
For the exponential Fourier Series g (t ) = ∑De
n =−∞
n
jnω0t

∑
2
Pg = Dn
n = −∞

For real g(t), |D-n|=|Dn|. Therefore,

∞
Pg = D + 2∑ Dn
2 2
0
n =1
EE 370 Communications Engineering I Chapter 2

Figure 2.15 (a) Unit length complex variable with positive frequency (rotating counterclockwise)
and (b) unit length complex variable with negative frequency (rotating clockwise).
EE 370: Communications Engineering I Chapter 2

A note on Negative Frequency

• All practical signals such as the ones generated in the laboratory are
real. For such a signal, amplitude spectrum has even symmetry and the
phase spectrum has odd symmetry. Thus the signal can be completely
defined by one side of the two sided spectrum.

• However, for the convenience of analysis, we may represent two different

real signals as real and imaginary components of a complex signal. This
will lead to a double sided spectrum which may not have the symmetry.
In such as a case both sides of the spectrum are needed to fully explain
the complex signal.

• A complex signal with two real signals as its real and imaginary
components carries twice as much information as a purely real or
imaginary signal.