EE 3010 Signals and

Dr. Dokhyl AlQahtani

Systems Analysis

1
Introduction and Definitions

2
 Definitions

Signals Systems Analysis Design

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 What is a signal? A signal can be defined as a physical
phenomenon that carries information.

Examples:
• Human voice.
• Electromagnetic waves.
Definitions • Sonar waves.
(1/3)
Note: Any quantity does not necessarily have to be in a
waveform type to be considered as a signal. For example,
Indians in ancient times used smoke to report any danger.

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 What is a system? A system is defined as an entity
that manipulates one or more signals to accomplish a
function, there by yielding new signals.

Examples:
Definitions • Electric Circuits.
(2/3) • Computer Program.
• Camera.

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 Problems that often arise when dealing
with signals and systems are of the form
of:
• Analysis: we are presented with a specific system and
are interested in charecterizing it in detail to
understand how it will respond to various inputs.
Definitions • Synthesis (Design): Our interest may be focused on
the problem of designing systems to process signals
(3/3) in particular ways.

Applications:
- Speech Processing.
- Image Processing.
- Biomedical engineering.
- Seismology.
- Sonar.
- Radar.
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 In telephone conversations, the speaker's voice (acoustic
signal), is transformed by a microphone (a system), into
an electrical signal (voice signal). The magnitude of the
Exapmle #1
voice signal may be too low and therefore it needs be
Telecommunications
raised up (or amplified) by an amplifier (a system) in
order for it to be audible at the listener's end (receiver
system).

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 Medical devices like lie detector, Electrocardiograph,
etc., (which are systems) pick up certain impulses from
Example #2 the body and convert them into electrical signals, which
Medical/Intelligence are used for medical diagnoses or for Intelligence
purposes.

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 Radar signals are used to detect presence, speed, or
Example #3
location of an object. This information can be used in
Military/Law
traffic control department, aviation control
Enforcements/Aviation
department, or in military communications.

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 ATM Pay points & bank machine exchange several
Example #4 Finance authorization signals to allow you convenient access
to bank accounts.

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 Example #5
Engineers at NASA control movements of robots from one
Advanced Robotics location to another in the MARS, remotely from Earth!, etc..
and Control systems

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Signals
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 • For electrical signal the value of voltage or current
changes with time, hence time is called independent
variable and voltage or current is called dependent
variable
THE • Independent variable can be continuous
INDEPENDENT — Trajectory of a space shuttle
— Mass density in a cross-section of a brain
VARIABLES
• Independent variable can be discrete
— DNA base sequence
— Digital image pixels

• Independent variable can be 1-D, 2-D, ••• N-D

• For this course: Focus on a single (1-D) independent

variable which we call “time”.

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 Continuous-Time (CT) signals

• Most of the signals in the physical world are CT signals—E.g. voltage &
current, pressure, temperature, velocity, etc.
• A continuous-time (CT) signal is one which is defined at every instant of time
over some time interval.
• Continuous-Time (CT) signals: x(t), t — continuous values
• They are functions of a continuous time variable.
• We often refer to a CT signal as 𝑥(𝑡).
• The independent variable is time 𝑡 and can have any real value, the function
𝑥(𝑡) is called a CT function because it is defined on a continuum of points in
time.

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 Discrete-Time (DT) signals (1/2)

• A discrete-time (DT) signal is one which is defined only at

discrete points in time and not between them.
• x[n], n — integer, time varies discretely.
• The independent variable takes have only a discrete set of values.
• Discrete-Time (DT) signals: x[n], n — integer values only.

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 Discrete-Time (DT) signals (2/2)

• It is often derived from continuous-time signal 𝑥(𝑡) by sampling it at a

uniform rate.
• Let T_s denote the sampling period and 𝑛 denote an integer.
• The symbol n denotes time for discrete time signal and [ ] is used to denote
discrete-value quantities.

xn  xnTs , n  0,  1,  2,....

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 Classification of CT and DT Signals

• We may identify four methods of classifying CT & DT signals

based on different features:

1. Deterministic & Random Signals

2. Periodic & Aperiodic Signals

3. Energy & Power Signals

4. Even & Odd Signals

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 Classification of CT and DT Signals

Deterministic & Random Signals

• Deterministic Signals: signals whose values are predictable at any time
(no uncertainty in knowing their values). i.e. we can predict the behavior
of the signal.
 e.g.:
𝑥 𝑡 = 𝑒 −𝑎𝑡
𝜋
𝑥 𝑛 = 𝑛 cos 𝑛
2
• Random Signals: signals whose values are random (unpredictable) at
any time and cannot be described by any mathematical function.
 e.g.: Noise

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 Classification of CT and DT Signals

Periodic & Aperiodic Signals

 A signal 𝑥(𝑡) is periodic if and only if
𝑥 𝑡 = 𝑥 𝑡 + 𝑇0 , −∞ < 𝑡 < ∞
 The smallest value of 𝑇0 that satisfies this condition is called the
fundamental period.
 Any deterministic signal which is not periodic is called aperiodic.
 The reciprocal of the fundamental period, 𝑇0 , is called the fundamental
frequency, 𝑓0
1
𝑓0 =
𝑇0
 Fundamental Angular Frequency:
2𝜋
𝜔0 = 2𝜋𝑓0 = radians/sec
𝑇0

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 Periodic CT Signals

 If 𝑥(𝑡) is periodic with period 𝑇0, then 𝑥 𝑡 = 𝑥 𝑡 + 𝑚𝑇0 for all 𝑡, and
for any integer 𝑚, 𝑚 = 1,2,3, …

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 Periodic CT Signals

Example #1:
With respect to the signal shown in Figure determine the fundamental frequency
and the fundamental angular frequency.
 It is clear that the fundamental period 𝑇0 = 0.2 𝑠𝑒𝑐.
 Thus,
1
 𝑓0 = 0.2 = 5 𝐻𝑧.
 𝜔0 = 2𝜋𝑓0 = 10 𝜋 radians/sec.

 It repeats itself 5 times in one second, which can be clearly seen in Figure.

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 Periodic CT Signals*

Example #2:
A real valued sinusoidal signal 𝑥(𝑡) can be expressed mathematically by:
𝑥 𝑡 = A sin (𝜔0 𝑡 + Ɵ)
Show that 𝑥(𝑡) is periodic.

In general sin (𝜔0 𝑡 + Ɵ) , cos (𝜔0 𝑡 + Ɵ) , are periodic for any values of 𝜔0 .
If a signal is not periodic, it is called Aperiodic signal. 22
 Sum of Periodic CT Signals

An important question for signal analysis is whether or not the sum of two
periodic signals is periodic.

For CT signals, sum of two periodic CT signals is not always periodic, will be
periodic if condition stated below is satisfied.

– The sum of two sinusoids is periodic if the ratio of their period is a rational
number. {e.g. 1, 2, 3, 3/5, etc}.
𝑇 𝑓2
– i.e. if 1 = = 𝑛, 𝑛 = 1,2,3, … , where 𝑇1 and 𝑇2 are the periods of the two
𝑇2 𝑓1
signals.

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 Sum of Periodic CT Signals *

Example:

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 Periodic DT Signals

A DT signal 𝑥[𝑛] is periodic with period 𝑁, where 𝑁 is an integer, if

𝑥[𝑛 + N] = 𝑥[𝑛] for all 𝑛.

The smallest value of 𝑁 that satisfies this condition is called the fundamental
period.

If the above condition holds, then 𝑥[𝑛] is also periodic with period: 2𝑁, 3𝑁,…

The fundamental angular frequency of a periodic DT signal 𝑥[𝑛] is

2𝜋
𝜔0 = 𝑟𝑎𝑑/𝑠𝑒𝑐
𝑁

For DT signals, sum of two periodic DT signals is always periodic.

If 𝑥[𝑛] is periodic with period 𝑁1 and y[𝑛] is periodic 𝑁2 , then

z 𝑛 = 𝑥 𝑛 + y[𝑛] is always periodic with period 𝑁1 𝑁2 / gcd(𝑁1 , 𝑁2 )
gcd = greatest common divisor. 25
 Periodic DT Signals

Example #1:
Assess the periodicity of the DT sinusoid 𝑥 𝑛 = sin 𝜔0 𝑛 .

𝑥 𝑛 is Periodic iff 𝑥 𝑛 = 𝑥[𝑛 + 𝑁]

⟹ sin 𝜔0 𝑛 + 𝑁 = sin 𝜔0 𝑛 + 𝜔0 𝑁

sin 𝜔0 𝑛 + 𝜔0 𝑁 = sin 𝜔0 𝑛 cos 𝜔0 𝑁 + cos 𝜔0 𝑛 sin 𝜔0 𝑁

⟹ periodic if cos 𝜔0 𝑁 = 1 and sin 𝜔0 𝑁 = 0

i.e. 𝜔0 𝑁 = 𝑘 ∙ 2𝜋 , which is satisfied if an integer 𝑁 exist such that

𝑘 ∙ 2𝜋
𝑁=
𝜔0
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 Periodic DT Signals *

Example #2:
Determine if 𝑥[𝑛] = sin3𝜋𝑛 is periodic and find its fundamental period.
Solution:
𝜔0 = 3𝜋

𝑘 ∙ 2𝜋 𝑘 ∙ 2𝜋
𝑁= = = 2, for k = 3
𝜔0 3𝜋

Therefore 𝑥[𝑛] is periodic with fundamental period 𝑁 = 2.

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 Periodic DT Signals *

Example #3:
Consider the signal 𝑥 𝑛 = sin 3𝑛 . Is 𝑥 𝑛 a periodic signal?

𝜔0 = 3

𝑘 ∙ 2𝜋 𝑘 ∙ 2𝜋
𝑁= =
𝜔0 3

As 𝑘 and 𝑁 are not relatively prime, hence the signal is aperiodic.

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 Periodic DT Signals *
Example #4:
𝜋𝑛2
Show that the signal 𝑥 𝑛 = cos is periodic with fundamental period N = 8.
8
Solution:
𝜋 𝑛+𝑁 2
𝑥 𝑛 + 𝑁 = cos
8
𝜋𝑛2 + 2𝜋𝑛𝑁 + 𝜋𝑁 2 𝜋𝑛2 2𝜋𝑛𝑁 + 𝜋𝑁 2
= cos = cos +
8 8 8

The smallest value of N for which 𝑥 𝑛 + 𝑁 = 𝑥 𝑛 is N = 8. that is

𝜋 𝑛+8 2
𝑥 𝑛 + 8 = cos
8
𝜋𝑛2 + 2𝜋𝑛8 + 𝜋82 𝜋𝑛2 𝜋𝑛2
= cos = cos + 2𝜋𝑛 + 8𝜋 = cos = 𝑥[𝑛]
8 8 8 29
 Sum of Periodic DT Signals *

Example #5:
𝜋 𝜋
𝑥 𝑛 = cos 𝑛 + sin 𝑛
3 4
Solution:
𝜋 𝜋 𝑘 ∙ 2𝜋
𝑥1 𝑛 = cos 𝑛 ⟹ 𝜔1 = ⟹ 𝑁1 = 𝜋 = 6, 𝑘=1
3 3
3
𝜋 𝜋 𝑘 ∙ 2𝜋
𝑥2 𝑛 = sin 𝑛 ⟹ 𝜔2 = ⟹ 𝑁2 = 𝜋 = 8, 𝑘=1
4 4
4
𝑁1 × 𝑁2 6×8 48
∴ 𝑁0 = = = = 24
gcd 𝑁1 , 𝑁2 gcd 6,8 2

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 Classification of CT and DT Signals

Energy and Power Signals:

• In electric circuits, a signal may be represented as a voltage, 𝑣(𝑡) or a current 𝑖(𝑡).

• Consider a voltage 𝑣(𝑡) developed across a resistor 𝑅, producing a current 𝑖(𝑡).

• The instantaneous power 𝑝(𝑡) dissipated in 𝑅 is defined by

𝑣 𝑡 2
2𝑅
𝑝 𝑡 = = 𝑖 𝑡
𝑅

2 2
• For 𝑅 = 1, we can write 𝑝 𝑡 = 𝑣 𝑡 = 𝑖 𝑡

• Regardless whether a given signal 𝑥(𝑡) represents a voltage or a current, we may express
𝑝(𝑡) associated with the signal 𝑥(𝑡) as

𝑝 𝑡 = 𝑥 2 (𝑡)

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 Classification of CT and DT Signals

• The total energy of a signal is

𝐸 = lim න 𝑥(𝑡) 2 𝑑𝑡 CT signal

𝑇→∞
−𝑇
𝑁
2
= lim ෍ 𝑥[𝑛] DT signal
𝑁→∞
𝑛=−𝑁

• The average power of a signal is

𝑇
1
𝑃𝑎𝑣𝑒 = lim න 𝑥(𝑡) 2 𝑑𝑡 CT signal
𝑇→∞ 2𝑇
−𝑇
𝑁
1 2
= lim ෍ 𝑥[𝑛] DT signal
𝑁→∞ 2𝑁 + 1
𝑛=−𝑁

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 Classification of CT and DT Signals

A signal is referred to as an energy signal if and only if the total energy of the signal
satisfies.
0 < 𝐸 < ∞ (non-zero & finite Energy)

A signal is referred to as power signal if and only if the average power of the signal
satisfies.
0 < 𝑃𝑎𝑣𝑒 < ∞ (non-zero & finite Power)

Energy and power classifications of signals are mutually exclusive {i.e., an energy
signal has zero average power and a power signal has infinite energy}.

Signals not satisfying the above conditions are classified as neither energy nor power
signals.

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 Classification of CT and DT Signals

Example: Determine the total energy and average power of the following signal
𝐴𝑒 −𝑎𝑡 , 𝑡 > 0, 𝑎 > 0
𝑥 𝑡 =ቊ
0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
Solution:
 Classification of CT and DT Signals

Example: Determine the average power of the following periodic signals:

1. 𝑥1 𝑡 = 𝐴𝑒 𝑗 𝜔0 𝑡+𝜃
2. 𝑥2 𝑡 = 𝐴cos 𝜔0 𝑡 + 𝜃
Solution:
 Classification of CT and DT Signals

Remarks:
• In general, periodic signals are power signals, the average power of a periodic
signal can be computed using:

where T0 and N are the fundamental periods of x(t) and x[n] respectively.
 Classification of CT and DT Signals

Even and Odd signals:

• A signal is referred to as even if
𝑥(−𝑡) = 𝑥(𝑡) CT
𝑥[−𝑛] = 𝑥[𝑛] DT
• A signal is referred to as odd if
𝑥(−𝑡) = −𝑥(𝑡) CT
𝑥[−𝑛] = −𝑥[𝑛] DT
• Any signal can be written as sum of two signals, one of which is even and one of
which is odd:
i.e. 𝑥(𝑡) = 𝑥𝑒𝑣𝑒𝑛(𝑡) + 𝑥𝑜𝑑𝑑(𝑡) , where
1
𝑥𝑒𝑣𝑒𝑛 𝑡 = 𝑥 𝑡 + 𝑥(−𝑡)
2
1
𝑥𝑜𝑑𝑑 𝑡 = 𝑥 𝑡 − 𝑥(−𝑡)
2
Similar results also hold for DT signals.
 Classification of CT and DT Signals

• Illustration:
 Classification of CT and DT Signals

Example:
Let
𝐴, 𝑛 = 0, 1 , 2, …
𝑥 𝑛 =ቊ
0, otherwise
Determine
• The even and odd parts of x[n].
• The total energy and average power of x[n]
Solution:
• The even and odd parts are
𝐴 𝑛=0
𝑥𝑒𝑣𝑒𝑛 𝑛 = ቐ𝐴
𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
2
𝐴
− , 𝑛<0
2
𝑥𝑜𝑑𝑑 𝑛 = 0, 𝑛=0
𝐴
, 𝑛>0
2
 Classification of CT and DT Signals

• The total energy and average power are:

𝑁 𝑁
2
𝐸 = lim ෍ 𝑥[𝑛] = lim ෍ 𝐴2 = lim 𝐴2 1 + 𝑁 = ∞
𝑁→∞ 𝑁→∞ 𝑁→∞
𝑛=−𝑁 𝑛=−𝑁
𝑁 𝑁
1 2
1 2 2
𝑁+1 𝐴2
𝑃𝑎𝑣𝑒 = lim ෍ 𝑥[𝑛] = lim ෍ 𝐴 = lim 𝐴 =
𝑁→∞ 2𝑁 + 1 𝑁→∞ 2𝑁 + 1 𝑁→∞ 2𝑁 + 1 2
𝑛=−𝑁 𝑛=−𝑁