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Signals and Systems

Chapter (1)Networks and Communication Department

Signals and Systems Analysis

21.1 What does a signal mean ?

A signal is a function representing a physical quantity or variable, typically is contains information about the behavior or nature of the phenomenon.

A system is viewed as transformation (mapping) of into , to process input signals to produce output signals.

Types of Signals in our lives and some examples:

• Electrical Signals:(e.g. voltage and current)

• Electromagnetic Signals:(e.g. radio, waves and light)

• Sound Signals:(e.g. human sounds and music)

SystemInput signal Output signal


31.2 Classifications of Signals

A.Continuous-Time and Discrete-Time Signals (Basic types of Signals)

Continuous-Time Signals: A signal is continuous-time signal if the independent variable t is continuous.

Discrete-Time Signals:• A signal is defined at discrete times, a discrete-time signal is

often identified as a sequence of numbers, denoted by • A very important class of discrete-time signals is obtained by

sampling a continuous-time signal


41.2 Classifications of Signals (Cont.)

B. Analog and Digital Signals Analog signals:

If a continuous-time signal can take on any value in the continuous interval (, b), where may be - and b may be +, then the continuous-time signal is called an analog signal.

Digital signals: It is a discrete or non-continuous waveform, that could take only a finite number of values; with examples such as computer 1s and 0s.



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E. Even and Odd Signals A signal or is referred to as an even signal if

= =



E. Even and Odd Signals (Cont.) A signal or is referred to as an odd signal if

= =



E. Even and Odd Signals (Cont.) Any signal x(t) or x[n] can be expressed as a sum of two signals, one of

which is even and one of which is odd. That is,

= e + o

= e[] + o[]

where e(t) = ½{ + } even part of

o(t) = ½{ - } odd part of

e[] = ½{ + } even part of

o[] = ½{ - } even part of



Periodic and Non-periodic signals:

Periodic signal:Pattern repeated over time.

Non-periodic signal(Aperiodic):Pattern not repeated over

time.



F. Periodic and Non-periodic Signals:A continuous-time signal is said to be periodic with period T if

there is a positive nonzero value of T for which

(t + T) = for all t

In other words, a periodic signal has the property that it is unchanged by a time shift of T (called period T).

An example of such a signal is shown below



F. Periodic and Non-periodic Signals (Cont.):From this equation

(t + T) =

it follows that

(t + mT) =

for all t and any integer m.

Note that this definition does not work for a constant signal .

Any continuous-time signal which is not periodic is called a non-periodic (or aperiodic ) signal.



F. Periodic and Non-periodic Signals (Cont.):



G. Exponential and Sinusoidal Signals:

atCetx )(C



1.2.1 Continuous-Time Complex Exponential and Sinusoidal Signals (Cont.):

Real Exponential Signals (Cont.):




Periodic Complex Exponential and Sinusoidal Signals:Euler's formula,

• An important property of this signal is that it is periodic.• The fundamental period ( ) is the smallest positive value of T,

tjtetx tj00 sincos)( 0

Real Part Imaginary Part

00

2

T

0T




Sinusoidal Signals:• A continuous-time sinusoidal signal can be expressed as

Where A is the amplitude (real), 0 is the radian frequency in radians per second, and is the phase angle in radians.

• Sinusoidal signal is periodic with fundamental period given by

)cos()( 0 tAtx

00

2

T




Sinusoidal Signals (Cont.)


191.2 Classifications of Signals (Cont.)1.2.1 Continuous-Time Complex Exponential and Sinusoidal Signals (Cont.):

Sinusoidal Signals (Cont.)The reciprocal of the fundamental period T0 is called the fundamental frequency 0:

The frequency is the number of times a signal makes a complete cycle within a given time frame.

From the previous two equations we can conclude the following relation:

0 is the fundamental angular frequency.

HzhertzT

f )(1

00

00 2 f



Frequency, Spectrum and Bandwidth:

Frequency is measured in Hertz (Hz), or cycles per second = A cos(t)e.g. S = 5cos (25t) here =5Hz.

Spectrum: Range of frequencies that a signal spans from minimum to maximum,

e.g. = + where =cos(25t) and =cos(27t). Here spectrum SP={5Hz,7Hz}.

Bandwidth : Absolute value of the difference between the lowest and

highest frequencies of a signal. In the above example bandwidth is BW=2Hz.



- Phase

The position of the waveform relative to a given moment of time or relative to time zero,

e.g. =cos(25t) and =cos(25t + /2).

Here has phase =0 and has phase = /2.

A change in phase can be any number of angles between 0 and 360 degrees.

Phase changes often occur on common angles,

such as /4=45, /2=90, 3/4=135, etc.

)cos()( 0 tAtx


221.2 Classifications of Signals (Cont.)1.2.1 Continuous-Time Complex Exponential and Sinusoidal Signals (Cont.):

The relationship between fundamental frequency and period for continuous-time sinusoidal signals.Here Which implies that


231.2 Classifications of Signals (Cont.)1.2.2 Discrete-Time Complex Exponential and Sinusoidal Signals:

Where and are complex (real) numbers.

Sinusoidal Sequence Signals:

• In order for the sequence to be periodic only if N > 0 must satisfy the following condition: (m is a positive integer)

nn CeCnx ][

C

)cos(][ nAnx o

N

mo

2

N is the fundamental

period

oo mN

2



1.2.2 Discrete-Time Complex Exponential and Sinusoidal Signals (Cont.):

Real Exponential Sequence Signals

> 1

0 < < 1

0- > > 1

- < 1


251.3 Transformations of the independent variable:

Reflection

Shifting


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1.3 Transformations of the independent variable (Cont.):

Scaling

Exercise :Sketch the signal x[n+2]


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1.4 The Unit Impulse and Unit Step Function

1.4.1 The Discrete-Time Unit Impulse

The shifted unit impulse (or sample) sequence is defined as:


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1.4 The Unit Impulse and Unit Step Function (Cont.)1.4.1 The Discrete-Time Unit Step (Cont.)

Note: that the value of u[n] at n = 0 is defined.

The shifted unit impulse (or sample) sequence is defined as:

Note: that the value of at n = 0 is defined.

step


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1.4 The Unit Impulse and Unit Step Function (Cont.)

1.4.2 The Continuous-Time Unit Step:

Note: The unit step is discontinuous at and that value is undefined.

Similarly, the shifted unit step function is defined as


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1.4 The Unit Impulse and Unit Step Function (Cont.)


311.5 Continuous-Time and Discrete-Time Systems

System Representation:

Continuous-Time and Discrete-Time Systems:



Cascade(series) Interconnection:

Parallel Interconnection:



Series-parallel Interconnection:

Feedback Interconnection:


341.6 Basic Systems Properties

System with Memory and without Memory: A system is said to be memoryless if the output at any time

depends on only the input at that same time. Otherwise, the system is said to have memory.

An example of a memoryless system:

An example of a system with memory:

)()( txRty

n

k

kxny ][][

1.6 Basic Systems Properties (Cont.)

Signals and Systems Analysis35

Inevitability and inverse systems: Invertible system: a system S is invertible if the the input signal

can always be uniquely recovered from the output signal Y(t)=2x(t)

The inverse system: formally written as ,such that the

cascade interconnection in the figure below is equivalent to the identity system, which leaves the input unchanged



Causality Systems: Causal Systems: if the output at any time depends on values of the

input at only the present and past time. Thus, in a causal system, it is not possible to obtain an output

before an input is applied to the system. All ”real time” systems must be causal, since they can not have

future inputs available to them. A system is called non-causal if it is not causal.

Examples of causal systems are:

Examples of non-causal systems are:

Note: that all memoryless systems are causal, but not vice versa.

)2(3)( txty

)1()( txty



Time-Invariant and Time-Varying Systems: A system is called time-invariant if a time shift (delay or advance)

in the input signal causes the same time shift in the output signal. Thus, for a continuous-time system, the system is time-invariant if for any real value of t.

For a discrete-time system, the system is time-invariant (or shift-invariant ) if for any integer k

A system that doesn’t satisfy this condition is called time-varying.

)()}({ tytxT

][]}[{ knyknxT



Linear Systems and Nonlinear Systems:

If the operator T in Equation y = Tx satisfies the following two conditions, then T is called a linear operator and the system represented by a linear operator T is called a linear system:

1- Additivity:

Given that Tx1 = y1 and Tx2 = y2, then

T{x1 + x2} = y1 + y2

for any signals x1 and x2.



Linear Systems and Nonlinear Systems (Cont.):

2- Homogeneity or (scaling):

Tax = ayfor any signal x and any scalar a.

Any system that does not satisfy both conditions is classified as a nonlinear system. Both conditions can be combined into a single condition as

T {a1x1 + a2x2} = a1y1 + a2y2

where a, and a, are arbitrary scalars. This final Equation is known as the superposition property.



Linear Time-Invariant System: A system is called Linear time-invariant (LTI) if this system is

both linear and time-invariant.

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