SIGNALS & SYSTEMS

Achmad Rizal Danisya/ achmad_rizal@st3telkom.ac.id

8838087/0601128301

Lecture rules 14 Meetings including 2 Quizes, Mid & Final Exam

Quizes = 20 %

Team Points = 15 %

Mid Exam = 25 %

Final Exam = 25 %

Assignments = 15 %

NO REMEDY

Assessment Mark Range Point Explaination

A > 80,0 4 Very Good

B 65,00-79,99 3 Good

C 50,00-64,99 2 Enough

D 30,00-49,99 1 Bad

E ≤ 29,99 0 Very Bad

Syllabus Pengertian Sinyal

Dasar sinyal berbasis waktu-kontinu.

Dasar sinyal berbasis waktu-diskret.

Beberapa sinyal penting.

Sistem & Klasifikasi Sistem

Realisasi sistem dalam bentuk blok diagram

Kestabilan sistem linear

Permasalahan dan penyelesaian persamaan differensial

Persamaan diferensial sistem dan fungsi sistem H(s).

Respon Impuls h(t).

Permasalahan dan penyelesaian beberapa sinyal

Operasi konvolusi sistem kontinu.

Tanggapan pulsa h(n).

Operasi konvolusi sistem diskret.

Syllabus (cont) Definisi dan contoh alih ragam Laplace.

Teorema alih ragam Laplace

Alih ragam Laplace balik.

Penerapan alih ragam Laplace dalam berbagai study kasus

Definisi dan evaluasi alih ragam - Z.

Teorema alih ragam - Z.

Penerapan alih ragam Z dalam berbagai study kasus

Alih ragam -Z balik.

Penerapan alih ragam -Z.

UTS

Syllabus (cont) Deret Fourier untuk sinyal periodik.

Karakteristik alih ragam Fourier

Perbandingan alih ragam Laplace dengan alih ragam Fourier.

Penerapan alih ragam Fourier dalam berbagai study kasus

Deret Fourier untuk sinyal periodik diskret.

Teorema pencuplikan dan masalah aliasing dan leakage.

Alih ragam Fourier diskret (DFT/ =Discrete Fourier Transform).

Karakteristik DFT

Membahas Fast Fourier Transform.

Membahas penerapan DFT dan FFT dalam penyelesaian masalah-masalah pengolahan sinyal digital.

14 Mei 2015 QUIZ 2

21 Mei 2015 Konsep dari keadaan (state).

Persamaan keadaan sistem kontinu.

28 Mei 2015 Persamaan keadaan sistem diskret.

Penerapan metode state space

Books to read Oppenheim, A.V., A.S. Willsky dan I.T. Young., 1983, Signals and Systems, Prentice-Hall,,

Englewoods Cliffs, new Jersey

Hsu, Hwei P., 1995, Schaum's outline of theory and problems of signals and systems,

McGraw-Hill Companies, Inc, United States of America

INTRODUCTION TO SIGNALS & SYSTEMS

(LECTURE 01)

What is a Signal ?

signals A Signal is the function of one or more independent

variables that carries some information to represent a

physical phenomenon.

Example : in a RC circuit the signal may represent the

voltage across the capacitor or the current flowing in the

Resistor.

Signal Examples Acoustic signals

– Acoustic pressure (sound) over time

Mechanical signals

– Velocity of a car over time

Video signals

– Intensity level of a pixel (camera, video) over time

Continuous-time Signal

• A continuous-time signal, also called an analog

signal, is defined along a continuum of time.

𝑥 𝑡 𝑑𝑡𝑡2

𝑡1

A discrete-time signal is defined at discrete

times.

Discrete time Signal

𝑥[𝑛]

𝑛2

𝑛=𝑛1

Elementary Signals

Sinusoidal & Exponential Signals • Sinusoids and exponentials are important in signal

and system analysis because they arise naturally in the solutions of the differential equations.

• Sinusoidal Signals can expressed in either of two ways :

cyclic frequency form- 𝐴 sin 2𝜋𝑓𝑜𝑡 = 𝐴 sin (2𝜋

𝑇𝑜

)𝑡

radian frequency form- 𝐴 sin 𝜔𝑜𝑡 𝜔𝑜 = 2𝜋𝑓𝑜 = 2𝜋/𝑇𝑜 To = Time Period of the Sinusoidal Wave

Signal’s Energy and Power For an instance, v(t) and i(t) are voltage and

current to a resistance R, then the Power of a

signal in a certain t time is :

𝑝 𝑡 = 𝑣 𝑡 𝑖 𝑡 =1

𝑅𝑣2(𝑡) (1.1)

So the total Energy for time interval 𝑡1 ≤ 𝑡 ≤𝑡2 is

𝑝 𝑡 𝑑𝑡 = 1

𝑅𝑣2 𝑡 𝑑𝑡

𝑡2

𝑡1

𝑡2

𝑡1 (1.2)

𝑥 𝑡 = 𝐴 sin 2П𝑓𝑜𝑡 + 𝜃 = 𝐴 sin (𝜔𝑜𝑡 + 𝜃)

𝑥(𝑡) = 𝐴𝑒𝑎𝑡 Real Exponential

= 𝐴𝑒𝑗𝜔 ̥𝑡 Complex Exponential

= 𝐴[cos (𝜔𝑜𝑡) + 𝑗 sin (𝜔𝑜𝑡)] θ = Phase of sinusoidal wave A = amplitude of a sinusoidal or exponential signal fo = fundamental cyclic frequency of sinusoidal signal ωo = radian frequency

Sinusoidal & Exponential Signals Contd.

Sinusoidal signal

Signal’s Energy and Power (with complex value)

the total Energy for a Continuous-time Signal with

Complex Properties ranging 𝑡1 ≤ 𝑡 ≤ 𝑡2 is

|𝑥 𝑡 |2𝑑𝑡𝑡2

𝑡1 (1.3)

the total Energy for a Discrete-time Signal with

Complex Properties ranging 𝑛1 ≤ 𝑛 ≤ 𝑛2 is

|𝑥 𝑛 |2𝑛2𝑛=𝑛1 (1.3)

Unit Step Function

1 , 0

u 1/ 2 , 0

0 , 0

t

t t

t

Precise Graph Commonly-Used Graph

Signum Function

1 , 0

sgn 0 , 0 2u 1

1 , 0

t

t t t

t

Precise Graph Commonly-Used Graph

The signum function, is closely related to the unit-step

function.

Unit Ramp Function

, 0

ramp u u0 , 0

tt tt d t t

t

•The unit ramp function is the integral of the unit step function.

•It is called the unit ramp function because for positive t, its

slope is one amplitude unit per time.

Rectangular Pulse or Gate Function

Rectangular pulse, 1/ , / 2

0 , / 2a

a t at

t a

Unit Impulse Function

As approaches zero, g approaches a unit

step andg approaches a unit impulse

a t

t

So unit impulse function is the derivative of the unit step

function or unit step is the integral of the unit impulse function

Functions that approach unit step and unit impulse

Representation of Impulse Function

The area under an impulse is called its strength or weight. It is

represented graphically by a vertical arrow. An impulse with a

strength of one is called a unit impulse.

Properties of the Impulse Function

0 0g gt t t dt t

The Sampling Property

0 0

1a t t t t

a

The Scaling Property

The Replication Property

g(t)⊗ δ(t) = g (t)

Unit Impulse Train

The unit impulse train is a sum of infinitely uniformly-

spaced impulses and is given by

, an integerT

n

t t nT n

The Unit Rectangle Function

The unit rectangle or gate signal can be represented as combination

of two shifted unit step signals as shown

The Unit Triangle Function

A triangular pulse whose height and area are both one but its base

width is not, is called unit triangle function. The unit triangle is

related to the unit rectangle through an operation called

convolution.

Sinc Function

sin

sinct

tt

Discrete-Time Signals

• Sampling is the acquisition of the values of a continuous-time signal at discrete points in time

• x(t) is a continuous-time signal, x[n] is a discrete-time signal

x x where is the time between sampless sn nT T

Discrete Time Exponential and Sinusoidal Signals

DT signals can be defined in a manner analogous to their continuous-time counter part

𝑥[𝑛] = 𝐴 sin (2𝜋𝑛/𝑁𝑜 + 𝜃) = 𝐴 sin (2𝜋𝐹𝑜𝑛 + 𝜃) 𝑥[𝑛] = 𝑒

𝑛 n = the discrete time A = amplitude θ = phase shifting radians, No = Discrete Period of the wave 1/N0 = Fo = Ωo/2𝜋 = Discrete Frequency

Discrete Time Sinusoidal Signal

Discrete Time Exponential Signal

Discrete Time Sinusoidal Signals

Discrete Time Unit Step Function or Unit Sequence Function

1 , 0

u0 , 0

nn

n

Discrete Time Unit Ramp Function

, 0

ramp u 10 , 0

n

m

n nn m

n

Discrete Time Unit Impulse Function or Unit Pulse Sequence

1 , 0

0 , 0

nn

n

for any non-zero, finite integer .n an a

Unit Pulse Sequence Contd.

The discrete-time unit impulse is a function in the ordinary sense in contrast with the continuous-time unit impulse.

It has a sampling property.

It has no scaling property i.e.

δ[n]= δ[an] for any non-zero finite integer ‘a’

Operations of Signals

Sometime a given mathematical function may completely describe a signal .

Different operations are required for different purposes of arbitrary signals.

The operations on signals can be Time Shifting Time Scaling Time Inversion or Time Folding

Time Shifting

The original signal x(t) is shifted by an amount tₒ.

X(t)X(t-to) Signal Delayed Shift to the right

Time Shifting Contd.

X(t)X(t+to) Signal Advanced Shift to the left

Time Scaling

For the given function x(t), x(at) is the time scaled version of x(t)

For a ˃ 1,period of function x(t) reduces and function speeds up. Graph of the function shrinks.

For a ˂ 1, the period of the x(t) increases and the function slows down. Graph of the function expands.

Time scaling Contd.

Example: Given x(t) and we are to find y(t) = x(2t).

The period of x(t) is 2 and the period of y(t) is 1,

Time scaling Contd.

Given y(t),

find w(t) = y(3t)

and v(t) = y(t/3).

Time Reversal

Time reversal is also called time folding

In Time reversal signal is reversed with respect to time i.e.

y(t) = x(-t) is obtained for the given function

Time reversal Contd.

0 0 , an integern n n n Time shifting

Operations of Discrete Time Functions

Operations of Discrete Functions Contd. Scaling; Signal Compression

n Kn K an integer > 1

KULSUM KUis keciL SepulUh Menit

1. 3 cos 𝜔0𝑡 𝛿 𝑡 𝑑𝑡 =?10

−5 dengan 𝑇0 =

1

2

2. Gambarkan x t + 2 dan 𝑥1

2𝑡 untuk

𝑥 𝑡 = (𝑡 + 2)(𝑢 𝑡 − 𝑢(𝑡 − 4)) 3. Buktikan bahwa 𝑥 𝑡 = cos (𝑡) adalah sinyal genap dengan penjumlahan