aydin adnan menderes university faculty of engineering

AYDIN ADNAN MENDERES UNIVERSITY

FACULTY OF ENGINEERINGDepartment of Electrical and Electronics Engineering

EE213 – TRANSFORM TECHNIQUESWITH COMPUTER APPLICATIONS

2020-2021, Fall(ONLINE)

Week 6

Dr. Adem Ükte

Basic Concepts in Signals & Systems

EE213 – Transform Techniques With Computer Applications Dr. Adem Ükte

A signal is a set of information or data that can be modeled as a function of one or more variables (generally time).

Ex: Speech, image, voltage in a circuit, sequence of daily stock prices in the financial market, …

What is a signal?

Continuous time (CT) signals

A signal that is specified for every real value of time

Time is continuous, that is it takes any value on the real axis

x(t)

t: Continuous time, sec

Discrete time (DT) signals

A signal that is specified only for discrete values of time

Time is discrete, that is it takes values at equally spaced intervals along the time axis (integer numbers)

x[n]

n: Discrete time, sample

Transformation of Signals


Time Reversal of CT Signal Time Reversal of DT Signal



Time Scaling in CT Signals, x(at) Time Scaling in DT Signals, x[an]

if |a|>1 → Compression

if |a|<1 → Expansion



Time Shifting in CT Signals, x(t-t0) Time Shifting in DT Signals, x[n-n0]

if t0>0 → x(t) is delayed in time, shifts to right

if t0<0 → x(t) is advanced in time, shifts to left

Even and Odd Signals


Periodic Signals in CT


x(t) is periodic with period T>0 if it satisfies x(t)=x(t+T)

The minimum value of T that satisfiesx(t)=x(t+T) is called fundamental period and denoted as T0

Fundamental frequency: f0=1/T0 Hertz (cycle/second) ω0=2π/T0 (radians/second)



If x1(t) is periodic with period T1 and x2(t) is periodic with period T2, then the sum of the two signals x1(t)+ x2(t) is periodicwith period equal to least common multiple (LCM) of T1 and T2, i.e. LCM(T1,T2)

Ex: 𝑥 𝑡 = 2 cos(𝜋

2𝑡) − sin(

𝜋

3𝑡)

𝜔1 =𝜋

2

𝑇1 =2𝜋

𝜔31= 4

𝑇 = lcm 𝑇1, 𝑇2 = lcm 4,6 = 12 sec. So 𝑥 𝑡 is periodic with 𝑇 = 12 seconds.

𝜔2 =𝜋

3

𝑇2 =2𝜋

𝜔32= 6



t=linspace(0,36,1000);

x1=2*cos(pi/2*t);

x2=-sin(pi/3*t);

x=x1+x2;

subplot(311),plot(t,x1),title('First Periodic Sinusoidal')

xlabel('t'),ylabel('2cos(t\pi/2)'),grid on,axis tight

subplot(312),plot(t,x2),title('Second Periodic Sinusoidal')

xlabel('t'),ylabel('-sin(t\pi/3)'),grid on,axis tight

subplot(313),plot(t,x),title('Sum of Periodic Sinusoidals')

xlabel('t'),ylabel('2cos(t\pi/2)-sin(t\pi/3)'),grid on,axis tight

Periodic Signals in DT


𝑥 𝑛 = 𝑥 𝑛 + 𝑁

Let 𝑥 𝑛 = 𝐶𝑒𝑗Ω0𝑛

Then 𝐶𝑒𝑗Ω0𝑛 = 𝐶𝑒𝑗Ω0 𝑛+𝑁 = 𝐶𝑒𝑗Ω0𝑛𝑒𝑗Ω0𝑁

𝑒𝑗Ω0𝑁 = 1

Ω0𝑁 = 2𝑘𝜋 where 𝑘:integer

Ω0

2𝜋=

𝑘

𝑁must be a rational number

Ex: 𝑥 𝑛 = sin(5𝜋𝑛) , 𝑁 =?

Ω0𝑁 = 2𝑘𝜋 5𝜋𝑛 = 2𝑘𝜋 𝑘

𝑁=

5

2𝑁 = 2



Ex: 𝑥 𝑛 = sin(5𝜋𝑛) + cos(5𝜋

6𝑛)

Ω12𝜋

=𝑘1𝑁1

5

2=𝑘1𝑁1

𝑁 = lcm 𝑁1, 𝑁2 = lcm 2,12 = 12 samples. So𝑥 𝑛 is periodic with 𝑁 = 12 samples.

Ω2

2𝜋=𝑘2𝑁2

5

12=𝑘2𝑁2

𝑁1 = 2 𝑁2 = 12



n=-40:40;

x=sin(5*pi*n)+cos(5*pi/6*n);

stem(n,x),title('x[n]=sin(5\pin)+cos(5\pin/6)')

xlabel('n'),ylabel('x[n]'),grid on

Practice 8:


A discrete time signal 𝑥 𝑛 is defined as,

Generate and plot the 𝑥 𝑛 and 𝑥 𝑛 − 4 signals using ‘stem’ function.

END OF WEEK 6


aydin adnan menderes university faculty of engineering

Documents