aydin adnan menderes university faculty of engineering
TRANSCRIPT
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
EE213 – Transform Techniques With Computer Applications Dr. Adem Ükte
Time Reversal of CT Signal Time Reversal of DT Signal
Transformation of Signals
EE213 – Transform Techniques With Computer Applications Dr. Adem Ükte
Time Scaling in CT Signals, x(at) Time Scaling in DT Signals, x[an]
if |a|>1 → Compression
if |a|<1 → Expansion
Transformation of Signals
EE213 – Transform Techniques With Computer Applications Dr. Adem Ükte
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
EE213 – Transform Techniques With Computer Applications Dr. Adem Ükte
Periodic Signals in CT
EE213 – Transform Techniques With Computer Applications Dr. Adem Ükte
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)
Periodic Signals in CT
EE213 – Transform Techniques With Computer Applications Dr. Adem Ükte
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
Periodic Signals in CT
EE213 – Transform Techniques With Computer Applications Dr. Adem Ükte
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 CT
EE213 – Transform Techniques With Computer Applications Dr. Adem Ükte
Periodic Signals in DT
EE213 – Transform Techniques With Computer Applications Dr. Adem Ükte
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
Periodic Signals in DT
EE213 – Transform Techniques With Computer Applications Dr. Adem Ükte
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
Periodic Signals in DT
EE213 – Transform Techniques With Computer Applications Dr. Adem Ükte
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:
EE213 – Transform Techniques With Computer Applications Dr. Adem Ükte
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
EE213 – Transform Techniques With Computer Applications Dr. Adem Ükte