signal and system

1. Dr. Alam – Updated by Mr. Asad – EE 207 – Semester 122 Page 1
Lecture 2
Signals and Systems Introduction
The material covered in this class will be as follows:
 Detailed analysis of sinusoidal signal
Sinusoidal Signal
In continuous-time domain, it is represented as
( ) cos( )ox t A t  
where A is the amplitude, o is the frequency in rad/sec,  is the phase angle in
radians. The period of the signal is
2
o
T


 .
Phasor representation
From Euler’s identity
cos sin
cos ( ) sin ( )
j
j j
e j
or e and e

 
 
 
 
   
Thus,
( )
( ) { } { } { }o o oj t j t j tj
x t Ae Ae e Xe   
     
where,
j
X Ae A
   is called the phasor. It represents magnitude and phase
of x(t).
The complex signal 𝑥̃(𝑡) and its conjugate 𝑥̃∗( 𝑡) can be written as
𝑥̃( 𝑡) = 𝐴𝑒𝑗( 𝜔 𝑜 𝑡+𝜃)
= 𝑋⃑ 𝑒𝑗 𝜔 𝑜 𝑡
= 𝐴[cos(𝜔 𝑜 𝑡 + 𝜃) + 𝑗 sin(𝜔 𝑜 𝑡 + 𝜃)]
𝑥̃∗( 𝑡) = 𝐴𝑒−𝑗( 𝜔 𝑜 𝑡+𝜃)
= 𝑋⃑ 𝑒−𝑗𝜔 𝑜 𝑡
= 𝐴[cos(𝜔 𝑜 𝑡 + 𝜃) − 𝑗 sin(𝜔 𝑜 𝑡 + 𝜃)]
Where 𝑋⃑ 𝑒𝑗 𝜔 𝑜 𝑡
is referred as rotating phasor.

2. Dr. Alam – Updated by Mr. Asad – EE 207 – Semester 122 Page 2
We conclude that the cosine signal x(t) can be expressed as one-half of the sum of
the complex signal ( )x t with positive frequency o and its conjugate
*
( )x t with
negative frequency -o, i.e.
𝑥( 𝑡) =
1
2
[ 𝑥̃( 𝑡) + 𝑥̃∗
(𝑡)]
Frequency Domain Spectra
An alternative way to visualize the sinusoidal signal 𝑥(𝑡) in the frequency domain
is in the form of two plots. One the amplitude 𝐴 as the function of frequency 𝑓, and
the other its phase angle  as a function of 𝑓. These plots are referred to as single-
sided spectrum. If the amplitude and phase angle plots are made for the oppositely
rotating phasors we obtain the so called double-sided spectra as shown.
Note that if a signal is represented as a sine function, before finding the signal
spectra it must be expressed in terms of a cosine function,
sin( ) cos( )
2
o ot t

      

3. Dr. Alam – Updated by Mr. Asad – EE 207 – Semester 122 Page 3
Lecture 1-2 Practice Problems
Introduction
Practice problems :Examples 1-6, 1-7, 1-8 in text book.
1. Determine whether or not each of the following signals is periodic. If a signal is periodic,
determine its fundamental period.
(a) 𝑥( 𝑡) = 𝑐𝑜𝑠
𝜋
3
𝑡 + 𝑠𝑖𝑛
𝜋
4
𝑡 (b) 𝑥( 𝑡) = 𝑐𝑜𝑠𝑡 + 𝑠𝑖𝑛√2𝑡 (c) 𝑥( 𝑡) = sin2
𝑡
Answer
(a) 𝑥( 𝑡) = 𝑐𝑜𝑠
𝜋
3
𝑡 + 𝑠𝑖𝑛
𝜋
4
𝑡 = 𝑥1( 𝑡) + 𝑥2( 𝑡)
(b)
(c)
2. Determine whether the following signals are energy signals, power signals, or neither.
(b) x(t) is periodic with period 𝑇𝑜 =
2𝜋
𝜔 𝑜
.
The average power is

4. Dr. Alam – Updated by Mr. Asad – EE 207 – Semester 122 Page 4
The signal energies of three typical pulses shown are
3. Find the signal energy for the following signals

5. Dr. Alam – Updated by Mr. Asad – EE 207 – Semester 122 Page 5
4. Find the total energy and the average power of the periodic signal x(t) = sin t
The average power is
5. For the signal ( ) 6cos(10 ) 4sin(18 )
6 6
x t t t
 
    
(a) Find the period and the fundamental frequency of the signal
(b) Write the signal as the real part of the sum of the rotating phasors.
(c) Write x(t) as the sum of counter rotating phasors
(d) Plot the single-sided amplitude and phase spectra
(e) Plot the 2-sided amplitude and phase spectra
Answer:
(a)
Converting the second term to a cosine function, we have
1 2
1
1 2
1 2 2
( ) 6cos(10 ) 4cos(18 ) ( ) ( )
6 3
2 2 1 2 2 1 1/5 9
; ;
10 5 18 9 1/9 5
x t t t x t x t
T
T T Now rational number
T
 
 
   
   
     
        
Period of the composite signal x(t) is 1 2
1
5 9 9 1 sec;
9
oT T T    
1
1o
o
Fundamental frequency f Hz
T
 
(b) x(t) in terms of the real part of the rotating phasors is
(10 ) (18 )
6 3
( ) 6 4
j t j t
x t e e
 
     
     
   

6. Dr. Alam – Updated by Mr. Asad – EE 207 – Semester 122 Page 6
,f Hz
Amplitude
9
6
50
4
( )Phaseshift rad
,f Hz9
6

50
3


(a)Single-sidedspectra
(c) x(t) in terms of the counter rotating phasors is
(10 ) (10 ) (18 ) (18 )
6 6 3 3
( ) 3 3 2 2
j t j t j t j t
x t e e e e
   
        
   
,f Hz
Amplitude
9
3
50
2
( )Phaseshift rad
,f Hz9
6

50
3


9
3
5
2
6


59
3

(b)Double-sidedspectra
6. A continuous-time signal 𝑥(𝑡) is shown in Fig. 1-27. Sketch and label each of the following
signals.
(a)
(b)

7. Dr. Alam – Updated by Mr. Asad – EE 207 – Semester 122 Page 7
(c)
Details of Answer (b):
𝑥( 𝑡) = 𝑢(𝑡 + 1) − 𝑢(𝑡) + 3[𝑢(𝑡) − 𝑢(𝑡 − 1)]+ 2[𝑢(𝑡 − 1) − 𝑢(𝑡 − 2)] + [𝑢(𝑡 − 2) − 𝑢(𝑡
− 3)]

8. Dr. Alam – Updated by Mr. Asad – EE 207 – Semester 122 Page 8
Answer: