This document provides an introduction to signals and systems. It begins by classifying different types of signals as continuous-time/discrete-time, analog/digital, deterministic/random, periodic/aperiodic, power/energy. It then discusses representations of signals in the time and frequency domains, including the Fourier series representation of periodic signals. Key concepts covered include the unit step, rectangular, triangular and sinc functions, as well as signal operations like time shifting, scaling and inversion. The document concludes by introducing Parseval's theorem relating the power of a signal to the power of its Fourier coefficients.

1. 1
Chapter 2
Introduction to
Signals and systems

2. 2
Outlines
• Classification of signals and systems
• Some useful signal operations
• Some useful signals.
• Frequency domain representation for
periodic signals
• Fourier Series Coefficients
• Power content of a periodic signal and
Parseval’ s theorem for the Fourier series

3. 3
Classification of Signals
• Continuous-time and discrete-time signals
• Analog and digital signals
• Deterministic and random signals
• Periodic and aperiodic signals
• Power and energy signals
• Causal and non-causal.
• Time-limited and band-limited.
• Base-band and band-pass.
• Wide-band and narrow-band.

4. 4
Continuous-time and discrete-time
periodic signals

5. 5
Continuous-time and discrete-time
aperiodic signals

6. 6
Analog & digital signals
• If a continuous-time signal can take on
any values in a continuous time interval, then
is called an analog signal.
• If a discrete-time signal can take on only a finite
number of distinct values, { }then the signal
is called a digital signal.
)(tg
)(tg
( )g n

7. 7
Analog and Digital Signals
0 1 1 1 1 0 1

8. 8
Deterministic signal
• A Deterministic signal is uniquely
described by a mathematical expression.
• They are reproducible, predictable and
well-behaved mathematically.
• Thus, everything is known about the signal
for all time.

9. 9
A deterministic signal

10. 10
Deterministic signal

11. 11
Random signal
• Random signals are unpredictable.
• They are generated by systems that
contain randomness.
• At any particular time, the signal is a
random variable, which may have well
defined average and variance, but is not
completely defined in value.

12. 12
A random signal

13. 13
Periodic and aperiodic Signals
• A signal is a periodic signal if
• Otherwise, it is aperiodic signal.
0( ) ( ), , is integer.x t x t nT t n= + ∀
( )x t
0
0
0
: period(second)
1
( ),fundamental frequency
2 (rad/sec), angulr(radian) frequency
T
f Hz
T
fω π
=
=

14. 14-2 0 2
-3
-2
-1
0
1
2
3
Time (s)
Square signal

15. 15
-1 0 1
-2
-1
0
1
2
Time (s)
Square signal

16. 16-1 0 1
-2
-1
0
1
2
Time (s)
Sawtooth signal

17. 17
• A simple harmonic oscillation is mathematically
described by
x(t)= A cos (ω t+ θ), for - ∞ < t < ∞
• This signal is completely characterized by three
parameters:
A: is the amplitude (peak value) of x(t).
ω: is the radial frequency in (rad/s),
θ: is the phase in radians (rad)

18. 18
Example:
Determine whether the following signals are
periodic. In case a signal is periodic,
specify its fundamental period.
a) x1(t)= 3 cos(3π t+π/6),
b) x2(t)= 2 sin(100π t),
c) x3(t)= x1(t)+ x2(t)
d) x4(t)= 3 cos(3π t+π/6) + 2 sin(30π t),
e) x5(t)= 2 exp(-j 20 π t)

19. 19
Power and Energy signals
• A signal with finite energy is an energy signal
• A signal with finite power is a power signal
∞<= ∫
+∞
∞−
dttgEg
2
)(
∞<= ∫
+
−
∞→
2/
2/
2
)(
1
lim
T
T
T
g dttg
T
P

20. 20
Power of a Periodic Signal
• The power of a periodic signal x(t) with period
T0 is defined as the mean- square value over
a period
0
0
/2
2
0 /2
1
( )
T
x
T
P x t dt
T
+
−
= ∫

21. 21
Example
• Determine whether the signal g(t) is power or
energy signals or neither
0 2 4 6 8
0
1
2
g(t)
2 exp(-t/2)

22. 22
Exercise
• Determine whether the signals are power or
energy signals or neither
1) x(t)= u(t)
2) y(t)= A sin t
3) s(t)= t u(t)
4)z(t)=
5)
6)
)(tδ
( ) cos(10 ) ( )v t t u tπ=
( ) sin 2 [ ( ) ( 2 )]w t t u t u tπ π= − −

23. 23
Exercise
• Determine whether the signals are power or energy
signals or neither
1)
2)
3)
1 1 2 2( ) cos( ) cos( )x t a t b tω θ ω θ= + + +
1 1 1 2( ) cos( ) cos( )x t a t b tω θ ω θ= + + +
1
( ) cos( )n n n
n
y t c tω θ
∞
=
= +∑

24. 24-1 0 1
-2
-1
0
1
2
Time (s)
Sawtooth signal
Determine the suitable measures for the signal x(t)

25. 25
Some Useful Functions
• Unit impulse function
• Unit step function
• Rectangular function
• Triangular function
• Sampling function
• Sinc function
• Sinusoidal, exponential and logarithmic
functions

26. 26
Unit impulse function
• The unit impulse function, also known as the
dirac delta function, δ(t), is defined by



≠
=∞
=
0,0
0,
)(
t
t
tδ 1)( =∫
+∞
∞−
dttand δ

27. 27
∆0

28. 28
• Multiplication of a function by δ(t)
• We can also prove that
)0()()( sdttts =∫
+∞
∞−
δ
)()0()()( tgttg δδ =
)()()()( τδττδ −=− tgttg
)()()( ττδ sdttts =−∫
+∞
∞−

29. 29
Unit step function
• The unit step function u(t) is
• u(t) is related to δ(t) by



<
≥
=
0,0
0,1
)(
t
t
tu
∫∞−
=
t
dtu ττδ )()( )(t
dt
du
δ=

30. 30
Unit step

31. 31
Rectangular function
• A single rectangular pulse is denoted by





>
=
<
=





2/,0
2/,5.0
2/,1
τ
τ
τ
τ
t
t
t
t
rect

32. 32-3 -2 -1 0 1 2 3
0
0.5
1
1.5
2
2.5
3
Time (s)
Rectangular signal

33. 33
Triangular function
• A triangular function is denoted by







>
<−
=





∆
2
1
,0
2
1
,21
τ
ττ
τ t
tt
t

34. 34
• Sinc function
• Sampling function
sin( )
sinc( )
x
x
x
π
π
=
( ) ( ), :samplig intervalsT s s
n
t t nT Tδ δ
∞
=−∞
= −∑

35. 35
-5 0 5
-0.5
0
0.5
1
1.5
2
2.5
3
Time (s)
Sinc signal

36. 36
Some Useful Signal Operations
• Time shifting
(shift right or delay)
(shift left or advance)
• Time scaling
( )g t τ−
( )g t τ+
t
a
t
a
( ), 1 is compression
( ), 1 is expansion
g( ), 1 is expansion
g( ), 1 is compression
g at a
g at a
a
a
f
p
f
p

37. 37
Signal operations cont.
• Time inversion
( ) : mirror image of ( ) about Y-axisg t g t−
( ) : shift right of ( )
( ) :shift left of ( )
g t g t
g t g t
τ
τ
− + −
− − −

38. 38-10 -5 0 5
0
1
2
3
Time (s)
g(t)
g(t-5)
g(t)
g(t-5)
g(t)
g(t-5)

39. 39-10 -5 0 5
0
1
2
3
Time (s)
g(t+5)

40. 40
Scaling
-5 0 5
0
2
4
-5 0 5
0
2
4
-5 0 5
0
2
4
g(t)
g(2t)
g(t/2)

41. 41
Time Inversion
-5 -4 -3 -2 -1 0 1 2 3 4 5
0
0.5
1
1.5
2
-5 -4 -3 -2 -1 0 1 2 3 4 5
0
0.5
1
1.5
2
g(t)
g(-t)

42. 42
Inner product of signals
• Inner product of two complex signals x(t), y(t) over
the interval [t1,t2] is
If inner product=0, x(t), y(t) are orthogonal.
2
1
( ( ), ( )) ( ) ( )
t
t
x t y t x t y t dt∗
= ∫

43. 43
Inner product cont.
• The approximation of x(t) by y(t) over the interval
is given by
• The optimum value of the constant C that minimize
the energy of the error signal
is given by
( ) ( ) ( )e t x t cy t= −
2
1
1
( ) ( )
t
y t
C x t y t dt
E
= ∫
1 2[ , ]t t
( ) ( )x t cy t=

44. 44
Power and energy of orthogonal
signals
• The power/energy of the sum of mutually
orthogonal signals is sum of their individual
powers/energies. i.e if
Such that are mutually orthogonal,
then
1
( ) ( )
n
i
i
x t g t
=
= ∑
( ), 1,....ig t i n=
1
i
n
x g
i
p p
=
= ∑

45. 45
Time and Frequency Domains
representations of signals
• Time domain: an oscilloscope displays the
amplitude versus time
• Frequency domain: a spectrum analyzer
displays the amplitude or power versus
frequency
• Frequency-domain display provides
information on bandwidth and harmonic
components of a signal.

46. 46
Benefit of Frequency Domain
Representation
• Distinguishing a signal from noise
x(t) = sin(2π 50t)+sin(2π 120t);
y(t) = x(t) + noise;
• Selecting frequency bands in
Telecommunication system

47. 470 10 20 30 40 50
-5
0
5
Signal Corrupted with Zero-Mean Random Noise
Time (seconds)

48. 480 200 400 600 800 1000
0
20
40
60
80
Frequency content of y
Frequency (Hz)

49. 49
Fourier Series Coefficients
• The frequency domain representation of a
periodic signal is obtained from the
Fourier series expansion.
• The frequency domain representation of a
non-periodic signal is obtained from the
Fourier transform.

50. 50
• The Fourier series is an effective technique for
describing periodic functions. It provides a
method for expressing a periodic function as a
linear combination of sinusoidal functions.
• Trigonometric Fourier Series
• Compact trigonometric Fourier Series
• Complex Fourier Series

51. 51
Trigonometric Fourier Series
0
0
0
2
( ) cos(2 )n
T
a x t nf t dt
T
π= ∫
( )0 0 0
1
( ) cos2 sin 2n n
n
x t a a nf t b nf tπ π
∞
=
= + +∑
0
0
0
2
( ) sin(2 )n
T
b x t nf t dt
T
π= ∫

52. 52
Trigonometric Fourier Series
cont.
0
0
0
1
( )
T
a x t dt
T
= ∫

53. 53
Compact trigonometric Fourier
series
0 0
1
2 2
0 0
1
( ) cos(2 )
,
tan
n n
k
n n n
n
n
n
x t c c nf t
c a b c a
b
a
π θ
θ
∞
=
−
= + +
= + =
 −
=  ÷
 
∑

54. 54
Complex Fourier Series
• If x(t) is a periodic signal with a
fundamental period T0=1/f0
• are called the Fourier coefficients
2
( ) oj n f t
n
n
x t D e π
∞
=−∞
= ∑
0
0
2
0
1
( ) j n f t
n
T
D x t e dt
T
π−
= ∫
nD

55. 55
Complex Fourier Series cont.
1
2
1
2
n
n
n n
j
n n
j
n n
j j
n n n n
D c e
D c e
D D e and D D e
θ
θ
θ θ
−
−
−
−
=
=
= =

56. 56
Frequency Spectra
• A plot of |Dn| versus the frequency is called the
amplitude spectrum of x(t).
• A plot of the phase versus the frequency is
called the phase spectrum of x(t).
• The frequency spectra of x(t) refers to the
amplitude spectrum and phase spectrum.
nθ

57. 57
Example
• Find the exponential Fourier series and sketch
the corresponding spectra for the sawtooth
signal with period 2 π
-10 -5 0 5 10
0
0.5
1
1.5
2

58. 58
• Dn= j/(π n); for n≠0
• D0= 1;
02
0
1
( )
o
j n f t
n
T
D x t e dt
T
π−
= ∫
( )12
−=∫ ta
a
e
dtet
ta
ta

59. 59
-5 0 5
0
0.5
1
1.5
Amplitude Spectrum
-5 0 5
-100
0
100 Phase spectrum

60. 60
Power Content of a Periodic Signal
• The power content of a periodic signal x(t)
with period T0 is defined as the mean- square
value over a period
∫
+
−
=
2/
2/
2
0
0
0
)(
1
T
T
dttx
T
P

61. 61
Parseval’s Power Theorem
• Parseval’ s power theorem series states that
if x(t) is a periodic signal with period T0, then
0
0
2
/ 2 2
2 2
0
10 / 2
2 2
2
0
1 1
1
( )
2
2 2
n
n
T
n
nT
n n
n n
D
c
x t dt c
T
a b
a
∞
=−∞
+ ∞
=−
∞ ∞
= =




= +


+ +

∑
∑∫
∑ ∑

62. 62
Example 1
• Compute the complex Fourier series coefficients for
the first ten positive harmonic frequencies of the
periodic signal f(t) which has a period of 2π and
defined as
( ) 5 ,0 2t
f t e t π−
= ≤ ≤

63. 63
Example 2
• Plot the spectra of x(t) if T1= T/4

64. 64
Example 3
• Plot the spectra of x(t).
0( ) ( )
n
x t t nTδ
∞
=−∞
= −∑

65. 65
Classification of systems
• Linear and non-linear:
-linear :if system i/o satisfies the superposition
principle. i.e.
1 2 1 2
1 1
2 2
[ ( ) ( )] ( ) ( )
where ( ) [ ( )]
and ( ) [ ( )]
F ax t bx t ay t by t
y t F x t
y t F x t
+ = +
=
=

66. 66
Classification of sys. Cont.
• Time-shift invariant and time varying
-invariant: delay i/p by the o/p delayed by same a
mount. i.e
0 0
if ( ) [ ( )]
then ( ) [ ( )]
y t F x t
y t t F x t t
=
− = −
0t

67. 67
Classification of sys. Cont.
• Causal and non-causal system
-causal: if the o/p at t=t0 only depends on the present
and previous values of the i/p. i.e
LTI system is causal if its impulse response is causal.
i.e.
0 0( ) [ ( ), ]y t F x t t t= ≤
( ) 0, 0h t t= ∀ p

68. 68
Suggested problems
• 2.1.1,2.1.2,2.1.4,2.1.8
• 2.3.1,2.3.3,2.3.4
• 2.4.2,2.4.3
• 2.5.2, 2.5.5
• 2.8.1,2.8.4,2.8.5
• 2.9.2,2.9.3