Chapter 2

1
Chapter 2
Introduction to
Signals and systems

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
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
Continuous-time and discrete-time
periodic signals

5
Continuous-time and discrete-time
aperiodic signals

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
Analog and Digital Signals
0 1 1 1 1 0 1

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.

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.

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
=
=

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

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

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

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
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
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
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
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
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
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
∞
=
= +∑

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

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

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

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

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=

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

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

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

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

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

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
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
− + −
− − −

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

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

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
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
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
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
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
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
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

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

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

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

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
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
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
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
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
• 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
-5 0 5
0
0.5
1
1.5
Amplitude Spectrum
-5 0 5
-100
0
100 Phase spectrum

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

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

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
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
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
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

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