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.

9

A deterministic signal

10

Deterministic signal

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

A random signal

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

00

: period(second)

1( ), fundamental frequency

2 (rad/sec), angulr(radian) frequency

T

f HzT

f

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 (t+ ), for - ∞ < t < ∞

• This signal is completely characterized by three parameters:

A: is the amplitude (peak value) of x(t).

is the radial frequency in (rad/s),

: 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(3 t+/6),

b) x2(t)= 2 sin(100 t),

c) x3(t)= x1(t)+ x2(t)

d) x4(t)= 3 cos(3 t+/6) + 2 sin(30 t),

e) x5(t)= 2 exp(-j 20 t)

19

Power and Energy signals

• A signal with finite energy is an energy signal

• A signal with finite power is a power signal

dttgEg2

)(

2/

2/

2)(

1lim

T

TT

g dttgT

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

/ 22

0 / 2

1( )

T

x

T

P x t dtT

21

Example• Determine whether the signal g(t) is power or

energy signals or neither

0 2 4 6 80

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)

)(t( ) cos(10 ) ( )v t t u t( ) sin 2 [ ( ) ( 2 )]w t t u t u t

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 t

1 1 1 2( ) cos( ) cos( )x t a t b t

1

( ) cos( )n n nn

y t c t

24-1 0 1

-2

-1

0

1

2

Time (s)

Sawtooth signalDetermine 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, (t), is defined by

0,0

0,)(

t

tt 1)(

dttand

27

0

28

• Multiplication of a function by (t)

• We can also prove that

)0()()( sdttts

)()0()()( tgttg )()()()( tgttg

)()()( sdttts

29

Unit step function

• The unit step function u(t) is

• u(t) is related to (t) by

0,0

0,1)(

t

ttu

t

dtu )()( )(tdt

du

30

Unit step

31

Rectangular function

• A single rectangular pulse is denoted by

2/,0

2/,5.0

2/,1

t

t

tt

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

t

tt

t

34

• Sinc function

• Sampling function

sin( )sinc( )

xx

x

( ) ( ), : samplig intervalsT s s

n

t t nT T

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

( )g t

ta

ta

( ), 1 is compression

( ), 1 is expansion

g( ), 1 is expansion

g( ), 1 is compression

g at a

g at a

a

a

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

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 50

0.5

1

1.5

2

-5 -4 -3 -2 -1 0 1 2 3 4 50

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 dtE

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

ii

x t g t

( ), 1,....ig t i n

1i

n

x gi

p p

45

Time and Frequency Domainsrepresentations 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(250t)+sin(2 120t);

y(t) = x(t) + noise;

• Selecting frequency bands in Telecommunication system

470 10 20 30 40 50-5

0

5Signal Corrupted with Zero-Mean Random Noise

Time (seconds)

480 200 400 600 800 10000

20

40

60

80Frequency 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

00

2( ) cos(2 )n

T

a x t n f t dtT

0 0 01

( ) cos 2 sin 2n nn

x t a a n f t b n f t

0

00

2( ) sin(2 )n

T

b x t n f t dtT

52

Trigonometric Fourier Seriescont.

0

00

1( )

T

a x t dtT

53

Compact trigonometric Fourier series

0 01

2 20 0

1

( ) cos(2 )

,

tan

n nk

n n n

nn

n

x t c c n f t

c a b c a

b

a

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 tn

n

x t D e

0

0

2

0

1( ) j n f t

n

T

D x t e dtT

nD

55

Complex Fourier Series cont.

1

21

2

n

n

n n

jn n

jn n

j jn n n n

D c e

D c e

D D e and D D e

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.

n

57

Example• Find the exponential Fourier series and sketch

the corresponding spectra for the sawtooth signal with period 2

-10 -5 0 5 100

0.5

1

1.5

2

58

• Dn= j/( n); for n0

• D0= 1;

02

0

1( )

o

j n f tn

T

D x t e dtT

12

taa

edtet

tata

59

-5 0 50

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

dttxT

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 22 2

010 / 2

2 220

1 1

1( )

2

2 2

nn

T

n

nT

n n

n n

D

cx t dt c

T

a ba

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 2 and defined as

( ) 5 ,0 2tf t e t

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 nT

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

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