Chapter 2

W
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
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
27
∆0
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=
30
Unit step
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
1 de 68

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Chapter 2

  • 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.
  • 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.
  • 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.
  • 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
  • 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=
  • 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 ∞ =−∞ = −∑
  • 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= ∫
  • 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

Notas do Editor

  1. Period= 1 sec
  2. In some other book, sinc(x)=sin(px)/px
  3. k in an integer
  4. Use Table of integral in Appendix D