1
Chapter 2
Introduction to Signals and systems
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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
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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.
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Continuous-time and discrete-time periodic signals
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Continuous-time and discrete-time aperiodic signals
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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
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Analog and Digital Signals
0 1 1 1 1 0 1
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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.
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A deterministic signal
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Deterministic signal
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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.
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A random signal
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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
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• 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)
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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)
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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
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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
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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)
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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
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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)
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Some Useful Functions
• Unit impulse function • Unit step function • Rectangular function • Triangular function• Sampling function• Sinc function• Sinusoidal, exponential and logarithmic
functions
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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
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0
28
• Multiplication of a function by (t)
• We can also prove that
)0()()( sdttts
)()0()()( tgttg )()()()( tgttg
)()()( sdttts
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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
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Unit step
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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
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Triangular function
• A triangular function is denoted by
2
1,0
2
1,21
t
tt
t
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• 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
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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
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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)
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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)
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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)
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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
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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
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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
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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.
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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)
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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.
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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
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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
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Trigonometric Fourier Seriescont.
0
00
1( )
T
a x t dtT
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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
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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
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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
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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
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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
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• 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
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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
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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
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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
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Example 2
• Plot the spectra of x(t) if T1= T/4
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Example 3
• Plot the spectra of x(t).
0( ) ( )n
x t t nT
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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
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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
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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
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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