Signals and Systems: Introduction
Prof. Miguel A. Muriel
Signals
g
and Systems:
y
Introduction
1 Introduction to Signals and Systems
12- Continuous-Time Signals
33- Discrete
Discrete-Time
Time Signals
4- Description of Systems
55- Time
Time-Domain
Domain System Analysis
6- Spectral Method
77- Fourier Transform
8- Sampling
99- Discrete Fourier Transform (DFT)
Miguel A. Muriel - Signals and Systems: Introduction - 2
1- Introduction to Signals
g
and Systems
y
Miguel A. Muriel - Signals and Systems: Introduction - 3
Introduction
-A signal is any physical phenomenon which conveys
i f
information.
i
-Systems respondd to signals
i l andd produce
d
new signals.
i l
-Excitation signals are applied at system inputs and response
signals are produced at system outputs.
Miguel A. Muriel - Signals and Systems: Introduction - 4
Signal
g types
yp
x(t)
Continuous-Time
Continuous-Value
Signal
x[n]
Discrete-Time
Continuous-Value
Signal
t
x(t)
Continuous-Time
Discrete-Value
Discrete
Value
Signal
n
x(t)
t
x(t)
Continuous-Time
Continuous-Value
C
ti
V l
Random Signal
Digital Signal
x(t)
t
Continuous-Time
Discrete-Value
Discrete
Value
Signal
t
Noisy Digital Signal
t
Miguel A. Muriel - Signals and Systems: Introduction - 5
Conversions Between Signal Types
Miguel A. Muriel - Signals and Systems: Introduction - 6
System
y
example
p
-A communication system has an information signal plus noise
signals.
g
Miguel A. Muriel - Signals and Systems: Introduction - 7
Voice Signal
g
Miguel A. Muriel - Signals and Systems: Introduction - 8
(a) 32 ms
(b) 256 samples
l
(a) Segment of a continuous-time speech signal x(t).
(b) Sequence
S
off samples
l x[n]=x(nT)
[ ] ( T) obtained
bt i d from
f
th signal
the
i l in
i partt (a)
( ) with
ith T =125
125 μs.
Miguel A. Muriel - Signals and Systems: Introduction - 9
2- Continuous-Time Signals
g
Miguel A. Muriel - Signals and Systems: Introduction - 10
-All
All continuous signals that are functions of time are
continuous-time , but not all continuous-time signals are
continuous functions of time.
Miguel A. Muriel - Signals and Systems: Introduction - 11
Impulse
p
 (t )
  t   0,
0 t0

   t  dt  1
1

 0

 
2 2
t
-The impulse
p
is not a function in the ordinaryy sense because its
value at the time of its occurrence is not defined.
-It is a functional.
Miguel A. Muriel - Signals and Systems: Introduction - 12
-It
It is
i represented
t d graphically
hi ll by
b a vertical
ti l arrow.
-Its
Its strength is either written beside it or is represented by its
length.
Miguel A. Muriel - Signals and Systems: Introduction - 13
Properties
p
of the Impulse
p
Sampling Property

 f  t    t  dt  f  0 


 f  t    t  t  dt  f  t 
0
0

-The sampling
p g property
p p y “extracts” the value of a function at a ppoint.
Miguel A. Muriel - Signals and Systems: Introduction - 14
Scaling Property
1
  a  t  t0      t  t0 
a
-This property illustrates that the impulse is different from
ordinary mathematical functions.
functions
Miguel A. Muriel - Signals and Systems: Introduction - 15
Periodic Impulse
p
T t  

   t  nT   n an integer 
n 
-The periodic impulse is a sum of infinitely many uniformlyspaced impulses.
impulses
Miguel A. Muriel - Signals and Systems: Introduction - 16
Miguel A. Muriel - Signals and Systems: Introduction - 17
Stepp
1 , t  0

u  t   1/ 2 , t  0
0 , t  0

Precise Graph
Commonly-Used Graph
Miguel A. Muriel - Signals and Systems: Introduction - 18
Signum
 1 , t  0


sgn  t    0 , t  0 
1 , t  0 


Precise Graph
Commonly-Used Graph
Miguel A. Muriel - Signals and Systems: Introduction - 19
Rampp
t , t  0  t
ramp  t   
   u   d  tu  t 
0 , t  0 
Miguel A. Muriel - Signals and Systems: Introduction - 20
Pulse (Rectangle)
(
g )
 1 , t  1/ 2 


rect  t   1/ 2 , t  1/ 2   u  t  1/ 2   u  t  1/ 2 


0
,
t

1/
2


Miguel A. Muriel - Signals and Systems: Introduction - 21
Real Sinusoid and Real Exponential
p
Miguel A. Muriel - Signals and Systems: Introduction - 22
Complex
p Exponential
p

t
g (t )  Ae  e j 2 ftf
 cos  2 ft   j sin  2 ft 
e
j 2 ft
e
 j 2 ft
 cos  2 ft   j sin  2 ft 
 e  j 2 ft
cos  2 ft  
2
j 2 ft
 e  j 2 ft
e
sin  2 ft  
2j
e
j 2 ft
Miguel A. Muriel - Signals and Systems: Introduction - 23
Miguel A. Muriel - Signals and Systems: Introduction - 24
Scaling and Shifting
 t  t0 
g  t   Ag 

 a 
Miguel A. Muriel - Signals and Systems: Introduction - 25
g  t   Ag  bt  t0 
Miguel A. Muriel - Signals and Systems: Introduction - 26
Differentiation
dx(t )
g t  
dt
Miguel A. Muriel - Signals and Systems: Introduction - 27
Integration
g
t
g (t ) 
 x   d

Miguel A. Muriel - Signals and Systems: Introduction - 28
Even and Odd Signals
g
Even Functions
g  t   g  t 
Odd Functions
g  t    g  t 
Miguel A. Muriel - Signals and Systems: Introduction - 29
Even and Odd Parts of Signals
g
g  t   ge  t   go  t 
The even part of a function is g e  t  
The odd part of a function is g o  t  
g  t   g  t 
2
g  t   g  t 
2
-The derivative of an even/odd function is odd/even.
-The integral of an even/odd function is an odd/even function,
pplus a constant.
Miguel A. Muriel - Signals and Systems: Introduction - 30
Periodic Signals
g
g  t   g  t  nT 
n is any integer
T is
i a period
i d off the
th function
f ti
-The minimum positive value of T for which g  t   g  t  T 
is called the fundamental period T0 of the function.
-The reciprocal
p
of the fundamental p
period
is the fundamental frequency f 0  1/ T0
Miguel A. Muriel - Signals and Systems: Introduction - 31
Examples of periodic functions with fundamental period T0
Miguel A. Muriel - Signals and Systems: Introduction - 32
Signal
g Energy
gy
The signal energy of a signal x  t  is

Ex 
 x t 
2
dt


Miguel A. Muriel - Signals and Systems: Introduction - 33
Signal
g Power
-Some signals have infinite signal energy.
-This usually occurs because the signal is not time limited.
-In that case it is more convenient to deal with average signal power.
-The average
g signal
g ppower of a signal
g x  t  is:
T /2
2
1
Px  lim
x  t  dt
T  T 
T / 2
Miguel A. Muriel - Signals and Systems: Introduction - 34
-For
For a periodic signal x  t  the average signal power is:
2
1
Px   x  t  dt
T T
where T is any period of x  t 
2
Miguel A. Muriel - Signals and Systems: Introduction - 35
-A
A signal
i l with
i h finite
fi i signal
i l energy is
i called
ll d an energy
signal.
-A signal with infinite signal energy and finite average
signal power is called a power signal.
-All
All periodic
i di signals
i l are power signals.
i l
Miguel A. Muriel - Signals and Systems: Introduction - 36
3- Discrete-Time Signals
Miguel A. Muriel - Signals and Systems: Introduction - 37
Sampling and Discrete Time
-Sampling is the acquisition of the values of a continuous-time signal
at discrete
di
points
i in
i time.
i
- x  t  is a continuous-time signal, x  n  is a discrete-time signal.
x  n   x  nTs 
(a) An ideal sampler
Ts
is the time between samples 
(b) An ideal sampler sampling uniformly
Miguel A. Muriel - Signals and Systems: Introduction - 38
Creating a discrete-time signal by sampling a continuous-time signal
Miguel A. Muriel - Signals and Systems: Introduction - 39
Unit Impulse
1 , n  0
  n  
0 , n  0
-The discrete-time unit impulse
p
is a function in the ordinaryy sense
(in contrast with the continuous-time impulse).
-It
It has a sampling property

 A  n  n  x  n  Ax  n 
0
0
n
-But no scaling property,   n     an  for any non-zero, finite integer a.
Miguel A. Muriel - Signals and Systems: Introduction - 40
Periodic Impulse
 N  n 

   n  mN 
m
Miguel A. Muriel - Signals and Systems: Introduction - 41
Unit Sequence
1 , n  0
u  n  
0 , n  0
Miguel A. Muriel - Signals and Systems: Introduction - 42
Signum
1 , n  0

sgn  n   0 , n  0
1 , n  0

Miguel A. Muriel - Signals and Systems: Introduction - 43
Unit Ramp
n , n  0 
ramp  n   
  nu  n 
0 , n  0 
Miguel A. Muriel - Signals and Systems: Introduction - 44
Exponentials
x  n   A n
Real 
 A is a real constant 
Complex 
Miguel A. Muriel - Signals and Systems: Introduction - 45
Sinusoids
g  n   A cos  2 F0 n   
A is a real constant,  is a real phase shift in radians
F0 is a real number, and n is discrete time
-Unlike a continuous-time sinusoid,
a discrete-time
di
t ti sinusoid
i
id is
i nott necessarily
il periodic.
i di
-g  n  periodic  F0 must be a ratio of integers (a rational number).
Miguel A. Muriel - Signals and Systems: Introduction - 46
Four discrete-time sinusoids
Miguel A. Muriel - Signals and Systems: Introduction - 47
-Two sinusoids whose analytical expressions look different
g1  n   A cos  2 F01n    and g 2  n   A cos  2 F02 n   
may actually be the same if F02  F01  m, where m is an integer.
A cos  2 F02 n     A cos  2 F01n  2 mn     A cos  2 F01n   
Miguel A. Muriel - Signals and Systems: Introduction - 48
Two cosines with different F’s but the same functional behavior
Miguel A. Muriel - Signals and Systems: Introduction - 49
A discrete-time sinusoid with frequency F repeats every time F changes by one
Miguel A. Muriel - Signals and Systems: Introduction - 50
Scaling and Shifting Functions
Let g  n be graphically defined by:
g  n  0 , n  15
Miguel A. Muriel - Signals and Systems: Introduction - 51
Time shifting
n  n  n0
 n0 an integer 
Miguel A. Muriel - Signals and Systems: Introduction - 52
Time compression
Time expansion
n  Kn
n  n/ K
 K an integer > 1
 K an integer > 1
-For all n such that n / K is an integer, g  n / K  is defined.
-For
F all
ll n suchh that
h n / K is
i not an integer,
i
g  n / K  is
i not defined.
d fi d
Miguel A. Muriel - Signals and Systems: Introduction - 53
Time compression for a discrete-time function
Miguel A. Muriel - Signals and Systems: Introduction - 54
Time expansion for a discrete-time function (K=2)
Miguel A. Muriel - Signals and Systems: Introduction - 55
Differencing
g  n  x  n   x  n  1
g  n  x  n  1  x  n
Miguel A. Muriel - Signals and Systems: Introduction - 56
Accumulation
g  n 
n
 h  m
m
m

Miguel A. Muriel - Signals and Systems: Introduction - 57
Even and Odd Signals
g  n   g   n
ge  n 
g  n   g  n 
2
g  n   g   n
go  n 
g  n  g  n
2
Miguel A. Muriel - Signals and Systems: Introduction - 58
Periodic Signals
g  n   g  n  mN 
m is any integer
N is
i a period
i d off the
th function
f ti
-The minimum positive value of N for which g  n   g  n  N 
is called the fundamental period N 0 of the function.
-The reciprocal
p
of the fundamental p
period
is the fundamental frequency F0  1/ N 0
Miguel A. Muriel - Signals and Systems: Introduction - 59
Examples of periodic functions with fundamental period N0
Miguel A. Muriel - Signals and Systems: Introduction - 60
Signal Energy
The signal energy of a signal x  n  is
Ex 

 x  n
2
n 
x  n  0 , n  31
Miguel A. Muriel - Signals and Systems: Introduction - 61
Signal Power
-Some
S
signals
i l have
h
infinite
i fi it signal
i l energy.
-This usually occurs because the signal is not time limited.
-In that case it is more convenient to deal with average signal power.
-The average signal power of a signal x  n  is
1
Px  lim
N  2 N
N 1
 x  n
2
n  N
Miguel A. Muriel - Signals and Systems: Introduction - 62
-For
F a periodic
i di signal
i l x  n  the
h average signal
i l power is:
i
1
Px 
N
 x  n
2
n N
 The notation  n N means the sum over any set of

 consecutive n 's exactly N in length





Miguel A. Muriel - Signals and Systems: Introduction - 63
-A
A signal
i l with
i h finite
fi i signal
i l energy is
i called
ll d an energy
signal.
-A signal with infinite signal energy and finite average
signal power is called a power signal.
-All
All periodic
i di signals
i l are power signals.
i l
Miguel A. Muriel - Signals and Systems: Introduction - 64
4- Description of Systems
Miguel A. Muriel - Signals and Systems: Introduction - 65
Systems
-Broadly speaking, a system is anything that responds when
stimulated or excited.
excited
-Engineering
Engineering system analysis is the application of mathematical
methods to the design and analysis of systems.
-Systems have inputs and outputs.
-Systems accept excitations or input signals at their inputs and
produce responses or output signals at their outputs
outputs.
Miguel A. Muriel - Signals and Systems: Introduction - 66
Systems
-Continuous-time
Continuous time systems are usually described by differential
equations.
-Discrete-time systems are usually described by difference
equations.
-The properties of discrete-time systems have the same meaning
as they do in continuous
continuous-time
time systems.
systems
Miguel A. Muriel - Signals and Systems: Introduction - 67
-Systems
y
are often represented
p
byy block diagrams
g
A single-input, single-output system block diagram
A multiple-input, multiple-output system block diagram
Miguel A. Muriel - Signals and Systems: Introduction - 68
Some Block Diagram Symbols
Three common block diagram symbols for an amplifier
Three common block diagram symbols for a summing junction
Miguel A. Muriel - Signals and Systems: Introduction - 69
-The
Th block
bl k diagram
di
symbols
b l for
f a summing
i junction
j ti andd an
amplifier are the same for discrete-time systems as they are for
continuous-time
continuous
time systems.
systems
Block diagram symbol (continuous-time systems) for an integrator
Block diagram symbol (discrete time systems) for a delay
Miguel A. Muriel - Signals and Systems: Introduction - 70
An Electrical Circuit Viewed as a System
-An RC lowpass filter is a simple electrical system.
-It is excited by a voltage vin  t  and responds with a voltage vout  t  .
-It can be viewed or modeled as a single-input, single-output system.
Miguel A. Muriel - Signals and Systems: Introduction - 71
Homogeneity
-In a homogeneous system, multiplying the excitation by
any constant (including
(i l di complex
l constants),
) multiplies
l i li
the response by the same constant.
Miguel A. Muriel - Signals and Systems: Introduction - 72
Additivity
-If a system when excited by an arbitrary x1 (t ) produces a response
y1 (t ),and
) and when excited by an arbitrary x2 (t ) produces a response y 2 (t )
and x1 (t )  x2 (t ) always produces the zero-state response y1 (t )  y 2 (t ),
th system
the
t is
i additive
dditi .
Miguel A. Muriel - Signals and Systems: Introduction - 73
Time Invariance
-If an arbitrary input signal x(t ) causes a response y (t ),
and an input signal x(t - t0 ) causes a response y(t - t0 ),
for any arbitrary t0 , the system is said to be time invariant.
Miguel A. Muriel - Signals and Systems: Introduction - 74
Linearity and LTI Systems
-If a system
y
is both homogeneous
g
and additive it is linear.
-If a system
y
is both linear and time-invariant it is called an LTI
system.
-The eigenfunctions of an LTI system are complex
exponentials.
-The eigenvalues of an LTI system are either real or, if
complex, occur in complex conjugate pairs.
Miguel A. Muriel - Signals and Systems: Introduction - 75
-Any
Any LTI system excited by a complex sinusoid respond with
another complex sinusoid of the same frequency, but generally
p
and pphase (multiplied
(
p
byy a complex
p
with different amplitud
constant).
-Using the principle of superposition for LTI systems, if the input
signal is an arbitrary function that is a linear combination of
complex sinusoids of various frequencies, then the output
signal is also a linear combination of complex sinusoids at
those same frequencies.
frequencies This idea is the basis for the methods
of Fourier transform analysis.
-All these statements are true of both continuous-time and
discrete-time
discrete
time systems.
Miguel A. Muriel - Signals and Systems: Introduction - 76
Causality
-Any system for which the response occurs only during or after
the time in which the excitation is applied is called a causal
system.
-Strictly speaking,
speaking all real physical systems are causal.
causal
Stability
-Any system for which the response is bounded for any
p
arbitraryy bounded excitation, is called a bounded-inputbounded-output (BIBO) stable system.
Miguel A. Muriel - Signals and Systems: Introduction - 77
5- Time-Domain System Analysis
Miguel A. Muriel - Signals and Systems: Introduction - 78
Continuous Time
Miguel A. Muriel - Signals and Systems: Introduction - 79
System
y
Response
p
Input signal
x t 
Output signal
System
y t  ?
-Once the response to an impulse is known, the response of any
system to any
a y arbitrary
a b t a y excitation
e c tat o can
ca be found.
ou d.
LTI syste
Miguel A. Muriel - Signals and Systems: Introduction - 80
Impulse
p
Response
p
h(t)
()
Impulse
 t 
Impulse response
System
h t 
-For an LTI system, the impulse response h(t) of the system
is a complete description of how it responds to any signal.
Miguel A. Muriel - Signals and Systems: Introduction - 81
Response of a linear shift-invariant system to impulses
Miguel A. Muriel - Signals and Systems: Introduction - 82
-If a continuous-time LTI system
y
is excited by
y an arbitrary
y
excitation, the response could be found approximately by
approximating the excitation as a sequence of continuous
rectangular pulses of width Tp
Exact
Excitation
Approximate
Excitation
Miguel A. Muriel - Signals and Systems: Introduction - 83
 t  nTp 
x  t    x  nTp  rect 


n 
 Tp 

 t  nTp 
1
x  t    Tp x  nTp  rect 


Tp
Tp 
n 



Shifted Unit Pulse
Unit pulse response
y t  

 T x  nT  h  t  nT 
p
p
p
p
n
Miguel A. Muriel - Signals and Systems: Introduction - 84
Tp  d

1
 t  nTp 
x t   
Tp x  nTp  rect 






Tp
n

 Tp 
   




d

 ( t - )


y t   
Tp x  nTp  hp  t  nTp 
    

n
  





d
h ( t - )

x(t ) 
 x( ) (t   )d

Sampling property of t)

y(t ) 
 x( )h(t   )d

Convolution integral
Miguel A. Muriel - Signals and Systems: Introduction - 85
Convolution Integral
g
y t  

 x( )h(t   )d  x  t  * h(t )

x t 
System
h t 
y  t   x  t  * h(t )
Miguel A. Muriel - Signals and Systems: Introduction - 86
Convolution Graphical
p
Example
p
-Let
Let x  t  be this smooth waveform
a eform and let it be appro
approximated
imated
by a sequence of rectangular pulses.
Miguel A. Muriel - Signals and Systems: Introduction - 87
The approximate excitation is a sum of rectangular pulses
Miguel A. Muriel - Signals and Systems: Introduction - 88
The approximate response is a sum of pulse responses.
Miguel A. Muriel - Signals and Systems: Introduction - 89
Exact and approximate excitation, unit-impulse response, unit-pulse response and exact
pp
system
y
response
p
with Tp=0,2
p , and 0.1
and approximate
Miguel A. Muriel - Signals and Systems: Introduction - 90
Miguel A. Muriel - Signals and Systems: Introduction - 91
Graphical
p
Illustration of the Convolution Integral
g
-The convolution integral
g is defined by
y
x t   h t  

 x   h  t    d

-For illustration purposes let the excitation x  t  and the
impulse response h  t  be the two functions below.
Miguel A. Muriel - Signals and Systems: Introduction - 92
Process of convolving
Miguel A. Muriel - Signals and Systems: Introduction - 93
Convolution Integral
g Properties
p
Convolution with an shifted impulse
p
x  t   A  t  t0   Ax  t  t0 
Commutativity
x t   y t   y t   x t 
A
Associativity
i ti it
 x  t   y  t    z  t   x  t    y  t   z  t  
Distributivity
 x  t   y  t    z  t   x  t   z  t   y  t   z  t 
Miguel A. Muriel - Signals and Systems: Introduction - 94
If y  t   x  t   h  t 
Differentiation Property
y  t   x  t   h  t   x  t   h  t 
Area Property
Area of y = (Area of x) x (Area of h)
S li Property
Scaling
P
y  at   a x  at   h  at 
Miguel A. Muriel - Signals and Systems: Introduction - 95
Miguel A. Muriel - Signals and Systems: Introduction - 96
System
y
Interconnections
-If the output signal from a LTI system is the input signal to a second
LTI system, the systems are said to be cascade connected.
-From
From the properties of convolution:
Cascade Connection
Miguel A. Muriel - Signals and Systems: Introduction - 97
-If two LTI systems are excited by the same signal and their
responses
p
are added they
y are said to be p
parallel connected.
-From properties of convolution:
Parallel Connection
Miguel A. Muriel - Signals and Systems: Introduction - 98
Stepp Response
p
-One of the most common signals used to test systems is the step
function The response of an LTI system to a unit step is:
function.
h1  t   h  t   u  t  

t


 h   u  t    d   h   d
-The response of an LTI system excited by a unit step is the
integral of the impulse response.
-As
As the unit step is the integral of the impulse,
impulse the unit-step
unit step
response is the integral of the unit-impulse response.
-In fact,, this relationship
p holds for anyy excitation.
-If any excitation is changed to its integral, the response also
changes to its integral.
Miguel A. Muriel - Signals and Systems: Introduction - 99
R l i
Relations
between
b
integrals
i
l andd derivatives
d i i
off excitations
i i
andd responses for
f an LTI system
Miguel A. Muriel - Signals and Systems: Introduction - 100
Discrete Time
Miguel A. Muriel - Signals and Systems: Introduction - 101
System
y
Response
p
I
Input
signal
i l
x  n
O
Output
signal
i l
System
y  n ?
-Once the response to a unit impulse is known, the response of
anyy LTI system
y
to any
y arbitraryy excitation can be found.
-Any arbitrary excitation is simply a sequence of amplitudescaled and time-shifted impulses.
p
-Therefore the response is simply a sequence of amplitudescaled and time-shifted impulse responses.
Miguel A. Muriel - Signals and Systems: Introduction - 102
Unit Impulse
p
Response
p
h[n]
[ ]
Unit Impulse
  n
Unit Impulse Response
System
h  n
-For an LTI system, the unit impulse response h[n] of the
system is a complete description of how it responds to any
signal.
Miguel A. Muriel - Signals and Systems: Introduction - 103
Convolution Sum
-The response
p
y  n  to an arbitraryy excitation x  n  is of the form
y  n    x  1 h  n  1  x  0 h  n   x 1 h  n  1  
-This can be written in a more compact form (Convolution Sum)
y  n 

 x  m h  n  m
m 

-Compare with
y(t ) 
 x( )h(t   )d

Miguel A. Muriel - Signals and Systems: Introduction - 104
y  n 
x  n

 x  m h  n  m  x  n * h  n
m
System
S
stem
h  n
y  n  x  n * h  n
Miguel A. Muriel - Signals and Systems: Introduction - 105
System
y
Response
p
Example
p
Miguel A. Muriel - Signals and Systems: Introduction - 106
Convolution Sum Properties
p
Convolution with a shifted unit impulse
x  n   A  n  n0   Ax  n  n0 
Commutativity
x  n  y  n  y  n  x  n
Associativity
 x  n   y  n   z  n   x  n    y  n   z  n 
Distributivity
 x  n   y  n   z  n   x  n   z  n   y  n   z  n 
Miguel A. Muriel - Signals and Systems: Introduction - 107
If
y  n  x  n * h  n
Differentiation Property
p y
y  n   y  n  1  x  n    h  n   h  n  1   x  n   x  n  1  h  n 
Sum Property (Sum of y = (Sum of x) x (Sum of h))
 

 

y  n    x  n  x   h  n 

n 
 n

 n


Miguel A. Muriel - Signals and Systems: Introduction - 108
System
y
Interconnections
-If the output signal from a LTI system is the input signal to a second
LTI system, the systems are said to be serie/cascade connected.
-From
From the properties of convolution:
Serie/Cascade Connection
Miguel A. Muriel - Signals and Systems: Introduction - 109
-If two LTI systems are excited by the same signal and their
responses
p
are added theyy are said to be p
parallel connected.
-From properties of convolution:
Parallel Connection
Miguel A. Muriel - Signals and Systems: Introduction - 110
6- Spectral
p
Method
Miguel A. Muriel - Signals and Systems: Introduction - 111
Introduction
Response of a linear shift-invariant system to impulses
Miguel A. Muriel - Signals and Systems: Introduction - 112
R
Response
off a li
linear shift-invariant
hift i
i t system
t to
t a harmonic
h
i function
f ti
Miguel A. Muriel - Signals and Systems: Introduction - 113
Representing
p
g a Signal
g
-The Spectral (Fourier) method represents a signal as a
linear combination of complex exponentials (Harmonic
functions).
-If an excitation can be expressed as a sum of complex
exponentials the response of an LTI system can be expressed
exponentials,
as the sum of responses to complex sinusoids too.
-This is the fundament of the Spectral Method.
Miguel A. Muriel - Signals and Systems: Introduction - 114
Miguel A. Muriel - Signals and Systems: Introduction - 115
Miguel A. Muriel - Signals and Systems: Introduction - 116
Miguel A. Muriel - Signals and Systems: Introduction - 117
Miguel A. Muriel - Signals and Systems: Introduction - 118
www.dsprelated.com
Miguel A. Muriel - Signals and Systems: Introduction - 119
Inner Product
-The inner product of two functions, is the integral of the
product of one function and the complex conjugate of the
other function over an interval.
-Inner Product of x1 (t ) and x2 (t ) on the interval t0  t  t0  T
t0 T
x1 (t ), x2 (t )   

Inner Product
x1 (t ) x2* (t )dt
t0
Miguel A. Muriel - Signals and Systems: Introduction - 120
Orthogonality
g
y
-Orthogonal means that the inner product of the two
functions of time
time, on some time interval,
interval is zero.
-Two functions x1 (t ) and x2 (t ) are orthogonal on the interval
t0  t  t0  T ,if
if :
t0 T
x1 (t ), x2 (t )   

Inner Product
x1 (t ) x2* (t ) dt  0
t0
Miguel A. Muriel - Signals and Systems: Introduction - 121
Orthogonality
g
y of complex
p exponentials
p
x1 (t )  e j1t  e j 2 f1t
x2 (t )  e j2t  e j 2 f2t




 e j1t , e j2t  0

 jt
e
dt




   j1t j2t

0
 e , e

1  2 
1  2 
e jt  Orthonormal ((Orthonormal and Normalized)) Basis
Miguel A. Muriel - Signals and Systems: Introduction - 122
7- Fourier Transform
Miguel A. Muriel - Signals and Systems: Introduction - 123
Definition
-Fourier Basis  e jt
-The Fourier Transform of x(t ) is the pprojection
j
of this function
onto the Fourier basis.
The Fourier Transform of x(t ) is defined as:
-The

F  x(t )   X ( )   x(t )e  jt dt

F
x(t )  X ( )
Miguel A. Muriel - Signals and Systems: Introduction - 124
-It follows that the Inverse Fourier Transform of X ( ) is:
1
F  X ( )   x(t ) 
2
1



X ( )e jt d
F1
X ( )  x(t )
Miguel A. Muriel - Signals and Systems: Introduction - 125
-The definitions based on frequency terms are:

X ( f )   x(t )e  j 2 ft dt


x(t )   X ( f )e jt df

Miguel A. Muriel - Signals and Systems: Introduction - 126
Miguel A. Muriel - Signals and Systems: Introduction - 127
Examples
p
Lowpass
Miguel A. Muriel - Signals and Systems: Introduction - 128
Highpass
Miguel A. Muriel - Signals and Systems: Introduction - 129
Bandpass
Miguel A. Muriel - Signals and Systems: Introduction - 130
Example: extracting frequency-domain information in the signal
Miguel A. Muriel - Signals and Systems: Introduction - 131
Example: blood pressure waveform (sampled at 200 points/s)
Miguel A. Muriel - Signals and Systems: Introduction - 132
FT Pairs
Miguel A. Muriel - Signals and Systems: Introduction - 133
Miguel A. Muriel - Signals and Systems: Introduction - 134
Miguel A. Muriel - Signals and Systems: Introduction - 135
Miguel A. Muriel - Signals and Systems: Introduction - 136
Miguel A. Muriel - Signals and Systems: Introduction - 137
Miguel A. Muriel - Signals and Systems: Introduction - 138
Sinc Function
sinc  t  
sin  t 
t
Miguel A. Muriel - Signals and Systems: Introduction - 139
Miguel A. Muriel - Signals and Systems: Introduction - 140
Miguel A. Muriel - Signals and Systems: Introduction - 141
Miguel A. Muriel - Signals and Systems: Introduction - 142
FT Properties
p
Lineality
F
 x  t    y  t  
 X  f    Y  f 
F
 x  t    y  t  
  X  j    Y  j 
Miguel A. Muriel - Signals and Systems: Introduction - 143
Domains Duality
F
F
X  t  
 x   f  and X  t  
 x f 
F
F
X  jt  
 2 x    and X   jt  
 2 x  
Miguel A. Muriel - Signals and Systems: Introduction - 144
Time Shifting
F
x  t  t0  
 X  f  e j 2 ft0
F
x  t  t0  

 X  j  e jt0
Miguel A. Muriel - Signals and Systems: Introduction - 145
Frequency Shifting
F
x  t  e  j 2 f0t 
 X  f  f0 
F
x  t  e j0t 
 X  j   0  
Miguel A. Muriel - Signals and Systems: Introduction - 146
The “Uncertainty”
y Principle
p
-The time and frequency scaling properties indicate that if a signal
is expanded
p
in one domain it is compressed
p
in the other domain.
-This is called the “uncertainty principle” of Fourier analysis.
Miguel A. Muriel - Signals and Systems: Introduction - 147
Time Scaling (Uncertainty Principle)
1  f 
x  at  

 X 
a a
F
F

x  at  
1  
X j 
a  a
Miguel A. Muriel - Signals and Systems: Introduction - 148
Frequency Scaling (Uncertainty Principle)
1
a
t F
x   
  X  af 
a
1
a
t F
 X  ja 
x   
a
Miguel A. Muriel - Signals and Systems: Introduction - 149
Time Differentiation
d
F
x  t   
 j 2 fX  f 

dt
d
x  t   
F  j X  j 

dt
Miguel A. Muriel - Signals and Systems: Introduction - 150
Time Integration
t
Xf
1
 x   d  j 2 f  2 X  0    f 
F
t
 x   d 
F

X  j 
j
  X  0    
Miguel A. Muriel - Signals and Systems: Introduction - 151
Total - Area Integrals
X  0 

 x  t  dt

x 0 

 X  f  dff

Miguel A. Muriel - Signals and Systems: Introduction - 152
Parseval’s Theorem

 x t 
2


 x t 



dt 
 Xf
2
df

2
1
dt 
2

 X  j 
2
df


Miguel A. Muriel - Signals and Systems: Introduction - 153
Multiplication Convolution Duality
F
x  t   y  t  
 X  f Y  f 
F
x  t   y  t  
 X  j  Y  j 
F
x  t  y  t  
 X  f  Y  f 
1
x  t  y  t  

X  j   Y  j 
2
F
Miguel A. Muriel - Signals and Systems: Introduction - 154
Time Domain (t )
Frequency Domain ( f )
y  xh
Y  XH
y  xh
h
Y  X *H
Miguel A. Muriel - Signals and Systems: Introduction - 155
y (t )

IFT
Y( f )
x(t )

FT

X(f )
h(t )
FT

H( f )
Miguel A. Muriel - Signals and Systems: Introduction - 156
Modulation
1
F
x  t  cos  2 f 0t  
  X  f  f 0   X  f  f 0  
2
1
x  t  cos 0t  
  X  j   0    X  j   0   
2
F
Miguel A. Muriel - Signals and Systems: Introduction - 157
Miguel A. Muriel - Signals and Systems: Introduction - 158
Miguel A. Muriel - Signals and Systems: Introduction - 159
Miguel A. Muriel - Signals and Systems: Introduction - 160
h t 
 t 
System
1
Hf 
Miguel A. Muriel - Signals and Systems: Introduction - 161
y  t   x  t  * h(t )
x t 
X(f )
System
h t  H  f 
Y  f   X  f H( f )
Miguel A. Muriel - Signals and Systems: Introduction - 162
System
y
Interconnections
-If the output signal from a LTI system is the input signal to a second
LTI system, the systems are said to be serie/cascade connected.
-In the frequency domain, the serie/cascade connection multiplies the
frequency responses instead of convolving the impulse responses.
Serie/Cascade Connection
Miguel A. Muriel - Signals and Systems: Introduction - 163
-If two LTI systems are excited by the same signal and their
responses are added they are said to be parallel connected.
H1 ( f )
Xf
+
Y  f   X  f  ( H1 ( f )  H 2 ( f ))
H2 ( f )
Xf
H1 ( f )  H 2 ( f )
Y  f   X  f  ( H1 ( f )  H 2 ( f ))
Parallel Connection
Miguel A. Muriel - Signals and Systems: Introduction - 164
8- Sampling
Miguel A. Muriel - Signals and Systems: Introduction - 165
Introduction
-The fundamental consideration in sampling theory is how fast to
sample a signal to be able to reconstruct the signal from the
samples.
Miguel A. Muriel - Signals and Systems: Introduction - 166
Miguel A. Muriel - Signals and Systems: Introduction - 167
Aliasing
1
fS 
TS
Miguel A. Muriel - Signals and Systems: Introduction - 168
Miguel A. Muriel - Signals and Systems: Introduction - 169
Nyquist rate
-If a continuous-time signal is sampled for all time at a rate fs
that is more than twice the bandlimit fm of the signal,
signal the original
continuous-time signal can be recovered exactly from the samples.
fS  2 fm
-The frequency
q
y 2 fm is called the Nyquist
yq
rate.
-A signal sampled at a rate less than the Nyquist rate is undersampled
-A signal
g
sampled
p at a rate ggreater than the Nyquist
yq
rate is oversampled.
p
Miguel A. Muriel - Signals and Systems: Introduction - 170
Miguel A. Muriel - Signals and Systems: Introduction - 171
Miguel A. Muriel - Signals and Systems: Introduction - 172
Miguel A. Muriel - Signals and Systems: Introduction - 173
Miguel A. Muriel - Signals and Systems: Introduction - 174
Interpolation
-Interpolation process for an ideal lowpass filter
-The corner frequency set to half the sampling rate
Miguel A. Muriel - Signals and Systems: Introduction - 175
-Sinc-function interpolation is theoretically perfect but it can never be
done in practice because it requires samples from the signal for all
time.
time
-Real interpolation must make causal compromises.
-The simplest realizable interpolation technique is what a DAC does.
Miguel A. Muriel - Signals and Systems: Introduction - 176
9- Discrete Fourier Transform (DFT)
Miguel A. Muriel - Signals and Systems: Introduction - 177
Intoduction
-A signal can be represented as a linear combination
off harmonic
h
i complex
l exponentials.
ti l
Miguel A. Muriel - Signals and Systems: Introduction - 178
Orthonormal ( Orthogonal and Normalized Basis)
e
j(
2
) nk
N
0  n  N 1
(t )   n 
0  k  N 1
( f )  k 
Miguel A. Muriel - Signals and Systems: Introduction - 179
Definition
-The
The most common definition of the Discrete Fourier Transform
(DFT) of x  n  is:
N 1
X  k    x  n e
n 0
 j(
2
) nk
N
DFT
x  n  
 X k 
N
x  n   N real values
X  k   N complex values  2 N real values
Miguel A. Muriel - Signals and Systems: Introduction - 180
-It follows that the Inverse Discrete Fourier Transform
(IDFT) of X  k  is:
2
j(
) nk
1 N 1
N
x  n   X  k  e
N n 0
X  k   x  n 
DFT-11
N
Miguel A. Muriel - Signals and Systems: Introduction - 181
DFT Matrix
WN  e
 j(
2
)
N
N 1
X [k ]   x[n]WNnk
n0
0  n  N 1
 X  0   WN0
WN0
WN0

  0
1
2
X
1
W
W
W



N
N

  N
 X  2    WN0
WN2
WN4

 




  
 X  N  1   W 0 W N -1  W 2( N -1)
N
N

  N



 WN2( N -1)1) 




 WN( N -1)( N -1) 


WN0
WNN -1
 x  0 


x
1
 

 x  2 





 x  N  1 


0  k  N 1
Miguel A. Muriel - Signals and Systems: Introduction - 182
Miguel A. Muriel - Signals and Systems: Introduction - 183
Miguel A. Muriel - Signals and Systems: Introduction - 184
Miguel A. Muriel - Signals and Systems: Introduction - 185
Miguel A. Muriel - Signals and Systems: Introduction - 186
Miguel A. Muriel - Signals and Systems: Introduction - 187
Miguel A. Muriel - Signals and Systems: Introduction - 188
Miguel A. Muriel - Signals and Systems: Introduction - 189
Miguel A. Muriel - Signals and Systems: Introduction - 190
Miguel A. Muriel - Signals and Systems: Introduction - 191
Miguel A. Muriel - Signals and Systems: Introduction - 192
2-point
p
DFT
DFT of vector ((a,b)
, )
N  2  W2  e  j  1
1 1  a   a  b 
1 1 b    a  b 

  

IDFT
1 1 1  a  b   a 
 



2 1 1 a  b   b 
Miguel A. Muriel - Signals and Systems: Introduction - 193
4-points
p
DFT
DFT off vector
t ((a,b,c,d)
b d)
N  4  W4  e
j

2
j
1 1 1 1  a   a  b  c  d 
1  j 1 j  b    a  c   j  b  d  


   
1 1 1 1  c    a  c    b  d  

   a  c  j b  d 
 

1 j 1  j  d   
Miguel A. Muriel - Signals and Systems: Introduction - 194
IDFT
1 1 1 1   a  b  c  d   a 
1 j 1  j    a  c   j  b  d    b 
1
 

4 1 1 1 1    a  c    b  d    c 

 a  c  j b  d   
 
 d 
1  j 1 j   
Miguel A. Muriel - Signals and Systems: Introduction - 195
DFT Properties
Miguel A. Muriel - Signals and Systems: Introduction - 196
Miguel A. Muriel - Signals and Systems: Introduction - 197
DFT Pairs
DFT
e j 2 n / N 
 mN  mN  k  m 
mN
DFT
cos  2 qn / N  
  mN / 2   mN  k  mq    mN  k  mq 
mN
DFT
sin  2 qn / N  

  jmN / 2   mNN  k  mq    mNN  k  mq 
mN
N
DFT
 N  n  
 m mN  k 
mN
1
DFT
 N N  k 
N

 u  n  n   u  n  n     n 
0
1
N
DFT
N
e
 j k  n1  n0  / N
 j k / N
 n1  n0  drcl  k / N , n1  n0 
e
DFT
2
tri  n / N w    N  n  

N
drcl
 k / N , N w  , N w an integer
w
N
DFT
sin
i c  n / w    N  n  
 wrect  wkk / N    N  k 
N
Miguel A. Muriel - Signals and Systems: Introduction - 198
Fast Fourier Transform (FFT)
(
)
-The DFT requires N2 complex multiplies and N(N-1) complex
additions.
-Algorithms that exploit computational savings are collectively
called Fast Fourier Transforms.
f
-They take advantage of the symmetry and periodicity of the
complex exponential.
Miguel A. Muriel - Signals and Systems: Introduction - 199
WN  e
 j(
2
)
N
N 1
X [k ]   x[n]WNnk
n 0
Symmetry  WN[ N n ]k  WN nk  (WNnk )*
Periodicity  WNnk  WN[ n N ]k  WNn[ k  N ]
N length DFT  N 2 multiplications
2
N2
N
N
2   lengths DFT  2   
multiplications
2
2
2
Miguel A. Muriel - Signals and Systems: Introduction - 200
N  2  W2  e  j  1
A 2-point
2 point DFT
Miguel A. Muriel - Signals and Systems: Introduction - 201
Miguel A. Muriel - Signals and Systems: Introduction - 202
An N-point DFT computed using two N/2-point DFT’s
Miguel A. Muriel - Signals and Systems: Introduction - 203
Textbook
-M.J.
M J Roberts
Roberts, "Signals
Signals and Systems Analysis Using Transform
Methods and MATLAB® ", 2nd Ed, McGraw-Hill (2012).
(Content and Figures are mainly from Textbook)
(M. J. Roberts - All Rights Reserved)
Miguel A. Muriel - Signals and Systems: Introduction - 204