Signals and Systems 1

1

• Signals and Systems 1
• Lecture 11
• Dr. Ali. A. Jalali
• September 13, 2002

2
Signals and Systems 1

• Lecture 11
• First TEST Review

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Signals and Systems
(Signals Systems)
Sequences
Signals
Systems
Electrocardiogram (ECG or EKG)
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Signals and Systems

1. A signal is the physical form of a waveform, like
   a sound wave or a radio wave.
2. Time is often the independent variable for
   signal.
3. The independent variable can be 1-D or 2-D (space
   x, y in image), 3-D or N-D
4. A system is an object or channel that changes a
   signal that passes through it.
5. Amplifiers are systems that increase the
   amplitude of signals passing through them.
6. Attenuators are systems that decrease the
   amplitude of signals passing through them.

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Signals and Systems

1. Signals
2. physical form of a waveform
3. e.g. a sound, electrical current, radio wave
4. Systems
5. a channel that changes a signal that passes
   through it
6. e.g. a telephone connection, a room, a vocal tract

System
Input Signal
Output Signal
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Signals

1. Classification of Signals
2. Deterministic and Stochastic signals
3. Periodic and Aperiodic signals
4. Continuous time (CT) and Discrete time (DT)
5. Causal and anti-causal signals
6. Right and left sided signals
7. Bounded and unbounded signals
8. Even and odd signals

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Building block signals Unit impulse definition

• The unit impulse ?(t), is an important signal of
  CT systems. The Dirac delta function, is not a
  function in the ordinary sense. It is defined by
  the integral relation
• And is called a generalized function.
• The unit impulse is not defined in terms of its
  values, but is defined by how it acts inside an
  integral when multiplied by a smooth function
  f(t). To see that the area of the unit impulse is
  1, choose f(t) 1 in the definition. We
  represent the unit impulse schematically as shown
  below the number next to the impulse is its area.

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Unit impulse- narrow pulse approximation

1. To obtain an intuitive feeling for the unit
   impulse, it is often helpful to imagine a set of
   rectangular pulses where each pulse has width ?
   and height 1/? so that its area is 1.

The unit impulse is the quintessential tall and
narrow pulse!
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Unit impulse- intuiting the definition

1. To obtain some intuition about the meaning of the
   integral definition of the impulse, we will use a
   tall rectangular pulse of unit area as an
   approximation to the unit impulse.
2. As the rectangular pulse gets taller and narrower,

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Unit Step Function

• Integration of the unit impulse yields the unit
  step function
• which is defined as

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F1_7b Signal g(t) multiplied by a pulse
functions
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Successive integration of the unit impulse

1. Successive integration of the unit impulse yields
   a family of functions.
2. Later we will talk about the successive
   derivatives of ?(t), but these are too horrible
   to contemplate in the first lecture.

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Building-block signals can be combined to make a
rich population of signals

1. Unit steps and ramps can he combined to produce
   pulse signals.

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Example

• Describe analytically the signals shown in
• Solution Signal is (A/2)t at ,
  turn on this signal at t 0 and turn it off
  again at t 2. This gives,

f(t)
A
t
0
2
See page 9 of text for more examples.
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Continuous Systems

1. Preview
2. A system is transforms input signals into output
   signals.
3. A continuous-time system receives an input signal
   x(t) and generates an output signals y(t).
4. y(t)h(t)x(t) means the system h(t) acts on
   input signal x(t) to produce output signal y(t).
5. We concentrate on systems with one input and one
   output signal, i.e., Single-input, single output
   (SISO) systems.
6. Systems often denoted by block diagram.
7. Lines with arrows denote signals (not wires).
   Arrows show inputs and outputs

Continuous-time System h(t)
x
(
t
)
y
(
t
)
Output
Input
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and Systems 1
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Continuous Systems

• Example Electric Network

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Systems

• Classifications of systems
• 1. Linear and nonlinear systems.
• 2. Time Invariant and time varying systems.
• 3. Causal, noncausal and anticausal systems.
• 4. Stable and unstable systems.
• 5. Memoryless systems and systems with memory.
• 6. Continuous and Discrete time systems.

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Continuous Systems

• Linearity
• principle of homogeneity (c is real
  constant).
• principle of additively
• homogeneity and additively (Principle of
  superposition)

x
(
t
)
C
x
(
t
)
y
(
t
)
C
y
(
t
)
x
(
t
)
y
(
t
)
LS
LS
1
1
1
1
1
1
x
(
t
)
y
(
t
)
LS
1
1
x
(
t
)

x
(
t
)
x
(
t
)
y
(
t
)

y
(
t
)
y
(
t
)
LS
1
2
1
2
x
(
t
)
y
(
t
)
LS
2
2
x
(
t
)
y
(
t
)
LS
1
1
y(t)C1y1(t)C2y2(t)
X(t)C1x1(t)C2x2(t)
LS
x
(
t
)
y
(
t
)
LS
2
2
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Continuous Systems

• Linearity example
• Let the response of a linear system at
  rest due to the system input
  be given by
  and let the response of the same
  system at rest due to another system input
  be
• Then the response of the same system at
  rest due to input given by
• Is simply obtained as

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Continuous Systems
Time Invariance
x(t)
y(t)
x(t)
y(t)
LS
t
1
t
1
3
y(t-T)
x(t-T)
x(t-T)
y(t-T)
LS
T
T
t
1
4
Shifted input Shifted output For All value of t
and T.
t
1
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Continuous Systems

• Time invariant example
• Let the response of a time-invariant
  linear system at rest due to
  be given by
  Then, the
  system response due to the shifted system input
  defined by
• Is

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Differential Equation Model

1. Many physical systems are described by linear
   differential equations.
2. Reducing differential equations to algebraic
   equations
3. Homogeneous solution, exponential solution and
   natural frequencies.
4. Particular solution, system function and
   poles-zeros
5. Total solution, initial condition and
   steady-state
6. Conclusion

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The Nth-order Differential Equation Model

1. The general linear constant-coefficient Nth-Order
   DE for SISO systems are

OR

• X(t) system input,
• Y(t) system output and our practical restriction
  order

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Initial Condition Solution of Differential
Equation

• This is Characteristic Equation
• Can be written as
• Or as factored form
• Characteristic roots are
• Where may be real
  or complex (conjugate pairs).

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Example Solution

• 1- Find CE?
• The homogeneous differential equation is
• Yielding the CE as
• 2- CR?
• Using the quadratic formula, CR are
• 3- Is this system stable?
• Both roots are negative real so the system is
  stable.
• 4- Algebraic form of IC response?
• 5- Constant IC solution?
  
• and
• Solving gives and Thus
  

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Unit Impulse Response

• The response of an LTI system to an input
  of unit impulse function is called the unit
  impulse response.
• Important When determining the unit impulse
• response h(t) of an LTI
  system, it is
• necessary to make all
  initial conditions
• zero. (output due to input
  not energy stored in system)

d
x
(
t
)

(
t
)
y
(
t
)

h
(
t
)
LTI

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Convolution Integral

1. The convolution integral is one of the most
   important results used in the study of the
   response of linear systems.
2. If we know the unit impulse response h(t) for a
   linear system, by using the convolution integral
   we can compute the system output for any known
   input x(t).
3. In the following integration integral h(t) is the
   systems unit impulse response.

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Example for total response of system

• Total response ZIR ZSR

R
y(t)
C
f(t)
-
IC response, Force response and Steady state
response
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f(t) -2u(t1)3u(t)-u(t-2)
EXAMPLE
f(t)
3u(t)
3
2
1
t
2
-1
0
-u(t-2)
-2
2u(t1)
f(t)
3u(t)
1
t
2
-u(t-2)
-2
2u(t1)
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f(t) -(t4)u(t4)(t2)u(t2)(t-2)u(t-2)-(t-4)u(
t-4)
EXAMPLE
f(t)
8
(t2)u(t2)
(t-2)u(t-2)
4
2
t
-4
4
2
-2
0
-2
-(t-4)u(t-4)
-(t4)u(t4)
-4
f(t)
4
-4
0
t
2
-2
-2
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Sketch the following sequence? 4u(n-3)-2(n-6)u(n-6
)2(n-8)u(n-8)
4
0
1
9
2
3
4
5
6
7
8
4
0
1
9
2
3
4
5
6
7
8
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Sketch the following sequence? 2(n5)u(n5)-3nu(n
)(n-10)u(n-10)
Slop2
Slop1
(n-10)u(n-10)
10
2(n5)u(n5)
n
0
1
9
2
3
4
5
6
7
8
10
-5
After n 10 the total slop is zero.
Slop-3
-3nu(n)
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Graphical Solution
EXAMPLE

1
1
t
t
0
0
1
1
-1
-1
34

-2lttlt-1
1
2
-2
0
1
-1
t
-2t
1
-1 lt t lt 0
2
-2
0
1
-1
t
-2t
-1t
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0 lt t lt 1
1
2
-2
0
1
-1
t
-2t
-1t
1
1 lt t lt 2
2
-2
0
1
-1
t
-2t
-1t
36
Convolution Plane
EXAMPLE

1
1
t
t
0
0
1
1
-1
-1
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Regions

1
g
h
d
f
-1
1
2
-2
0
c
e
t
a
b
-1
38

• Integral arrangement

y(t)
Region
Integral
t lt -2
a
-2 lt t lt-1
c
d
b
-1 lt t lt 0


e
g
f
0 lt t lt 1


1 lt t lt 2
h
t gt 2