Signals and Systems Analysis

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Signals and Systems Analysis

• NET 351
• Instructor Dr. Amer El-Khairy
• ?. ???? ??????

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Brief description of basic learning outcomes

• Understand basic knowledge of signal analysis and
  processing
• Deal with different domains and systems
• Understand basic knowledge of sampling theory
• Solve signal problems using correlation
• Solve signal problems using Fourier, and Laplace
  Transforms
• Apply signal analysis and processing on some
  applications
• Use some analytical tools (e.g. MATLAB)

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Course description

• Introduction to the course content, text
  book(s), reference(s) and course plane.
• Introduction to Signal Processing and its
  Applications.
• Types of Signals its Properties.
• Singularity Functions.
• Signals in the Time Frequency Domains.
• Continuous-Time Linear Time System.
• Correlation Convolution Theory.

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Course description (continued)

• Fourier series.
• Fourier Transform.
• Fourier Applications.
• Laplace Transform.
• Inverse Laplace Transform.
• Sampling Theory.
• Signal Reconstruction.
• Introduction to some Analytical Tools (e.g.
  MATLAB).

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Assessment
Assessment weight () Due week The nature of the evaluation function (e.g. article, quiz, group project, etc.) index
20 Week 7 First exam 1
20 Week 12 Second exam 2
10 Weeks 5 10 Theoretical assignments 3
50 After Week 15 Final exam 4
100 100 100 Total
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Textbook and References

• Textbook Oppenheim Willsky, and Nawab, "Signals
  and Systems", Printice-Hall, The Latest Edition.
• Reference Won Y. Yang, Tae G. Chang, Ik H. Song,
  "Signals and Systems with MATLAB",
  Springer-Verlag Berlin Heidelberg 2009.

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Recommended Reading Material

• Signals and Systems, Oppenheim Willsky
• Signals and Systems, Haykin Van Veen
• Mastering Matlab 6
• Mastering Simulink 4
• Many other introductory sources available. Some
  background reading at the start of the course
  will pay dividends when things get more
  difficult.

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What is a Signal?

• A signal is a pattern of variation of some form
• Signals are variables that carry information
• Examples of signal include
• Electrical signals
• Voltages and currents in a circuit
• Acoustic signals
• Acoustic pressure (sound) over time
• Mechanical signals
• Velocity of a car over time
• Video signals
• Intensity level of a pixel (camera, video) over
  time

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How is a Signal Represented?

• Mathematically, signals are represented as a
  function of one or more independent variables.
• For instance a black white video signal
  intensity is dependent on x, y coordinates and
  time t f(x,y,t)
• On this course, we shall be exclusively concerned
  with signals that are a function of a single
  variable time

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What a Signal is?

• A signal is a mathematical representation that
  describes a physical phenomenon.
• Examples
• Speech signal ? one-dimensional signal that
  descries the acoustic pressure variation as a
  function of time, t
• Picture signal ? two-dimensional signal that
  describes the gray level as a function of spatial
  coordinates, x and y.
• Only one-dimensional signals are considered in
  this course. The independent variable is referred
  to as the time.

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Continuous Time (CT) and Discrete Time (DT)
Signals

• A signal is a continuous-time (CT) signal if it
  is defined for a continuum of values of the
  independent variable, t.
• A signal is discrete-time (DT) if it is defined
  only at discrete times and the independent
  variable takes on only a discrete set of values

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Analog vs. Digital

• If a continuous-time signal x(l) can take on any
  value in the continuous interval (a, b), where a
  may be - ? and b may be ?, then the
  continuous-time signal x(t) is called an analog
  signal.

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Analog vs. Digital (continued)
Digital is a discrete or non-continuous waveform
with examples such as computer 1s and 0s
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Analog vs. Digital (continued)
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Real and complex signals

• A signal x(t) is a real signal if its value is a
  real number, and a signal x(t) is a complex
  signal if its value is a complex number. A
  general complex signal x(t) is a function of the
  form
• x( t ) x1( t ) j x2( t )
• where x1( t ) and x2( t ) are real signals and j
  .

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Deterministic and Random Signals

• Deterministic signals are those signals whose
  values are completely specified for any given
  time. Thus, a deterministic signal can be modeled
  by a known function of time t.
• Random signals are those signals that take random
  values at any given time and must be
  characterized statistically. Random signals will
  not be discussed in this text

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Even and Odd Signals

• A signal x(t) or xn is referred to as an even
  signal if
• x(-t) x(t)
• x-n xn

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Even and Odd Signals (continued)

• A signal x(t) or xn is referred to as an odd
  signal if
• x(-t) -x(t)
• x-n -xn

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Even and Odd Signals (continued)

• Any signal x(t) or xn can be expressed as a sum
  of two signals, one of which is even and one of
  which is odd. That is,
• x(t) xe(t) xo(t)
• xn xen xon
• where xe(t) ½ x(t) x(-t)
• xo(t) ½ x(t) - x(-t)
• xen ½ xn x-n
• xon ½ xn - x-n

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Periodic and Non-periodic Signals

• A continuous-time signal x(t) is said to be
  periodic with period T if there is a positive
  nonzero value of T for which
• x(t T) x(t) for all t
• An example of such a signal is shown below

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Periodic and Non-periodic Signals (continued)

• From this equation
• x(t T) x(t)
• it follows that
• x(t mT) x(t)
• for all t and any integer m.
• Note that this definition does not work for a
  constant signal x(t).
• Any continuous-time signal which is not periodic
  is called a non-periodic (or aperiodic ) signal.

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Periodic and Non-periodic Signals (continued)

• Periodic discrete-time signals are defined
  analogously. A sequence (discrete-time signal)
  xn is periodic with period N if there is a
  positive integer N for which
• xn N xn for all n

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Periodic and Non-periodic Signals (continued)

• From the following equation
• xn N xn
• It follows that
• xn mN xn
• for all n and any integer m.
• Any sequence which is not periodic is called a
  non-periodic (or aperiodic) sequence.

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Causal vs. Anticausal vs. Noncausal Signals

• Causal signals are signals that are zero for all
  negative time.

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Causal vs. Anticausal vs. Noncausal Signals

• Anti-causal are signals that are zero for all
  positive time.

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Causal vs. Anticausal vs. Noncausal Signals

• Non-causal signals are signals that have nonzero
  values in both positive and negative time.

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Signal Energy and Power

• The total energy over an interval t1lttltt2 in a
  continuous time signal x(t) is defined as ?
• For discrete-time sequence xn ?

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Signal Energy and Power

• A signal is called energy signal if
• A signal is called Power signal if

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Transformation of Independent Variables

• Reflection
• Shifting

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• Scaling
• Example

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Basic continuous-time signals

• The Unit Step Function
• The unit step function u(t) is defined as

Note that it is discontinuous at t 0 and that
the value at t 0 is undefined.
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Basic continuous-time signals (continued)

• Similarly, the shifted unit step function u(t -
  to) is defined as

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Basic continuous-time signals (continued)
The Unit Impulse Function The unit impulse
function d(t), also known as the Dirac delta
function, plays a central role in system
analysis. Traditionally, d(t) is often defined as
the limit of a suitably chosen conventional
function having unity area over an infinitesimal
time interval
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Basic continuous-time signals (continued)

• The unit impulse function d(t) properties

But an ordinary function which is everywhere 0
except at a single point must have the integral
0. Thus, d(t) cannot be an ordinary function
and mathematically it is defined by
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Basic continuous-time signals (continued)

• Similarly, the delayed delta function
• d(t t0) is defined by

where f (t) is any regular function continuous at
t t0. For convenience, d(t) and d(t-t0) are
depicted graphically
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Basic continuous-time signals (continued)

• Some additional properties of d(t) are

Sifting property
if x(t) is continuous at t0
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Basic continuous-time signals (continued)

• using the two equations

and
We can conclude that any signal x(t) can be
expressed as follows
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Basic continuous-time signals (continued)

• Complex Exponential Signals

Using Euler's formula, this signal can be defined
as
Thus, x(t) is a complex signal whose real part is
cos w0t and imaginary part is sin w0t.
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Basic continuous-time signals (continued)

• Complex Exponential Signals

x(t) is a periodic signal whose fundamental
period is
w0 is called the fundamental frequency.
General Complex Exponential Signals Let s s
jw be a complex number. We define x(t) as
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Basic continuous-time signals (continued)
Signal x(t) known as a general complex
exponential signal whose real part est coswt and
imaginary part est sinwt are exponentially
increasing (s gt 0) or decreasing (s lt 0)
sinusoidal signals
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Basic continuous-time signals (continued)
Real Exponential Signals if s s (a real
number), then
reduces to a real exponential signal. If s gt 0
then x(t) is a growing exponential. If s lt 0
then x(t) is a decaying exponential.
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Basic continuous-time signals (continued)
Sinusoidal Signals A continuous-time sinusoidal
signal can be expressed as
where A is the amplitude (real), w0 is the radian
frequency in radians per second, and f is the
phase angle in radians. The sinusoidal signal
x(t) is periodic with fundamental period
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Basic continuous-time signals (continued)
Sinusoidal Signals The reciprocal of the
fundamental period T0 is called the fundamental
frequency f0 From the previous two equations
we can conclude the following relation w0 is
the fundamental angular frequency.
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Basic continuous-time signals (continued)
Sinusoidal Signals Using Euler's formula, the
sinusoidal signal can be expressed as
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Basic continuous-time signals (continued)

• f - Frequency
• The number of times a signal makes a complete
  cycle within a given time frame frequency is
  measured in Hertz (Hz), or cycles per second e.g.
  S5cos(2?5t) here f5Hz
• Spectrum Range of frequencies that a signal
  spans from minimum to maximum e.g. SS1S2 where
  S1cos(2?5t) and S2cos(2?7t). Here spectrum
  SP5Hz,7Hz.
• Bandwidth Absolute value of the difference
  between the lowest and highest frequencies of a
  signal. In the above example bandwidth is BW2Hz.
• Consider an average voice
• The average voice has a frequency range of
  roughly 300 Hz to 3100 Hz
• The spectrum would be 300 3100 Hz
• The bandwidth would be 2800 Hz

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Basic continuous-time signals (continued)
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Basic continuous-time signals (continued)

• ? - Phase
• The position of the waveform relative to a given
  moment of time or relative to time zero, e.g.
  S1cos(2?5t) and S2cos(2?5t ?/2). Here S1 has
  phase ?0 and S2 has phase ? ?/2.
• A change in phase can be any number of angles
  between 0 and 360 degrees
• Phase changes often occur on common angles, such
  as ?/445, ?/290, 3?/4135, etc

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Basic continuous-time signals (continued)
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Basic continuous-time signals (continued)
Ramp Function The ramp function is closely
related to the unit-step discussed above. Where
the unit-step goes from zero to one
instantaneously, the ramp function better
resembles a real-world signal, where there is
some time needed for the signal to increase from
zero to its set value, one in this case. We
define a ramp function as follows
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Basic continuous-time signals (continued)
Ramp Function
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Basic Discrete-time signals
The Unit Step Sequence The unit step sequence
un is defined as
Note that the value of un at n 0 is defined
unlike the continuous-time step function u(t) at
t 0 and equals unity.
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Basic Discrete-time signals
The Unit Step Sequence The shifted unit step
sequence un-k is defined as
Note that the value of un at n 0 is defined
unlike the continuous-time step function u(t) at
t 0 and equals unity.
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Basic Discrete-time signals
The Unit impulse Sequence The unit impulse or
unit sample sequence dn is defined as
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Basic Discrete-time signals
The Unit impulse Sequence The shifted unit
impulse or shifted unit sample sequence dn-k is
defined as
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Basic Discrete-time signals
The Unit impulse Sequence Unlike the
continuous-time unit impulse function d(t), dn
is defined without mathematical complication or
difficulty. From above definitions it is readily
seen that
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Basic Discrete-time signals
The Unit impulse Sequence Unlike the
continuous-time unit impulse function d(t), dn
is defined without mathematical complication or
difficulty. From above definitions it is readily
seen that
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Basic Discrete-time signals
The Sinusoidal Sequence A sinusoidal sequence
can be expressed as If n is dimensionless, then
both Wo, and q have units of radians. In order
for the sequence to be periodic with period N gt
0, Wo must satisfy the following condition (m
is a positive integer)
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Basic Discrete-time signals
The Sinusoidal Sequence (continued) Thus the
sequence is not periodic for any value of Wo. It
is periodic only if Wo / 2p is a rational
number. Thus, if Wo satisfies the periodicity
condition, Wo ? 0, N and m have no factors in
common, then the fundamental period of the
sequence xn is No given by
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Basic Discrete-time signals
The following sequence is periodic
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Basic Discrete-time signals
The following sequence is non-periodic
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Basic Discrete-time signals
Complex Exponential Sequences The complex
exponential sequence is of the form Again,
using Euler's formula, xn can be expressed
as In order for xn to be periodic with period
N gt 0, Wo must satisfy the following condition
(m is a positive integer)
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Basic Discrete-time signals
Complex Exponential Sequences (continued) Thus
the sequence is not periodic for any value of Wo.
It is periodic only if Wo / 2p is a rational
number. Thus, if Wo satisfies the periodicity
condition, Wo ? 0, N and m have no factors in
common, then the fundamental period of the
sequence xn is No given by
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Basic Discrete-time signals
Complex Exponential Sequences (continued) in
dealing with discrete-time exponentials, we need
only consider an interval of length 2p in which
to choose Wo. Usually, we will use the interval
0 Wolt 2p or the interval -p Wolt p.
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Basic Discrete-time signals
General Complex Exponential Sequences The most
general complex exponential sequence is often
defined as xn Can where C and a are in
general complex numbers. Note that Equation
is a special case of the above
equation with C 1 and a
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Basic Discrete-time signals

• Real Exponential Sequences
• If C and a are both real, then xn is a real
  exponential sequence.
• Four distinct cases can be identified
• a gt 1,
• 0 lt a lt 1,
• -1 lt a lt 0, and
• a lt - 1.
• Note that if a 1, xn is a constant sequence,
  whereas if a - 1, xn alternates in value
  between C and -C.

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Basic Discrete-time signals
Real Exponential Sequences (continued)
a gt 1
0 lt a lt 1
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Basic Discrete-time signals
Real Exponential Sequences (continued)
0 gt a gt -1
a lt -1
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What is a System?

• Systems process input signals to produce output
  signals
• Examples
• A CD player takes the signal on the CD and
  transforms it into a signal sent to the loud
  speaker
• A system takes a signal as an input and
  transforms it into another signal

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What is a System(continued)?
Let x and y be the input and output signals,
respectively, of a system. Then the system is
viewed as a transformation (or mapping) of x into
y. This transformation is represented by the
mathematical notation y Tx where T is the
operator representing some well-defined rule by
which x is transformed into y.
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Classifications of System
Continuous-Time and Discrete-Time Systems If the
input and output signals x and y are
continuous-time signals, then the system is
called a continuous-time system. If the input and
output signals are discrete-time signals or
sequences, then the system is called a
discrete-time system.
continuous-time system
discrete-time system
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Classifications of System
Systems with Memory and without Memory A system
is said to be memoryless if the output at any
time depends on only the input at that same time.
Otherwise, the system is said to have memory. An
example of a memoryless system An example of a
system with memory
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Classifications of System
Causal and Noncausal Systems A system is called
causal if its output y(t) at an arbitrary time t
to, depends on only the input x(t) for t to.
That is, the output of a causal system at the
present time depends on only the present and/or
past values of the input, not on its future
values. Thus, in a causal system, it is not
possible to obtain an output before an input is
applied to the system. A system is called
noncausal if it is not causal.
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Classifications of System
Causal and Noncausal Systems All realtime
systems must be causal, since they can not have
future inputs available to them. Examples of
noncausal systems are Examples of causal
systems are Note that all memoryless systems
are causal, but not vice versa.
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Classifications of System(cntd.)
Linear Systems and Nonlinear Systems If the
operator T in Equation y Tx satisfies the
following two conditions, then T is called a
linear operator and the system represented by a
linear operator T is called a linear system 1-
Additivity Given that Tx1 y1 and Tx2 y2,
then Tx1 x2 y1 y2 for any signals x1
and x2.
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Classifications of System(cntd.)
Linear Systems and Nonlinear Systems 2-
Homogeneity or (scaling) Tax ay for any
signal x and any scalar a. Any system that does
not satisfy both conditions is classified as a
nonlinear system. Both conditions can be combined
into a single condition as T a1x1 a2x2
a1y1 a2y2 where a, and a, are arbitrary
scalars. This final Equation is known as the
superposition property.
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Classifications of System(cntd.)
Linear Systems and Nonlinear Systems
(illustration using diagrams) 1- Additivity
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Classifications of System(cntd.)
Linear Systems and Nonlinear Systems
(illustration using diagrams) 2- Homogeneity or
(scaling)
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Classifications of System(cntd.)
Time-Invariant and Time-Varying Systems A system
is called time-invariant if a time shift (delay
or advance) in the input signal causes the same
time shift in the output signal. Thus, for a
continuous-time system, the system is
time-invariant if for any real value of t. For
a discrete-time system, the system is
time-invariant (or shift-invariant ) if for any
integer k A system that doesnt satisfy this
condition is called time-varying.
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Classifications of System(cntd.)
Linear Time-Invariant System A system is called
Linear time-invariant (LTI) if this system is
both linear and time-invariant. Stable
Systems A system is bounded-input/bounded-output
(BIBO) stable if for any bounded input x defined
by the corresponding output y is also
bounded defined by where k1 and k2
are finite real constants.
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Classifications of System(cntd.)
Stable Systems (Graphical illustration)
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Classifications of System(cntd.)
Invertible Systems a system S is invertible if
the input signal can always be uniquely
recovered from the output signal. The inverse
system, formally written as S1 (this is not the
arithmetic inverse), is such that the cascade
interconnection in Figure below is equivalent to
the identity system, which leaves the input
unchanged.
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System block diagrams(Interconnections)
Systems may be interconnections of other systems.
For example, the discrete-time system. For
example, the discrete-time system shown as a
block diagram in the Figure below can be
described by the following system equations
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System block diagrams(Interconnections)
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System block interconnections
Cascade Interconnection The cascade
interconnection shown in the Figure below is a
successive application of two (or more) systems
on an input signal
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System block interconnections
Parallel Interconnection The parallel
interconnection shown in the Figure below is an
application of two (or more) systems to the same
input signal, and the output is taken as the sum
of the outputs of the individual systems.
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System block interconnections
Feedback Interconnection The feedback
interconnection of two systems as shown in Figure
below is a feedback of the output of system G1 to
its input, through system G2.