EGR 277

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Lecture 8 EGR 261 Signals and Systems
Read Ch. 1, Sect. 1-4, 6-8 in Linear Signals
Systems, 2nd Ed. by Lathi

• Classification of Signals
• Several classes of signals are considered in this
  course
• Continuous-time and discrete-time signals
• Analog and digital signals
• Periodic and aperiodic signals
• Energy and power signals
• Deterministic and probabilistic signals

Continuous-time and discrete-time
signals Continuous-time signals are signals that
are specified for a continuum of time values
(i.e., all values of time over a specified
range). Discrete-time signals are signals that
are only defined at discrete times (i.e., for a
specific set of time values.) For example, a
discrete-time signal may have values defined once
per millisecond. A discrete-time signal might be
a set of data point measured at specific time
intervals, such as annual population figures. A
discrete-time signal might also be formed by
sampling a continuous-time signal (measuring
its value at specific time intervals to produce a
sequence of data points).

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Lecture 8 EGR 261 Signals and Systems
Example Continuous-time signal
x(t) is defined for any value of t. In this case
a function x(t) Csin(wt - ?) describes the
signal.
Example Discrete-time signal
A series of data points (x0, x1,
x2,) represents the signal.

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Lecture 8 EGR 261 Signals and Systems
Analog and Digital Signals Analog signals are
signals that can have any amplitude. Digital
signals are signals that can only have specific
amplitudes (such as binary signals that can only
have values 0 or 1). Analog and digital signals
are sometimes confused with continuous-time and
discrete-time signals. The difference can be
summarized by the following

• Continuous-time signals can have any value of
  time (any x value).
• Discrete-time signals can have only specific
  values of time (only a set of x values).
• Analog signals can have any amplitude (any y
  value).
• Digital signals can have only specific amplitudes
  (only a set of y values).

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Lecture 8 EGR 261 Signals and Systems

Reference Linear Signals and Systems, 2nd
Edition, by Lathi.
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Lecture 8 EGR 261 Signals and Systems
Periodic and Aperiodic Signals A signal x(t) is
said to be periodic if for some positive constant
To x(t) x(t To) for all t The
smallest value of To that satisfies the equation
above is the fundamental period of x(t). If a
signal is not periodic, then it is aperiodic.
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Lecture 8 EGR 261 Signals and Systems

• Notes on periodic signals
• A periodic signal by definition
• Remains unchanged when time-shifted by N periods
  (where N is an integer)
• Is an everlasting signal (exists over the range
  -? lt t lt ?)
• The area under the curve for a periodic signal is
  the same for any interval of duration To

Causal, noncausal, and anticausal signals A
signal that does not start before t 0 is a
causal signal. A signal that starts before t 0
is a noncausal signal. A signal that is zero for
all t gt 0 is an anticausal signal. Note that all
periodic signals are non-causal. Examples
Identify each signal below as causal, noncausal,
or anticausal.
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Lecture 8 EGR 261 Signals and Systems

• Energy signals and power signals
• Earlier we discussed how to calculate signal
  energy and signal power.
• To summarize
• A signal with finite energy is an energy signal.
• A signal with finite and nonzero power is a power
  signal.
• A signal cannot be both an energy signal and a
  power signal since
• A signal with finite energy has zero power.
• A signal with finite power has infinite energy.
• All periodic signals are power signals.
• Some signals are neither energy signals nor power
  signals.
• Examples x(t) t, x(t) KtN (for N gt 1),
  x(t) Ke-at (defined for all time)

Deterministic signals and random signals Signals
that can be completely described mathematically
or graphically are deterministic signals. Signals
that cannot be predicted precisely, but are known
only in terms of probabilistic description (such
as mean value), are random signals. Examples of
random signals include noise, atmospheric
disturbances, stock market values, etc. This
course will only deal with deterministic signals.
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Lecture 8 EGR 261 Signals and Systems

• Useful signal models
• Three important functions are used commonly in
  the area of signals and systems
• Unit step function, u(t)
• Impulse function, ?(t)
• Exponential function, est

Unit step function, u(t) The unit step function
should be familiar from EGR 260 Circuit
Analysis. A brief summary is shown below.
Definition u(t) unit step function
where and u(t) is represented by the graph shown
below.
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Lecture 8 EGR 261 Signals and Systems
Two useful skills using unit step
functions 1) Determining the function that
represents a given graph Approach Represent
each unique portion of the function using unit
step windows 2) Graphing a function specified
by unit steps Approach As each unit step
function turns on, graph the cumulative
function.
Example Represent x(t) shown below using unit
step functions.

Example Graph the function x(t) 2tu(t)
(4-2t)u(t-2) (8-2t)u(t-4) (2t-12)u(t-6)
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Lecture 8 EGR 261 Signals and Systems
Impulse function, ?(t) Recall that the impulse
function was defined earlier (and is repeated
here). ?(t) impulse function (also called the
Dirac delta function) The impulse function is
defined as Graphically ?(t) is shown as
Illustration To illustrate the concept that the
area under ?(t) 1 (not the height 1), consider
the function f(t)
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Lecture 8 EGR 261 Signals and Systems
Important relationships related to the impulse
function If a function ?(t) is continuous at t
0 and since ?(t) 0 for t ? 0, then
?(t)?(t) ?(0)?(t)
So the result is an impulse of strength
?(0). Illustration
Similarly, since ?(t - T) 0 for t ? T, then
?(t)?(t - T) ?(T)?(t - T)
So the result is an impulse of strength
?(T). Illustration
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Lecture 8 EGR 261 Signals and Systems
Sifting Property Since we have just seen that
?(t)?(t) ?(0)?(t) it follows that
(Sifting Property)
(Note that we will see more of this useful
property)
Similarly
Proof
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Lecture 8 EGR 261 Signals and Systems
Example Evaluate each integral below
Relationship between u(t) and ?(t) We can use
integration by parts to show that
But the sifting property states that
This yields the following important result
or
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Lecture 8 EGR 261 Signals and Systems
Exponential function, est Recall that s complex
frequency where s ? jw so est e(?
jw)t e?tejwt e?tcos(wt) jsin(wt)
(Eq. 1) Also note that since s ? - jw (the
complex conjugate of s) then est e(? - jw)t
e?te-jwt e?tcos(wt) - jsin(wt) (Eq.
2) Combining Eq. 1 and Eq. 2 above
yields e?tcos(wt) ½est est
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Lecture 8 EGR 261 Signals and Systems

• Functions represented by est
• Note that est represents four types of functions
  (show the form of each and sketch)
• Constants
• Exponential functions
• Sinusoids
• Exponentially varying sinusoids