ECE 3231

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Communication Principles

• ECE 3231
• Semester II, 2005/2006
• Lecture 2 Introduction to Signals

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Outline

• Introduction to Signals
• Classification of Signals
• Some Useful Signal Operations
• Unit Impulse Function ?(t)
• Unit Step Function u(t)
• Phasor Signals and Spectra
• Parsevals Theorem
• Reading Assignment for Lecture 2
• Assignment

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1. Introduction to Signals
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• Signal
• A single-valued function of time that conveys
  information for every instant of time there is a
  unique value of the function (This value may be a
  real number, in which case we have a real-valued
  signal, or it may be a complex number, in which
  case we have a complex-valued signal)
• Usually time is the independent variable, which
  is real-valued
• e.g. telephone or television signal (signal with
  a function of space, e.g. electrical charge
  distributed over a surface)
• 4 different method of dividing signals into two
  classes

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1. Classification of Signals
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• (a) Periodic/aperiodic signals
• (b) Deterministic/random signals
• (c) Energy/power signals
• (d) Analogue/digital signals

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1(a) Periodic/aperiodic signals
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• A periodic signal g(t) is a function that
  satisfies the
• condition
• for all t, where t denotes time and T0 is a
  constant.
• The smallest value of T0 that satisfies this
  condition is called the period of g(t).
• The period T0 defines the duration of one
  complete cycle of g(t).
• Any signal for which there is no value of T0 to
  satisfy the above equation is called a
  nonperiodic or aperiodic signal.

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1(a) Periodic/aperiodic signals (cont.)
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1(b) Deterministic/random signals
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• A deterministic signal is a signal about which
  there is no uncertainty with respect to its value
  at any time.
• Deterministic signals may be modeled as
  completely specified functions of time.
• A random signal is a signal about which there is
  uncertainty before its actual occurrence.
• A random signal may be viewed as belonging to an
  ensemble of signals, with each signal in the
  collection having a different waveform and a
  certain probability of occurrence.

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1(c) Energy/power signals
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• A signal may represent a voltage or a current.
• The instantaneous power dissipated in a given
  resistor at a given time
• or
• If R 1?, then both equation take on the same
  mathematical form (In signal analysis, it is
  customary to work with a 1-ohm resistor, so that
  regardless of whether a given signal g(t)
  represents a voltage or a current, the
  instantaneous power associated with the signal
  can be represented as follows)

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1(c) Energy/power signals (cont.)
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• Thus, the total energy of a signal g(t) can be
  defined as
• And its average power
• The signal g(t) is a energy signal if and only if
  the total energy of the signal satisfies the
  condition 0ltElt?
• The signal g(t) is a power signal if and only if
  the average power of the signal satisfies the
  condition 0ltPlt?
• The energy and power classifications of signals
  are mutually exclusive (an energy signal has zero
  average power, whereas a power signal has
  infinite energy)
• Usually, periodic signals and random signals are
  power signals, whereas signals that are both
  deterministic and nonperiodic are energy signals
• The measure of energy or power is
  indicative of the energy (or power) capability of
  the signal, and not the actual energy. Thus, the
  concepts of conservation of energy should not be
  applied. These measures are merely convenient
  indicators of the signal size.

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1(d) Analogue/digital signals
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• An analogue signal is a signal with an amplitude
  that varies continuously for all time i.e. both
  amplitude and time are continuous over their
  respective intervals. E.g. when a physical
  waveform such as an acoustic wave or a light wave
  is converted into an electrical signal by means
  of a transducer
• A discrete-time/digital signal is defined only a
  discrete instants of time. Thus, the independent
  variable takes on only discrete values, which are
  usually uniformly spaced. Discrete-time signals
  are described as sequences of samples that may
  take on a continuum of values. E.g. the output of
  a digital computer.
• An analogue signal may be converted into digital
  form by sampling in time, then quantizing and
  coding.

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2. Some Useful Signal Operations
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• Time Shifting
• Time Scaling
• Time Inversion/Reversal

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2(a) Time Shifting
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2(b) Time Scaling
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2(c) Time Inversion/Reversal
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3. Unit Impulse Function ?(t)
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• One of the most important function in the study
  of signals and systems. First defined by P.A.M.
  Dirac when
• Thus
• Multiplication of a Function by an Impulse
• When ?(t) is multiplied by a function ?(t) that
  is
• known to be continuous at t0, we obtain
• ?(t) ?(t) ?(0) ?(t)
• because the impulse only exist at t0 and the
• value of ?(t) at t0 is ?(0).

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3. Unit Impulse Function ?(t) (cont.)
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• Similarly if ?(t) is multiplied by an impulse
  ?(t-T), then
• ?(t) ?(t-T)?(T) ?(t-T)

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3. Unit Impulse Function ?(t) (cont.)
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• Sampling Property of ?(t)
• provided that ?(t) is continuous at t0. This
  means that the area under the product of a
  function with an impulse ?(t) is equal to the
  value of that function at the instant where the
  unit impulse is located.
• This property is VERY IMPORTANT and useful, and
  is known as sampling or sifting property of the
  unit impulse. Thus, it follows that

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4. Unit Step Function u(t)
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• Defined as
• A signal that does not start before t0 is
• called a causal signal
• Consequently,
• Thus it can be seen that

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5. Phasor Signals and Spectra
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• A sinusoidal or ac waveform can be represented
  as
• Where
• A peak value or amplitude
• ?0 radian frequency
• phase angle (represent the fact that
• the peak has been shifted away
• from the time origin and occurs at
• t -?/?0
• Cyclical frequency,

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5. Phasor Signals and Spectra (cont.)
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• A sinusoid is usually represented by a complex
  exponential or phasor form
• Eulers Theorem
• where and ? is an arbitrary
  angle
• Let , then any sinusoid can
  be written as the real part of a complex
  exponential

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5. Phasor Signals and Spectra (cont.)
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• The diagram shows a phasor representation of a
  signal because the term inside the
• brackets may be viewed as a rotating vector in a
  complex plane whose axes are the real
• and imaginary parts.
• The phasor has length A, rotate counter-clockwise
• at a rate f0 revolution per second, and at time
• t 0 makes an angle ? with respect to the
  positive
• real axis.
• The three parameters that completely specifies a
• phasor
• amplitude
• phase angle and
• rotational frequency.

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5. Phasor Signals and Spectra (cont.)
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• To describe the same phasor in the frequency
  domain, the corresponding
• amplitude and phase must be associated with the
  particular frequency, f0,
• giving us the LINE SPECTRA. (Line spectra have
  great conceptual value
• when extended to more complicated signals.)

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5. Phasor Signals and Spectra (cont.)
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• Given that
• Where z is any complex quantity with complex
  conjugate z.
• Thus
• The corresponding phasor diagram ?

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5. Phasor Signals and Spectra (cont.)
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• The phasor diagram consists of two phasors with
  equal lengths but opposite angles and direction
  of
• rotation.
• The phasor sum always fall along the real axis to
  yield
• The line spectrum is two-sided since it must
  include negative frequencies to allow for the
  opposite
• rotational directions, and one-half of the
  original amplitude is associated with each of the
  two
• frequencies
• The amplitude spectrum has even symmetry while
  the phase spectrum has odd symmetry.

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Four conventions regarding Line Spectra
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• In all spectral drawings, the independent
  variable will be cyclical frequency f Hertz an
  specific frequency will be identified by a
  subscript (e.g. f0)
• Phase angle will be measured w.r.t. cosine waves
  or, equivalently w.r.t. the positive real axis of
  the phasor diagram (convert sine waves to cosine
  via the identity
  )
• Regard amplitude as always being a positive
  quantity. When negative sign appears, they must
  be absorbed in the phase using
• Phase angles are usually expressed in degrees.

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6. Parsevals Theorem
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• The energy of the sum of orthogonal signals is
  equal to the sum of their energies
• Thus, the energy of the right-hand side of the
  above equation is the sum of energies of the
  individual orthogonal components

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Reading Assignment
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• Textbook
• B.P. Lathi, Modern Digital and Analog
  Communication Systems, 3rd Edition
• Chapter 2 Introduction to Signals
• pg. 15-17, 20-30

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Assignment 1

• Due on 23th December 2005 (Submit during lab
  session)
• 2.1-1, 2.1-5, 2.1-8
• 2.3-2, 2.3-3
• 2.4-1(a,c,e)

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Thank you

• End of Lecture 2
• Any Questions?