SIGNALS AND INFORMATION THEORY Lecture 1

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SIGNALS AND INFORMATION THEORYLecture 1
Associate Professor PhD Carmen GERIGAN TRANSILVANI
A UNIVERSITY OF BRASOV second semester -
2003/2004
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WHAT IS A SIGNAL? Signals are variables that
carry information.

• EXAPLES
• A telephone or television signal
• (continuous-time)
• Monthly sales of a corporation or daily closing
  prices of a stock market
• (discrete-time)

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SYGNALS AND INFORMATION THEORYThis subject
deals with mathematical methods used to describe
signals.

• SYSTEMS
• Systems process input signals
• to produce output signals.

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OUTLINE

• Clasification of signals
• Some useful functions
• Some useful operations

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CLASSIFICATION OF SIGNALS

• Continuous-time and discrete-time signals
• Analogue and digital signals
• Deterministic and random signals
• Periodic and nonperiodic signals
• Real and complex signals
• Causal and anti-causal signals
• Bounded and unbounded signals
• Even and odd signals
• Power and energy signals

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CLASSIFICATION OF SIGNALS

• Continuous-time and discrete-time signals
• Analogue and digital signals
• Deterministic and random signals
• Periodic and nonperiodic signals
• Real and complex signals
• Causal and anti-causal signals
• Bounded and unbounded signals
• Even and odd signals
• Power and energy signals

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CONTINUOUS-TIMEDISCRETE-TIMESIGNALS

• A signal x(t) is a continuous-time signal if t is
  a continuous variable.
• If t is a discrete variable, x(t) is defined at
  discrete moments of time, then the signal x(t) is
  a discrete-time signal (often identified as a
  sequence of numbers)
• plots

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CLASSIFICATION OF SIGNALS

• Continuous-time and discrete-time signals
• Continuous-time and discrete-time signals
• Analogue and digital signals
• Deterministic and random signals
• Periodic and nonperiodic signals
• Real and complex signals
• Causal and anti-causal signals
• Bounded and unbounded signals
• Even and odd signals
• Power and energy signals

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ANALOGUE AND DIGITAL SIGNALS

• If a continuous-time signal can take on any
  values in continuous time interval, then the
  signal x(t) is called analogue signal.
• If a discrete-time signal can take on only a
  finite number of distinct values, then the signal
  is called a digital signal.
• plot

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CLASSIFICATION OF SIGNALS

• Continuous-time and discrete-time signals
• Analogue and digital signals
• Deterministic and random signals
• Periodic and nonperiodic signals
• Real and complex signals
• Causal and anti-causal signals
• Bounded and unbounded signals
• Even and odd signals
• Power and energy signals

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DETERMINISTIC AND RANDOM SIGNALS

• Deterministic signals are those signals whose
  values are completely specified for any given
  time.
• (subject of the first part - SIGNALS THEORY)
• Random signals are those signals that can take
  random values at any given time.
• (subject of the second part - INFORMATION
  THEORY)

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CLASSIFICATION OF SIGNALS

• Continuous-time and discrete-time signals
• Analogue and digital signals
• Deterministic and random signals
• Periodic and nonperiodic signals
• Real and complex signals
• Causal and anti-causal signals
• Bounded and unbounded signals
• Even and odd signals
• Power and energy signals

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PERIODIC AND NONPERIODIC SIGNALS

• Periodic signals have the property that
• x(tT)x(t)
• for all t. The smallest value of T that
  satisfies the definition is called period.
• If , x(tT) x(t)
• the signal is a nonperiodic or an aperiodic
  signal.

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CLASSIFICATION OF SIGNALS

• Continuous-time and discrete-time signals
• Analogue and digital signals
• Deterministic and random signals
• Periodic and nonperiodic signals
• Real and complex signals
• Causal and anti-causal signals
• Bounded and unbounded signals
• Even and odd signals
• Power and energy signals

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REAL AND COMPLEX SIGNALS

• Signals can be real, imaginary or complex.
• An important class of signals are the complex
  exponetials.
• x(t)est, where s is a complex number.
• xnzn, where z is a complex number.
• Q. Why do we deal with complex signals?
• A. They are often analitically simpler to deal
  with than real signals.

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CLASSIFICATION OF SIGNALS

• Continuous-time and discrete-time signals
• Analogue and digital signals
• Deterministic and random signals
• Periodic and nonperiodic signals
• Real and complex signals
• Causal and anti-causal signals
• Bounded and unbounded signals
• Even and odd signals
• Power and energy signals

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CAUSAL AND ANTI-CAUSAL SIGNALS

• a causal signal is zero for tlt0
• an anti-causal signal is zero for tgt0
• plots

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CLASSIFICATION OF SIGNALS

• Continuous-time and discrete-time signals
• Analogue and digital signals
• Deterministic and random signals
• Periodic and nonperiodic signals
• Real and complex signals
• Causal and anti-causal signals
• Bounded and unbounded signals
• Even and odd signals
• Power and energy signals

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BOUNDED AND UNBOUNDED SIGNALS

• plots

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CLASSIFICATION OF SIGNALS

• Continuous-time and discrete-time signals
• Analogue and digital signals
• Deterministic and random signals
• Periodic and nonperiodic signals
• Real and complex signals
• Causal and anti-causal signals
• Bounded and unbounded signals
• Even and odd signals
• Power and energy signals

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EVEN AND ODD SIGNALS (1)

• An even signal is defined by
• xe(t) xe(-t)
• An odd signal is defined by
• xo(t) -xo(-t)
• plots

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EVEN AND ODD SIGNALS (2)

• Any signal is a sum of unique odd and even
  signals
• x(t) xe(t)xo(t) and x(-t) xe(t)-xo(t)
• yields
• xe(t) 1/2 (x(t) x(-t))
• and
• xo(t) 1/2 (x(t) - x(-t))

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CLASSIFICATION OF SIGNALS

• Continuous-time and discrete-time signals
• Analogue and digital signals
• Deterministic and random signals
• Periodic and nonperiodic signals
• Real and complex signals
• Causal and anti-causal signals
• Bounded and unbounded signals
• Even and odd signals
• Power and energy signals

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POWER AND ENERGY SIGNALS

• A complex signal x(t) is a power signal if the
  average normalised power P is finite
• formula 1
• A complex signal x(t) is an energy signal if the
  normalised energy E is finite.
• formula 2

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USEFUL FUNCTIONS

• Unit impulse function
• Unit step function
• Sampling function
• Sinc function
• Rectangular function
• Triangular function

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UNIT IMPULS FUNCTION

• (dirac delta function)

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UNIT STEP FUNCTION

• u(t)

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SAMPLING FUNCTION
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SINC FUNCTION
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RECTANGULAR FUNCTION
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TRIANGULAR FUNCTION
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SOME USEFUL OPERATIONS

• Time average
• Direct current value (dc)
• Power and energy
• Cross-corelation
• Auto-corelation
• Convolution

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TIME AVERAGE OPERATOR

• any signal
• periodic signal

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DIRECT CURRENT VALUE

• (dc)

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POWER AND ENERGY

• instantaneous power
• average power
• root mean square (rms)
• normalised power
• average normalised power
• total normalised energy
• decibel gain

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CROSS-CORRELATION

• two real-valued power waveforms (any)
• with the same period
• two real-valued energy waveforms
• two complex waveforms
• correlation is a useful operation to measure the
  similarity between two waveforms.

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AUTO-CORRELATION

• a real-valued power waveform
• periodic waveform
• a real-valued energy waveforms
• a complex waveforms

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CONVOLUTION
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