Lecture 2 Signals and Systems (I)

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Lecture 2Signals and Systems (I)

• Principles of Communications
• Fall 2008
• NCTU EE Tzu-Hsien Sang

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Outlines

• Signal Models Classifications
• Signal Space Orthogonal Basis
• Fourier Series Transform
• Power Spectral Density Correlation
• Signals Linear Systems
• Sampling Theory
• DFT FFT

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Signal Models and Classifications

• The first step to knowledge classify things.
• What is a signal?
• Usually we think of one-dimensional signals can
  our scheme extend to higher dimensions?
• How about representing something uncertain, say,
  a noise?
• Random variables/processes mathematical models
  for signals

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• Deterministic signals completely specified
  functions of time. Predictable, no uncertainty,
  e.g. , with A and
  are fixed.
• Random signals (stochastic signals) take on
  random values at any given time instant and
  characterized by pdf (probability density
  function). Not completely predictable, with
  uncertainty, e.g. x(n) dice value at the n-th
  toss.

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• Periodic vs. Aperiodic signals
• Phasors and why are we obsessed with sinusoids?

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Singularity functions (they are not functions at
all!!!)

• Unit impulse function
• Defined by
• It defines a precise sample point of x(t) at the
  incidence t0
• Basic function for linearly constructing a time
  signal
• Properties

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• What is precisely? some intuitive ways of
  imaging it
• Unit step funcgtion

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Energy Signals Power Signals

• Energy
• Power
• Energy signals iff
• Power signals iff
• Examples

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• If x(t) is periodic, then it is meaningless to
  find its energy we only need to check its power.
• Noise is often persistent and is often a power
  signal.
• Deterministic and aperiodic signals are often
  energy signals.
• A realizable LTI system can be represented by a
  signal and mostly is a energy signal.
• Power measure is useful for signal and noise
  analysis.
• The energy and power classifications of signals
  are mutually exclusive (cannot be both at the
  same time). But a signal can be neither energy
  nor power signal.

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Signal Spaces Orthogonal Basis

• The consequence of linearity N-dimensional basis
  vectors
• Degree of freedom and independence For example,
  in geometry, any 2-D vector can be decomposed
  into components along two orthogonal basis
  vectors, (or expanded by these two vectors)
• Meaning of linear in linear algebra

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• A general function can also be expanded by a set
  of basis functions (in an approximation sense)
• or more feasibly
• Define the inner product as (arbitrarily)
•  
• and the basis is orthogonal
• then

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• Examples cosine waves
• What good are they?
• Taylors expansion orthogonal basis?
• Using calculus can show that function
  approximation expansion by orthogonal basis
  functions is an optimal LSE approximation.
• Is there a very good set of orthogonal basis
  functions?
• Concept and relationship of spectrum, bandwidth
  and infinite continuous basis functions.

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Fourier Series Fourier Transform

• Fourier Series
• Sinusoids (when?)
• If x(t) is real,

Notice the integral bounds.
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• Or, use both cosine and sine
• with
• Yet another formulation

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• Some Properties
• Linearity If x(t) ? ak and y(t) ? bk
• then Ax(t)By(t) ? Aak Bbk 
• Time Reversal
• If x(t) ? ak then x(-t) ? a-k
• Time Shifting
• Time Scaling
• x(at) ? ak But the fundamental frequency
  changes
• Multiplication x(t)y(t) ?
• Conjugation and Conjugate Symmetry
• x(t) ? ak and x(t) ? a-k
• If x(t) is real ? a-k ak

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• Parsevals Theorem
• Power in time domain power in frequency domain

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Some Examples
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Extension to Aperiodic Signals

• Aperiodic signals can be viewed as having periods
  that are infinitely long.
• Rigorous treatments are way beyond our abilities.
  Lets use our intuition.
• If the period is infinitely long. What can we say
  about the fundamental frequency.
• The number of basis functions would leap from
  countably infinite to uncountably infinite.
• The synthesis is now an integration..
• Remember, both cases are purely mathematical
  construction.

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The wisdom is to tell the minute differences
between similar-looking things and to find the
common features of seemingly-unrelated ones
Fourier Series Fourier Transform
Good orthogonal basis functions for a periodic function Intuitively, basis functions should be also periodic. Intuitively, periods of the basis functions should be equal to the period or integer fractions of the target signal. Fourier found that sinusoidal functions are good and smooth functions to expand a periodic function.   Good orthogonal basis functions for a aperiodic function Already know sinusoidal functions are good choice. Sinusoidal components should not be in a fundamental harmonic relationship. Aperiodic signals are mostly finite duration. Consider the aperiodic function as a special case of periodic function with infinite period  
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Synthesis analysis (reconstruction projection) Given periodic with period , and , it can be synthesized as Spectra coefficient, spectra amplitude response   Before synthesizing it, we must first analyze it first and find out . By orthogonality   Synthesis analysis (reconstruction projection) Given aperiodic with period , and , we can synthesize it as   By orthogonality   Hence,  
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Frequency components Decompose a periodic signal into countable frequency components. Has a fundamental freq. and many other harmonics. Discrete line spectra   Power Spectral Density and (by Parsevals equality) Frequency components Decompose an aperiodic signal into uncountable frequency components No fundamental freq. and contain all possible freq. Continuous spectral density     Energy Spectral Density and    
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In real basis functions   note that for real x(t). Exercises!
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Conditions of Existence
Expansion by orthogonal basis functions can be shown is equivalent to finding using the LSE (or MSE) cost function Would as ? This requires square integrable condition (for the power signal) and not necessarily Dirichlets conditions finite no. of finite discontinuities finite no. of finite max min. absolute integrable Dirichlets condition implies convergence almost everywhere, except at some discontinuities. Expansion by orthogonal basis functions can be shown is equivalent to finding using the LSE (or MSE) cost function Would as ? This requires square integrable condition (for the energy signal) Dirichlets conditions finite no. of finite discontinuities finite no. of finite max min. absolute integrable Dirichlets condition implies convergence almost everywhere, except at some discontinuities.