FOURIER SERIES Jean-Baptiste Fourier (France, 1768 - 1830

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FOURIER SERIES

• Jean-Baptiste Fourier (France, 1768 - 1830)
  proved that almost any period function can be
  represented as the sum of sinusoids with
  integrally related frequencies.
• The Fourier series is one example of an
  orthogonal set of basis functions, as a very
  important example for engineers.
• Trigonometric form of Fourier Series
• Let us map the functions 1, and by the
  following
• The purpose of this nothing deeper than to map
  the conventional Fourier series onto the notation
  we have derived for orthogonal functions

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• The Fourier series is a special case of the more
  general theory of orthogonal functions.
• Now calculate the value of lm from
• ie
• The value of lm for is simply the
  power in a sinewave (cosine wave)
• The value of l0 is the power in a DC signal of
  unit amplitude.
• Now we can derive immediately the Euler formula
  from equation
• by substituting in the values of and lm from
  the above equations then

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• In the Fourier series, instead of using the term
  a-m as the coefficient of the cosine terms in the
  Fourier expansion we usually use the term am with
  bm reserved for the sine terms.
• The important point to realize here is that the
  Fourier series expansion is only a special case
  of an expansion in terms of orthogonal functions
• There are many other function (e.g. Walsh
  function), so using the Fourier series as an
  example, try and understand the more general
  orthogonal function approach
• When we write a periodic function using a Fourier
  series expansion in terms of a DC term and sine
  and cosine terms the problem which remains is to
  determine the coefficients a0 , am and bm

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• Solving a problem using Fourier series
• Consider a sawtooth wave which rises from -2V to
  2V in a second. It passes through a linear time
  invariant communication channel which does not
  pass frequencies greater than 5.5 Hz.
• What is the power lost in the channel ? Assume
  the output and input impedance are the same. (Use
  sine or cosine Fourier series).

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• The first step is set up the problem
  mathematically.
• The time origin has not been specified in the
  problem.
• Since the system is time invariant it doesn't
  matter when t 0 is located since it is will not
  change the form of the output.
• Choose time t 0 at the center of the rise of
  the sawtooth because it makes the function which
  we now call u(t), into an odd function.
• Since the system is specified in terms of it
  frequency response, i.e. what it will do if a
  sinwave of a given frequency is input, it makes a
  lot of sense to express as a sum of either sines
  and cosines or complex exponentials since as we
  know what happens to these functions.
• If it's a sinewave or cosine wave and has a
  frequency less than 5.5 Hz it is transmitted,
  otherwise it is eliminated.
• The situation with complex exponential is a
  little trickier, if its in the range -5.5,5.5
  Hz then will be transmitted otherwise will be
  eliminated.
• It would do no good to find the response to each
  sinewave individually, because we could not then
  add up these individual response to form the
  total output, because that would require
  superposition to hold.

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• Let us calculate the Fourier series for a
  sawtooth wave or arbitrary period and amplitude.
  Now with the choice of t 0, we can write the
  input as mathematically within the period as
• We don't need to worry that outside the range
  -T/2 lt t lt T/2 the above formula is incorrect
  since all the calculation are done within the
  range -T/2 lt t lt T/2.
• As we strict to the range given the mathematical
  description is identical to the sawtooth and all
  will be well.
• We want the input to written in the form

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• Go back to Euler formula for Fourier series which
  have derived earlier from the general
  orthogonality conditions
• Now for the DC value of the sawtooth
• For an the coefficients of the cosine terms

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• The next step is use integration by parts, i.e.
• Therefore

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• This is a lot of work for 0, when it is fairly
  obvious that the integral of the product of an
  odd function (sawtooth) and an even function (the
  cosine term) is always zero when we integrate
  from -v,v whatever value of v, as shown below
• The procedure for calculating for is almost
  identical, the final answer is
• Now we know a0 , an and bn , we can write down
  the Fourier series representation for u(t) after
  substituting T 1 and A 2
• The series has all the sine terms present, i.e.bm
  is never zero and there are no cosine terms.

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• How can be check that our calculations for u(t)
  is correct ?
• We can calculate the power in the signal by
• and also by Parseval's theorem when applied to
  sine and cosine functions
• Where the 0.5 came from ?
• Remember that is lm which has the basis function
  was equal to sin(t) and the power in the
  sinewave, lm 0.5 .
• Having calculated the Fourier series and having
  checked it using the Parseval's theorem it only
  remains to calculate the power in the first 5
  harmonics, i.e. those with a frequency less than
  5.5 Hz

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• Thus the power transmitted by the channel is
• The power loss is therefore 1.3333 - 1.1863
  0.1470, and the power gain in dB is thus -0.51
  dB.
• The final answer is that the channel attenuates
  the signal by 0.51 dB.