Teleteknikk Digital kommunikasjon Forelesning 19.01.2004

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Teleteknikk Digital kommunikasjon Forelesning
19.01.2004

• Klassekontakt for 2ET Tore Jønvik
  tore.jonvik_at_iu.hio.no eller tore-erling.jonvik_at_tel
  enor.com Mobil 90199176.
• Møter med tillitsvalg og faglærere
• Hvem er tillitsvalgt i 2 ET?
• Obligatorisk laboratoriearbeid
• Godkjent lab en forutsetning for å gå opp til
  eksamen
• Må ha godkjent Elektronikk lab for å kunne delta
  på lab
• Alle som skal ha lab må melde seg til Henry
  Johansen innen 30.1.2004
• Telelab starter i uke 7 (10 og 11 februar)
• Forstette med Fourier rekker

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Fourier series

• The representation of a time domain signal by a
  summation of sine or cosine components is usually
  referred to as the spectrum of the waveform. It
  is traditional to draw the spectrum as discrete
  lines on a graph, with the position of the lines
  on the x-axis representing the frequency of the
  component, and the height of the line
  representing the amplitude.

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Fourier series

• It is common practice to represent only the
  absolute value or magnitude of each component in
  the spectrum, however, one must not forget that
  in fact each term in the Fourier Series expansion
  could be made up of both sine and cosine terms at
  any given frequency and hence both magnitude and
  phase are required for a complete representation
  of the time domain signal
• http//users.ece.gatech.edu/slabaugh/java/fourier
  /fourier.html

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Trigonometric expansion

• Where

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Complex exponential expansion
where

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If we now consider the Fourier series expansion
of a train of pulses representing successive data
bits, we find that the amplitudes of the
frequency components are all constrained by a
general spectral envelope which passes through
zero at multiples of the data pulse width .
This spectral envelope is given by the equation
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Example 1 Find the trigonometrical Fourier
series expansion for the following waveform
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If the waveform is shifted by T / 4, the Fourier
series expansion would be represented by cosine
terms as follows
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Example 2 Find the trigonometrical Fourier series
expansion for the following waveform
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The general expression for the trigonometrical
Fourier series expansion of a function x(t) is
given by
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It is interesting to note that all of the Fourier
series components for a train of pulses (data
bits) with width t (except for the dc component)
are bounded by the sinc(nt / T) envelope.
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Example 3 Find the complex Fourier series
expansion for the following waveform
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Factors affecting signal bandwidth
It is perhaps instinctively obvious that a
waveform which has sharp transitions in the time
domain will have a much higher harmonic content
than one with smooth transitions. This is because
the sharp changes in waveform can only be
constructed from a large number of low-level high
frequency sinusoids in a Fourier series
expansion.Hence, modulation formats that
possess smooth pulse shapes or smooth phase
transitions between symbol states are to be
favoured when bandwidth is limited
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Not only is the shape of the waveform important
in determining the amplitude of the frequency
components within the Fourier series expansion,
but the width of the data pulses also plays an
important role.As can be seen here, reducing
the width of the pulse but keeping the period of
the waveform constant results in an increase in
the level of the higher harmonics at the expense
of the lower harmonic levels. Overall, the energy
content in the waveform has also gone down and so
the combined power of the harmonics must also be
reduced.In the limit, as the pulse width tends
to zero, that is, a delta function, we can expect
the amplitude of each harmonic to approach a
constant yet diminishing value.
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Spectrum of a data pulse

• As the fundamental period of the time waveform
  increases, the fundamental frequency of the
  Fourier series components making up the waveform
  decreases and the harmonics become more closely
  spaced. In the limit, as the time between
  pulses approaches infinity, the harmonic spacing
  becomes infinitely small and the spectrum is in
  fact continuous and bounded by the sinc function
  as shown.A single pulse is not of course a
  periodic time function and the spectrum cannot
  strictly be evaluated using the Fourier series
  expansion. Instead the more general Fourier
  transform should be used.

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The Fourier transform
 
The Fourier transform is widely used for
converting any mathematical description of a time
domain waveform into the frequency domain
equivalent. As such it can be viewed as a
generalization of the Fourier series expansion.
There is also an inverse Fourier transform which
will convert from the frequency domain to the
time domain.The Fourier transform X(f) of a
time function x(t) is defined as
                                                  
   and the inverse Fourier transform is
given by