Signal Processing

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Signal Processing

• Mike Doggett
• Staffordshire University

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FOURIER SERIES (F.S.)

• Fouriers Theorem states that any periodic
  function of time, f(t), (i.e. a periodic signal)
  can be expressed as a Fourier Series consisting
  of
• A DC component the average value of f(t).
• A component at a fundamental frequency and
  harmonically related components, collectively the
  AC components.
• ie f(t) DC AC components.

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• The Fourier Series for a periodic signal may be
  expressed by

AC components Fundamental frequency (n1) at ?
rads per second.
DC or average component
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• a0, an and bn are coefficients given by

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• NOTE
• The function must be periodic, i.e. f(t)
  f(tT). Periodic time T. Frequency f Hz.
• If f(t) f(-t) the function is EVEN and only
  cosine terms (and a0) will be present in the F.S.

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• If f(t) -f(-t) the function is ODD and only
  sine terms (and a0) will be present in the F.S.
• The coefficients an and bn are the amplitudes of
  the sinusoidal components.
• For example, in general, an cos n?t

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• The component at the lowest frequency (excluding
  the DC component) is when n 1,
• i.e. a1 cos ?t
• This is called the fundamental or first
  harmonic. The component for n 2 is called the
  second harmonic, n 3 is the third harmonic and
  so on.

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FOURIER SERIES FOR A UNIPOLAR RECTANGULAR PULSE
TRAIN

• Consider the rectangular pulse train below.

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• Pulse width t seconds, periodic time T seconds,
  amplitude E volts (unipolar).
• As shown, the function is chosen to be even, ie
  f(t) f(-t) so that a DC term and cosine terms
  only will be present in the F.S.
• We define f(t) E,
• And f(t) 0, elsewhere

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• As noted, the Fourier Series for a periodic
  signal may be expressed by
• Applying to find

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• The an coefficients are given by

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• Since sin(-A) -sinA

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• In this case it may be show that bn 0 (because
  the choice of t 0 gives an even function).
• Hence
• and

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• Simplifying, by noting
• substituting back into the F.S. equation

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• Fourier Series for a unipolar pulse train.
• But NOTE, it is more usual to convert this to a
  Sinc function.
• ie Sinc(X)

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• Note the trick, i.e multiply by
• This reduces to

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• Hence
• This is an important result, the F.S. for a
  periodic pulse train and gives a spectrum of the
  form shown below

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FOURIER SERIES (F.S.) Review

• We have discussed that the general FS for an Even
  function is
• Fourier Series for a unipolar pulse train.

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• The Sinc function gives an envelope for the
  amplitudes of the harmonics.
• The Sinc function, in conjunction with
• gives the amplitudes of the harmonics.
• Note that Sinc(0) 1. (As an exercise, justify
  this statement).

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• The amplitudes of the harmonic components are
  given by
• To calculate, it is usually easier to use the
  form

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• The harmonics occur at frequencies n? radians per
  second.
• We normally prefer to think of frequency in
  Hertz, and since ? 2pf, we can consider
  harmonics at frequencies nf Hz.
• The periodic time, T, and frequency are related
• by f Hz.

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Rules of Thumb

• The following rules of thumb may be deduced for
  a pulse train, illustrated in the waveform below.

E volts
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• Harmonics occur at intervals of f
• OR f, 2f, 3f, etc.
• Nulls occur at intervals of
• If x is integer, then nulls occur
  every xth
• harmonic.

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• For example if T 10 ms and t 2.5 ms, then
• 4 and there will be nulls at the 4th
• harmonic, the 8th harmonic, the 12th harmonic
  and so on at every 4th harmonic.
• As t is reduced, ie the pulse gets narrower, the
  first and subsequent nulls move to a higher
  frequency.
• As T increases, ie the pulse frequency gets
  lower, the first harmonic moves to a lower
  frequency and the spacing between the harmonics
  reduces, ie they move closer together.

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Exercise Q1. Label the axes and draw the pulse
waveform corresponding to the spectrum below.
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Q2. What pulse characteristic would give this
spectrum?
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• Q3.
• Suppose a triac firing circuit produces a narrow
  pulse, with 1 nanosecond pulse width, and a
  repetition rate of 50 pulses per second.
• What is the frequency spacing between the
  harmonics?
• At what frequency is the first null in the
  spectrum?
• Why might this be a nuisance for radio reception?

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COMPLEX FOURIER SERIES

• Up until now we have been considering
  trigonometric Fourier Series.
• An alternative way of expressing f(t) is in terms
  of complex quantities, using the relationships

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• Since the trig form of F.S. is
• , then this may be written in the complex form

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• The complex F.S. may be written as
• where

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• When n 0, C0 ej0 C0 is the average value.
• n 1, n 2, n 3 etc represent pairs of
  harmonics.
• These are general for any periodic function.

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• In particular, for a periodic unipolar pulse
  waveform, we have
• OR

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• Hence
• Alternative forms of complex F.S. for pulse train

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Example

• Express the equation below (for a periodic pulse
  train) in complex form.
• NOTE, we change the cos term, We DONT change
  the Sinc term.

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• Since
• By changing the sign of the -n and summing from
  -8 to -1, this may be written as

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• We the have and
• We want
• We need to include the term for n 0 and may
• show that for n 0, the term results.

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• Consider when n 0
• Sinc(0) 1 and ej0 1,
• ie when n 0
• Hence we may write

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• Comments
• Fourier Series apply only to periodic functions.
• Two main forms of F.S., Trig F.S. and Complex
  F.S. which are equivalent.
• Either form may be represented on an Argand
  diagram, and as a single-sided or two-sided
  (bilateral) spectrum.
• The F.S. for a periodic function effectively
  allows a time-domain signal (waveform) to be
  represented in the frequency domain, (spectrum).

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Exercise

• Q1.
• A pulse waveform has a ratio of
• 5.
• Sketch the spectrum up to the second null using
  the rules of thumb.

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• Q2.
• A pulse has a periodic time of T 4 ms and a
  pulse width t 1 ms.
• Sketch, but do not calculate in detail, the
  single-sided and two-sided spectrum up to the
  second null, showing frequencies in Hz.

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• Q3.
• With T 4 ms and t 1 ms as in Q2, now
  calculate, tabulate and sketch the single-sided
  and two-sided spectrum.

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• Q4.
• Convert the trig FS to complex by using the
  substitution