4 Fourier Transform

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• 4 Fourier Transform
• 4-1 Formulations
• If f(t) is periodic, it can be represented by
  Fourier series (FS)
• If f(t) is not periodic, it can only be
  represented by an FS over a specified interval
• Outside the specified interval, the FS does NOT
  necessarily represent f(t) (Recall Figure 3.7 (a)
  (b) in p. 192)
• We want to find a way to represent a
  non-periodic f(t) everywhere
• Solution Fourier Transform (FT)

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• Consider a non-periodic f(t) below

T

• is periodical
• Applying a limiting process, we have

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• Recall Fourier Series of

• In the limit, as

• Namely, takes on all real values of
  frequencies. Then,

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• Consequently,

• (1) is called the Fourier transform
• (2) is called the inverse Fourier transform
• In general, is a complex-valued
  function of Thus, it has the magnitude and
  phase spectra

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• Existence of FT and Examples
• Sufficient condition
• f(t) is an energy signal
• In any finite interval, f(t) may have only a
  finite number of maximums and minimums, and a
  finite number of finite discontinuities
• If these conditions are satisfied, then
• At all continuous points, the RHS of formulation
  (2) converges to f(t)
• At the discontinuous point the RHS of (2)
  converges to

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• Example 4.1
• If a signal does not satisfy the sufficient
  conditions (e.g., power signals), its FT may
  still exist
• 4.2 FT of Some Basic Functions
• Unit gate function
• Unit triangle function

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• Interpolation function

• Example 4.2
• Example 4.3 FT of the impulse signal

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• Example 4.4 FT of f(t) 1

• Example 4.5, 4.6

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• Example 4.7

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• 4-3 Properties of FT
• 4-3-1 Linearity

• Example

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• 4-3-2 Symmetry

• Proof (see whiteboard)

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• Exercise E4.5 (p. 255)
• Want to show

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• 4-3-3 Time Scaling

• Proof
• Time expansion implies frequency compression
  (see Fig. 4.18)
• In other words If f(t) is wider, then its
  spectrum is narrower, and vice versa

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• Let a -1, then we have
• Time and Frequency Inversion (p. 256)

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• 4-3-4 Time Shifting

• Proof
• This is to say Shifting in time results in a
  phase shift in frequency domain, but does not
  change the amplitude spectrum
• More questions

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• 4-3-5 Frequency Shifting

• Proof

• Extension 1

• Extension 2

• The multiplication of by a sinusoid of
  frequency in the time domain results in a
  shifting by in the frequency domain

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• This kind of multiplication is known as
  amplitude modulation
• The sinusoid is the carrier
• is the modulating signal
• is the modulated signal
• How to sketch? Observe that

• and act as envelopes for