32 Exponential Fourier Series EFS

1

• 3-2 Exponential Fourier Series (EFS)
• With compact trigonometric FS we obtain

• How to directly determine ?

2

• Therefore,

• Example 3.6 (p. 208)

3

• 3-2-1 Exponential Fourier Spectra
• is a complex number

• Magnitude spectrum
• Angle spectrum
• For the series in Example 3.6, we have (see the
  whiteboard)

4

• Negative frequency
• The spectrum at means that
  the component exists
• The component exists because
• The Bandwidth of a signal is the difference
  between the highest and lowest (non-negative)
  frequencies of its spectrum
• It is more readily to use the trigonometric form
  to look at the bandwidth

5

• Example 3.8 (p. 212)
• Eqs. (3.80) and (3.81) are instructive for many
  issues in signal processing

6

• Parsevals Theorem (for FS)
• Recall Trigonometric Fourier series expression

• The power (Ex. 1.2, p. 55)
• Therefore

• The power of a periodic signal is the sum of the
  powers of its FS components

7

• Recall Exponential Fourier series expression

• The power
• Therefore
• Plotting shows how powers
  of the basis functions is distributed with
  frequency
• This is called the power spectrum

8

• 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)

9

• Consider a non-periodic f(t) below

T

• is periodical
• Applying a limiting process, we have

10

• Recall Fourier Series of

• In the limit, as

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

11

• 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

12

• 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