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Lecture 10 - Fourier Transform

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Lecture 10 - Fourier Transform Powered By Docstoc
					                                                                                                                    Definition of Fourier Transform

                                                                                                   !    The forward and inverse Fourier Transform are defined for aperiodic
                                                                                                         signal as:
                                 Lecture 10

                         Fourier Transform
                                 (Lathi 7.1-7.3)

                                                                                                   !    Already covered in Year 1 Communication course (Lecture 5).
                                                                                                   !    Fourier series is used for periodic signals.
                                      Peter Cheung
                     Department of Electrical & Electronic Engineering
                                Imperial College London

                URL: www.ee.imperial.ac.uk/pcheung/teaching/ee2_signals
                           E-mail: p.cheung@imperial.ac.uk                                                                                                                      L7.1 p678

PYKC Feb-8-10                       E2.5 Signals & Linear Systems           Lecture 10 Slide 1   PYKC Feb-8-10                       E2.5 Signals & Linear Systems         Lecture 10 Slide 2




         Connection between Fourier Transform and Laplace
                            Transform
                                                                                                                      Define three useful functions

  !    Compare Fourier Transform:                                                                  !    A unit rectangular window (also called a unit gate) function rect(x):



  !    With Laplace Transform:

  !    Setting s = j! in this equation yield:                                                      !    A unit triangle function !(x):




  !    Is it true that:                     ?
  !    Yes only if x(t) is absolutely integrable, i.e. has finite energy:
                                                                                                   !    Interpolation function sinc(x):
                                                                                                                        or
                                                                             L7.2-1 p697                                                                                    L7.2-1 p687

PYKC Feb-8-10                       E2.5 Signals & Linear Systems           Lecture 10 Slide 3   PYKC Feb-8-10                       E2.5 Signals & Linear Systems         Lecture 10 Slide 4
                      More about sinc(x) function                                                       Fourier Transform of                          x(t) = rect(t/#)

!    sinc(x) is an even function of x.                                                  !    Evaluation:
!    sinc(x) = 0 when sin(x) = 0
      except when x=0, i.e. x = ±",
      ±2", ±3"…..                                                                       !    Since rect(t/#) = 1 for -#/2 < t < #/2 and 0 otherwise
!    sinc(0) = 1 (derived with
      L’Hôpital’s rule)
!    sinc(x) is the product of an
      oscillating signal sin(x) and a
      monotonically decreasing
      function 1/x. Therefore it is a                                                                                                                       Bandwidth $ 2"/#
      damping oscillation with period
      of 2" with amplitude
      decreasing as 1/x.


                                                                     L7.2 p688                                                                                             L7.2 p689

PYKC Feb-8-10                      E2.5 Signals & Linear Systems   Lecture 10 Slide 5   PYKC Feb-8-10                      E2.5 Signals & Linear Systems                 Lecture 10 Slide 6




          Fourier Transform of unit impulse x(t) = %(t)                                                    Inverse Fourier Transform of %(&)

!    Using the sampling property of the impulse, we get:                                !    Using the sampling property of the impulse, we get:




!    IMPORTANT – Unit impulse contains COMPONENT AT EVERY FREQUENCY.                    !    Spectrum of a constant (i.e. d.c.) signal x(t)=1 is an impulse 2"%(&).

                                                                                                                      or




                                                                     L7.2 p691                                                                                             L7.2 p691

PYKC Feb-8-10                      E2.5 Signals & Linear Systems   Lecture 10 Slide 7   PYKC Feb-8-10                      E2.5 Signals & Linear Systems                 Lecture 10 Slide 8
                 Inverse Fourier Transform of %(& - &0)                                                       Fourier Transform of everlasting sinusoid cos &0t

!    Using the sampling property of the impulse, we get:                                          !    Remember Euler formula:

                                                                                                  !    Use results from slide 9, we get:

!    Spectrum of an everlasting exponential ej&0t is a single impulse at &=&0.
                                                                                                  !    Spectrum of cosine signal has two impulses at positive and negative
                                                                                                        frequencies.

                                      or



                                     and


                                                                               L7.2 p692                                                                                     L7.2 p693

PYKC Feb-8-10                       E2.5 Signals & Linear Systems            Lecture 10 Slide 9   PYKC Feb-8-10                      E2.5 Signals & Linear Systems       Lecture 10 Slide 10




                Fourier Transform of any periodic signal                                                          Fourier Transform of a unit impulse train

!    Fourier series of a periodic signal x(t) with period T0 is given by:                         !    Consider an impulse train

                                                                                                  !    The Fourier series of this impulse train can be shown to be:

!    Take Fourier transform of both sides, we get:
                                                                                                  !    Therefore using results from the last slide (slide 11), we get:



!    This is rather obvious!




                                                                               L7.2 p693                                                                                     L7.2 p694

PYKC Feb-8-10                       E2.5 Signals & Linear Systems           Lecture 10 Slide 11   PYKC Feb-8-10                      E2.5 Signals & Linear Systems       Lecture 10 Slide 12
                Fourier Transform Table (1)                                                   Fourier Transform Table (2)




                                                            L7.3 p702                                                                     L7.3 p702

PYKC Feb-8-10           E2.5 Signals & Linear Systems   Lecture 10 Slide 13   PYKC Feb-8-10           E2.5 Signals & Linear Systems   Lecture 10 Slide 14




                Fourier Transform Table (3)




                                                            L7.3 p702

PYKC Feb-8-10           E2.5 Signals & Linear Systems   Lecture 10 Slide 15

				
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