# Lecture 10 - Fourier Transform by dfsiopmhy6

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```									                                                                                                                    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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