The Fourier Transform

Dr John MitchellDepartment of Electronic & Electrical EngineeringUniversity College LondonEmail: j.mitchell@ee.ucl.ac.uk

Reminder of the Fourier Series

Fourier: Jean Baptiste Joseph Fourier (1768-1830), orphaned at eight, was a child genius. By 16 he was a math teacher, and at 20 he joined Napoleon on his Egyptian campaign as the first scientific advisor.

Any periodic signal can be described as a sum of sinusoids at different frequencies

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Sums of sinusoidal signals

A square wave may be synthesised as a sum of sinusoidal components comprising the fundamental component and all odd harmonics, with amplitudes decreasing with harmonic number

Here we show the sum of just the fundamental and the third harmonic; the emergence of a square wave is clear even from this most limited sum!

t

x(t)

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Fourier Series

A periodic signal is one that repeats at equal intervals of T. Formally we can say that:

where n is any integer

( ) ( )v t v t nT=

2 /

/ 22 /

/ 2

( )

1 ( )

j nt Tn

nT

j nt Tn

T

x t c e

c x t e dtT

=

=

=

4

Consider the square pulse again

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1 - 1T

F(n0)= Sinc (n/T)

T=2

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-50 -40 -30 -20 -10 0 10 20 30 40 50-0.5

0

0.5

1

n0

Four

ier c

oeffi

cient

c n

T

T=2

T=4

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-50 -40 -30 -20 -10 0 10 20 30 40 50-0.5

0

0.5

1

n0

Four

ier c

oeffi

cien

t cn

T

T=4

T=8

8

-50 -40 -30 -20 -10 0 10 20 30 40 50-0.5

0

0.5

1

n0

Four

ier c

oeffi

cien

t cn

T

T=8

What happens as T ?

The spacing between frequency components gets smaller.

When T= then the spectrum becomes a continuum.

Note: the shape of the curve seems to be independent of T

we can replace n0 by a continuous variable and the summation by integration

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The Fourier Transform

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We denote F and F-1as the Fourier transform and the inverse Fourier transform respectively. Notice that Fourier transform and the inverse Fourier transform are similar in form, albeit with the exception of the 2 scale factor and the different sign in the complex exponential.

However, we like to work with f

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Fourier Transform

Inverse Fourier Transform

Definitions The forward and inverse transforms relate a time signal x(t) and its (Fourier) spectrum X(f)

Commonly x(t) is real and X(f) complex, although in general both x(t) and X(f) may be complex.

Note the use of f as the frequency variable, rather than =2f

Produce a continuous spectrum as they have no well-defined period

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Examples

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Sinc Spectrum T=1

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Delta Function

Imagine a pulse

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t

t

What happens as t

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Properties of the Dirac Delta Function

Shifting property

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Fourier Transform and the Dirac Delta

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t

(t)1

f

Fourier Transform and the Dirac Delta Function

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(f)1

t f

22

(f-F)

f

Fourier Pairs - Cosine

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cos(2Ft) 12 f + F( )+12 f F( )

12 t + T( )+

12 t T( ) cos(2fT)

t f

-T T

t fF-F

Fourier Pairs - Rectangle

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rect(t) sinc(f)t f

t f

sinc(t) rect(f)

Exercise

What is the Fourier Transform of an Decaying Exponential Function?

Find the FT of

where a>0

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Exercise

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t

x(t)

e-at

Exercise

27

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a=4

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MATLAB Codef=linspace(-5,5);v=abs(1./(a+(i.*pi.*f*2)));phase=angle(1./(a+(i.*pi.*f*2)));figure(1)plot(f,v)figure(2)plot(f,phase)

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A Gaussian Shape

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-5 -4 -3 -2 -1 0 1 2 3 4 5

0

0.2

0.4

tf(t)

=(2

)-1/2

exp

(-t2 /

2)

Gaussian Shape

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the Fourier transform of a Gaussian is also a Gaussian.

If f(t) is narrow, then its spectrum F(), is wide. Similarly if f(t) is wide, then its spectrum F() is narrow.

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