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
PYKC 8-Feb-11
E2.5 Signals & Linear Systems
L7.1 p678
Lecture 10 Slide 1
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Connection between Fourier Transform and Laplace
Transform
Compare Fourier Transform:
With Laplace Transform:
Setting s = jω in this equation yield:
E2.5 Signals & Linear Systems
Lecture 10 Slide 2
Define three useful functions
A unit rectangular window (also called a unit gate) function rect(x):
A unit triangle function Δ(x):
Is it true that:
?
Yes only if x(t) is absolutely integrable, i.e. has finite energy:
Interpolation function sinc(x):
or
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Lecture 10 Slide 3
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Lecture 10 Slide 4
More about sinc(x) function
Fourier Transform of
sinc(x) is an even function of x.
sinc(x) = 0 when sin(x) = 0
except when x=0, i.e. x = ±π,
±2π, ±3π…..
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
damping oscillation with period
of 2π with amplitude decreasing
as 1/x.
x(t) = rect(t/τ)
Evaluation:
Since rect(t/τ) = 1 for -τ/2 < t < τ/2 and 0 otherwise
Bandwidth ≈ 2π/τ
⇔
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Lecture 10 Slide 5
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Fourier Transform of unit impulse x(t) = δ(t)
E2.5 Signals & Linear Systems
Lecture 10 Slide 6
Inverse Fourier Transform of δ(ω)
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 2πδ(ω).
or
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E2.5 Signals & Linear Systems
Lecture 10 Slide 7
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Lecture 10 Slide 8
Inverse Fourier Transform of δ(ω - ω0)
Fourier Transform of everlasting sinusoid cos ω0t
Using the sampling property of the impulse, we get:
Remember Euler formula:
Use results from slide 9, we get:
Spectrum of an everlasting exponential ejω0t is a single impulse at ω=ω0.
Spectrum of cosine signal has two impulses at positive and negative
frequencies.
or
and
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Lecture 10 Slide 9
E2.5 Signals & Linear Systems
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Fourier Transform of any periodic signal
Lecture 10 Slide 10
E2.5 Signals & Linear Systems
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
∞
δ T (t ) = ∑ δ (t − nT0 )
0
−∞
The Fourier series of this impulse train can be shown to be:
∞
δT (t ) = ∑ Dn e jnω t where ω0 =
0
Take Fourier transform of both sides, we get:
0
−∞
2π
T0
and Dn =
1
T0
Therefore using results from the last slide (slide 11), we get:
This is rather obvious!
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Lecture 10 Slide 11
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Lecture 10 Slide 12
Fourier Transform Table (1)
Fourier Transform Table (2)
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E2.5 Signals & Linear Systems
Lecture 10 Slide 13
Fourier Transform Table (3)
L7.3 p702
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E2.5 Signals & Linear Systems
Lecture 10 Slide 15
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E2.5 Signals & Linear Systems
Lecture 10 Slide 14