Lecture 10
Fourier Transform
(Lathi 7.1-7.3)
Peter Cheung
Department of Electrical & Electronic Engineering
Imperial College London
URL: www.ee.imperial.ac.uk/pcheung/teaching/ee2_signals
E-mail: [email protected]
PYKC 10-Feb-08
E2.5 Signals & Linear Systems
Lecture 10 Slide 1
Definition of Fourier Transform
X
X
X
The forward and inverse Fourier Transform are defined for aperiodic
signal as:
Already covered in Year 1 Communication course (Lecture 5).
Fourier series is used for periodic signals.
L7.1 p678
PYKC 10-Feb-08
E2.5 Signals & Linear Systems
Lecture 10 Slide 2
Connection between Fourier Transform and Laplace
Transform
X
Compare Fourier Transform:
X
With Laplace Transform:
X
Setting s = jω in this equation yield:
X
X
Is it true that:
?
Yes only if x(t) is absolutely integrable, i.e. has finite energy:
L7.2-1 p697
PYKC 10-Feb-08
E2.5 Signals & Linear Systems
Lecture 10 Slide 3
Define three useful functions
X
A unit rectangular window (also called a unit gate) function rect(x):
X
A unit triangle function Δ(x):
X
Interpolation function sinc(x):
or
L7.2-1 p687
PYKC 10-Feb-08
E2.5 Signals & Linear Systems
Lecture 10 Slide 4
More about sinc(x) function
X
X
X
X
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.
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PYKC 10-Feb-08
Lecture 10 Slide 5
E2.5 Signals & Linear Systems
Fourier Transform of
x(t) = rect(t/τ)
X
Evaluation:
X
Since rect(t/τ) = 1 for -τ/2 < t < τ/2 and 0 otherwise
Bandwidth ≈ 2π/τ
⇔
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PYKC 10-Feb-08
E2.5 Signals & Linear Systems
Lecture 10 Slide 6
Fourier Transform of unit impulse x(t) = δ(t)
X
Using the sampling property of the impulse, we get:
X
IMPORTANT – Unit impulse contains COMPONENT AT EVERY FREQUENCY.
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PYKC 10-Feb-08
E2.5 Signals & Linear Systems
Lecture 10 Slide 7
Inverse Fourier Transform of δ(ω)
X
Using the sampling property of the impulse, we get:
X
Spectrum of a constant (i.e. d.c.) signal x(t)=1 is an impulse 2πδ(ω).
or
L7.2 p691
PYKC 10-Feb-08
E2.5 Signals & Linear Systems
Lecture 10 Slide 8
Inverse Fourier Transform of δ(ω - ω0)
X
Using the sampling property of the impulse, we get:
X
Spectrum of an everlasting exponential ejω0t is a single impulse at ω=ω0.
or
and
L7.2 p692
PYKC 10-Feb-08
E2.5 Signals & Linear Systems
Lecture 10 Slide 9
Fourier Transform of everlasting sinusoid cos ω0t
X
Remember Euler formula:
X
Use results from slide 9, we get:
X
Spectrum of cosine signal has two impulses at positive and negative
frequencies.
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PYKC 10-Feb-08
E2.5 Signals & Linear Systems
Lecture 10 Slide 10
Fourier Transform of any periodic signal
X
Fourier series of a periodic signal x(t) with period T0 is given by:
X
Take Fourier transform of both sides, we get:
X
This is rather obvious!
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PYKC 10-Feb-08
E2.5 Signals & Linear Systems
Lecture 10 Slide 11
Fourier Transform of a unit impulse train
X
Consider an impulse train
∞
δ T (t ) = ∑ δ (t − nT0 )
0
X
X
−∞
The Fourier series of this impulse train can be shown to be:
∞
2π
1
δ T0 (t ) = ∑ Dn e jnω0t where ω0 =
and Dn =
T0
T0
−∞
Therefore using results from the last slide (slide 11), we get:
L7.2 p694
PYKC 10-Feb-08
E2.5 Signals & Linear Systems
Lecture 10 Slide 12
Fourier Transform Table (1)
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PYKC 10-Feb-08
E2.5 Signals & Linear Systems
Lecture 10 Slide 13
Fourier Transform Table (2)
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PYKC 10-Feb-08
E2.5 Signals & Linear Systems
Lecture 10 Slide 14
Fourier Transform Table (3)
L7.3 p702
PYKC 10-Feb-08
E2.5 Signals & Linear Systems
Lecture 10 Slide 15