Fourier Transform Properties

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Fourier Transform Properties - PowerPoint PPT Presentation

Fourier Transform Properties. t. F ( w ). w. - 6 p. - 4 p. - 2 p. 2 p. 4 p. 6 p. t. t. t. t. t. t. 0. Duality. Forward/inverse transforms are similar Example: rect(t/ t )  t sinc( w t / 2) Apply duality t sinc(t t /2)  2 p rect(- w / t )

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Fourier Transform Properties

t

F(w)

w

-6p

-4p

-2p

2p

4p

6p

t

t

t

t

t

t

0

Duality
• Forward/inverse transforms are similar
• Example: rect(t/t)  t sinc(wt / 2)

Apply duality t sinc(t t/2)  2 p rect(-w/t)

rect(·) is even t sinc(t t /2)  2 p rect(w/t)

f(t)

1

t

-t/2

0

t/2

Scaling
• Given and that a 0

|a| > 1: compress time axis, expand frequency axis

|a| < 1: expand time axis, compress frequency axis

• Extent in time domain is inversely proportional to extent in frequency domain (a.k.a bandwidth)

f(t) is wider  spectrum is narrower

f(t) is narrower  spectrum is wider

Shifting in Time
• Shift in time

Does not change magnitude of the Fourier transform

Shifts phase of Fourier transform by -wt0(so t0 is the slope of the linear phase)

• Derivation

Let u = t – t0, so du = dt and integration limits stay same

F(w)

1

w

-w1

w1

0

Y(w)

1/2 F(w+w0)

1/2 F(w-w0)

1/2

w

-w0 - w1

-w0 + w1

w0 - w1

w0 + w1

0

-w0

w0

Sinusoidal Amplitude Modulation
• Example: y(t) = f(t) cos(w0 t)

f(t) is an ideal lowpass signal

Assume w1 << w0

• Demodulation (i.e. recovery of f(t) from y(t)) is modulation followed by lowpass filtering
• Similar derivation for modulation with sin(w0 t)
Conditions

f(t)  0 when |t| 

f(t) is differentiable

Derivation of property:Given f(t) F(w)

Time Differentiation Property
Summary
• Definition of Fourier Transform
• Two ways to find Fourier Transform

Use definition

Use transform pairs and properties