Fourier Transform Saravanan Vijayakumaran sarva@ee.iitb.ac.in - - PowerPoint PPT Presentation

Fourier Transform

Saravanan Vijayakumaran sarva@ee.iitb.ac.in

Department of Electrical Engineering Indian Institute of Technology Bombay

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Definition

• Fourier transform of a signal s(t)

S(f) = ∞

−∞

s(t)e−j2πft dt

• Inverse Fourier transform

s(t) = ∞

−∞

S(f)e j2πft df

• Notation

s(t) ↔ S(f)

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Properties of Fourier Transform

• Linearity

as1(t) + bs2(t) ↔ aS1(f) + bS2(f)

• Duality

S(t) ↔ s(−f)

• Conjugation in time corresponds to conjugation and

reflection in frequency, and vice versa s∗(t) ↔ S∗(−f) s∗(−t) ↔ S∗(f)

• Real-valued signals have conjugate symmetric Fourier

transforms s(t) = s∗(t) = ⇒ S(f) = S∗(−f)

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Properties of Fourier Transform

• Time scaling

s(at) ↔ 1 |a|S f a

• Time shift

s(t − t0) ↔ S(f)e−j2πft0

• Modulation

s(t)e j2πf0t ↔ S(f − f0)

• Convolution

s1(t) ∗ s2(t) ↔ S1(f)S2(f)

• Multiplication

s1(t)s2(t) ↔ S1(f) ∗ S2(f)

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Fourier Transforms using Dirac Function

• DC Signal

1 ↔ δ(f)

• Complex Exponential

e j2πfct ↔ δ(f − fc)

• Sinusoidal Functions

cos(2πfct) ↔ 1 2[δ(f − fc) + δ(f + fc)] sin(2πfct) ↔ 1 2j [δ(f − fc) − δ(f + fc)]

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Properties of Fourier Transform

• Parseval’s identity

∞

−∞

s1(t)s∗

2(t) dt =

∞

−∞

S1(f)S∗

2(f) df

• Energy is independent of representation

Es = s2 = ∞

−∞

|s(t)|2 dt = ∞

−∞

|S(f)|2 df

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Signum Function

sgn(t) =    +1, t > 0 0, t = 0 −1, t < 0

−2 −1 1 2 −2 −1 1 2 t

Fourier Transform sgn(t) ↔ 1 jπf

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Signum Function

g(t) =    e−at, t > 0 0, t = 0 −eat, t < 0

−2 −1 1 2 −2 −1 1 2 t a=1.5 a=0.5

sgn(t) = lim

a→0+ g(t)

G(f) = −j4πf a2 + (2πf)2

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Unit Step Function

u(t) =    1, t > 0

1 2,

t = 0 0, t < 0

−2 −1 1 2 −2 −1 1 2 t

Fourier Transform u(t) = 1 2[sgn(t) + 1] u(t) ↔ 1 j2πf + 1 2δ(f)

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Rectangular Pulse

I[− T

2 , T 2 ](t) =

• 1,

|t| ≤ T

2

0, |t| > T

2

− T

2 T 2

−1 1 2 t

I[− T

2 , T 2 ] (t) ↔ Tsinc(fT)

− 4

T

− 2

T 2 T 4 T

−T T 2T f

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References

• pp 13 — 14, Section 2.1, Fundamentals of Digital

Communication, Upamanyu Madhow, 2008

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