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CONTINUOUS FOURIER TRANSFORM Lecture Notes Ahmet Ademoglu, PhD

Ahmet Ademoglu, PhD CONTINUOUS FOURIER TRANSFORM Lecture Notes

Continuous Fourier Transform and its Inverse Pair Special case of Laplace Transform X (s) =

R∞

x(t)e −st dt

−∞

where s = jω

Z∞ X (jω) =

x(t)e −jωt dt

−∞

1 x(t) = 2π

Z∞

X (jω)e jωt dω

−∞ F

x(t) ←→ X (jω)

Ahmet Ademoglu, PhD CONTINUOUS FOURIER TRANSFORM Lecture Notes

Example x(t) = e −at u(t)

a > 0, X (ω) =

x(t) = e −a|t| u(t)

1 a + jω

a > 0, X (ω) =

Ahmet Ademoglu, PhD CONTINUOUS FOURIER TRANSFORM Lecture Notes

a2

2a + ω2

Example

x(t) =

  1 

0

X (ω) = 2

Ahmet Ademoglu, PhD CONTINUOUS FOURIER TRANSFORM Lecture Notes

|t| < T1 , |t| > T1 sin(ωT1 ) ω

Example

X (ω) =

  1 

0

x(t) =

Ahmet Ademoglu, PhD CONTINUOUS FOURIER TRANSFORM Lecture Notes

|ω| < W , |ω| > W sin(Wt) πt

Inverse Relation between the Time and Frequency Domains

Ahmet Ademoglu, PhD CONTINUOUS FOURIER TRANSFORM Lecture Notes

Fourier Transform of Periodic Signals

X (ω) = 2πδ(ω − ω0 ) Z∞

1 x(t) = 2π

2πδ(ω − ω0 )e jωt dω = e jω0 t

−∞

Example: x(t) = sin(ω0 t) X (ω) =

π π δ(ω − ω0 ) − δ(ω + ω0 ) j j

Ahmet Ademoglu, PhD CONTINUOUS FOURIER TRANSFORM Lecture Notes

Properties of Continuous Fourier Transform F

Linearity : ax(t) + by (t) ←→ aX (ω) + bY (ω) F

Time Shifting: x(t − t0 ) ←→ e −jωt0 X (ω) Example: x(t) = 12 x1 (t − 2.5) − x2(t − 2.5)

X (ω) = e −j5ω/2

n

sin(ω/2+2sin(3ω/2)) ω

Ahmet Ademoglu, PhD CONTINUOUS FOURIER TRANSFORM Lecture Notes

o

Properties of Continuous Fourier Transform

Conjugation and Conjugate Symmetry x ∗ (t) = X ∗ (ω) If x(t) is real, X (ω) = X ∗ (ω) If x(t) is even, X (−ω) = X (ω) If x(t) is odd, X (−ω) = −X (ω) F

Even{x(t)} ←→ Re{X (ω)} F

Odd{x(t)} ←→ jIm{X (ω)} Example: at −at u(t) x(t) = e −a|t| = e at u(−t) + e −at u(t) = 2 e u(−t)+e 2 1 X (ω) = 2Re{ a+jω } = a22a +ω 2

Ahmet Ademoglu, PhD CONTINUOUS FOURIER TRANSFORM Lecture Notes

Properties of Continuous Fourier Transform F

Convolution: x(t) ∗ y (t) ←→ X (ω)Y (ω) F

Differentiation: dx(t) dt ←→ jωX (ω) Rt F 1 x(τ )dτ ←→ jω X (ω) + πX (0)δ(ω) Integration : Rt

−∞ R∞

x(τ )dτ =

−∞

F

x(τ )u(t − τ )dτ ←→ X (ω)U(ω)

−∞ F

2 u(t) = 12 (sgn(t) + 1) ←→ 12 ( jω + 2πδ(ω)) t R F 1 x(τ )dτ ←→ X (ω)( jω + πδ(ω))

−∞

F

Time and Frequency Scaling : x(at) ←→ Ahmet Ademoglu, PhD CONTINUOUS FOURIER TRANSFORM Lecture Notes

1 ω |a| X ( a )

Properties of Continuous Fourier Transform

F

F

2 2 |ω| Example: If e −|t| ←→ 1+ω 2 then 1+t 2 ←→ 2πe R∞ 2 jωt R∞ 2 jωt 1 −|t| = e dω←→2πe e dω e −|t| = 2π 2 1+ω 1+ω 2

2πe −|ω| =

−∞ R∞

−∞

−∞

2 e −jωt dt 1+t 2

Ahmet Ademoglu, PhD CONTINUOUS FOURIER TRANSFORM Lecture Notes

Properties of Continuous Fourier Transform F

Differentiation in Frequency: −jtx(t) ←→

dX (ω) dω

F

Shift in Frequency: e jω0 t x(t) ←→ X (ω − ω0 ) R∞ R∞ 1 |x(t)|2 dt = 2π Parseval’s Relation: |X (ω)|2 dω −∞

F

z(t) = x(t)y (t) ←→ Z (ω) = F

1 2π

R∞

−∞

X (θ)Y (ω − θ)dθ

−∞

Example: p(t) = cos(ω0 t) ←→ P(ω) = πδ(ω − ω0 ) + πδ(ω − ω0 )

Ahmet Ademoglu, PhD CONTINUOUS FOURIER TRANSFORM Lecture Notes

Systems Characterized by Linear Differential Equations Example: 1

dy (t) dt

+ ay (t) = x(t) with a > 0 H(ω) =

1 a + jω

h(t) = e −at u(t) 2

d 2 y (t) dt 2

+ 4 dydt(t) + 3y (t) =

dx(t) dt

+ 2x(t) with a > 0

x(t) = e −at u(t), X (ω) =

1 a + jω

1/4 1/2 1/4 + − 1 + jω (1 + jω)2 3 + jω 1 1 1 y (t) = ( e −t + te −t − e −3t )u(t) 4 2 4

Y (ω) =

Ahmet Ademoglu, PhD CONTINUOUS FOURIER TRANSFORM Lecture Notes