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6: Fourier Transform

6: Fourier Transform•Fourier Series as

T→ ∞

•Fourier Transform •Fourier Transform

Examples

•Dirac Delta Function •Dirac Delta Function:

Scaling and Translation

•Dirac Delta Function:

Products and Integrals

•Periodic Signals •Duality •Time Shifting and Scaling •Gaussian Pulse •Summary E1.10 Fourier Series and Transforms (2014-5559)Fourier Transform: 6 - 1 / 12

Fourier Series asT→ ∞

6: Fourier Transform•Fourier Series as

T→ ∞•Fourier Transform

•Fourier Transform

Examples

•Dirac Delta Function •Dirac Delta Function:

Scaling and Translation

•Dirac Delta Function:

Products and Integrals

•Periodic Signals •Duality •Time Shifting and Scaling •Gaussian Pulse •Summary E1.10 Fourier Series and Transforms (2014-5559)Fourier Transform: 6 - 2 / 12

Fourier Series:

u(t) =?∞n=-∞Unei2πnFt

Fourier Series asT→ ∞

6: Fourier Transform•Fourier Series as

T→ ∞•Fourier Transform

•Fourier Transform

Examples

•Dirac Delta Function •Dirac Delta Function:

Scaling and Translation

•Dirac Delta Function:

Products and Integrals

•Periodic Signals •Duality •Time Shifting and Scaling •Gaussian Pulse •Summary E1.10 Fourier Series and Transforms (2014-5559)Fourier Transform: 6 - 2 / 12

Fourier Series:

u(t) =?∞n=-∞Unei2πnFt

The harmonic frequencies arenF?nand are spacedF=1

Tapart.

Fourier Series asT→ ∞

6: Fourier Transform•Fourier Series as

T→ ∞•Fourier Transform

•Fourier Transform

Examples

•Dirac Delta Function •Dirac Delta Function:

Scaling and Translation

•Dirac Delta Function:

Products and Integrals

•Periodic Signals •Duality •Time Shifting and Scaling •Gaussian Pulse •Summary E1.10 Fourier Series and Transforms (2014-5559)Fourier Transform: 6 - 2 / 12

Fourier Series:

u(t) =?∞n=-∞Unei2πnFt

The harmonic frequencies arenF?nand are spacedF=1

Tapart.

AsTgets larger, the harmonic spacing becomes smaller. e.g.T= 1s?F= 1Hz

Fourier Series asT→ ∞

6: Fourier Transform•Fourier Series as

T→ ∞•Fourier Transform

•Fourier Transform

Examples

•Dirac Delta Function •Dirac Delta Function:

Scaling and Translation

•Dirac Delta Function:

Products and Integrals

•Periodic Signals •Duality •Time Shifting and Scaling •Gaussian Pulse •Summary E1.10 Fourier Series and Transforms (2014-5559)Fourier Transform: 6 - 2 / 12

Fourier Series:

u(t) =?∞n=-∞Unei2πnFt

The harmonic frequencies arenF?nand are spacedF=1

Tapart.

AsTgets larger, the harmonic spacing becomes smaller. e.g.T= 1s?F= 1Hz

T= 1day?F= 11.57μHz

Fourier Series asT→ ∞

6: Fourier Transform•Fourier Series as

T→ ∞•Fourier Transform

•Fourier Transform

Examples

•Dirac Delta Function •Dirac Delta Function:

Scaling and Translation

•Dirac Delta Function:

Products and Integrals

•Periodic Signals •Duality •Time Shifting and Scaling •Gaussian Pulse •Summary E1.10 Fourier Series and Transforms (2014-5559)Fourier Transform: 6 - 2 / 12

Fourier Series:

u(t) =?∞n=-∞Unei2πnFt

The harmonic frequencies arenF?nand are spacedF=1

Tapart.

AsTgets larger, the harmonic spacing becomes smaller. e.g.T= 1s?F= 1Hz

T= 1day?F= 11.57μHz

IfT→ ∞then the harmonic spacing becomes zero, the sum becomes an integral and we get the

Fourier Transform

u(t) =?+∞ f=-∞U(f)ei2πftdf

Fourier Series asT→ ∞

6: Fourier Transform•Fourier Series as

T→ ∞•Fourier Transform

•Fourier Transform

Examples

•Dirac Delta Function •Dirac Delta Function:

Scaling and Translation

•Dirac Delta Function:

Products and Integrals

•Periodic Signals •Duality •Time Shifting and Scaling •Gaussian Pulse •Summary E1.10 Fourier Series and Transforms (2014-5559)Fourier Transform: 6 - 2 / 12

Fourier Series:

u(t) =?∞n=-∞Unei2πnFt

The harmonic frequencies arenF?nand are spacedF=1

Tapart.

AsTgets larger, the harmonic spacing becomes smaller. e.g.T= 1s?F= 1Hz

T= 1day?F= 11.57μHz

IfT→ ∞then the harmonic spacing becomes zero, the sum becomes an integral and we get the

Fourier Transform

u(t) =?+∞ f=-∞U(f)ei2πftdf

Here,U(f), is the

spectral density ofu(t).

Fourier Series asT→ ∞

6: Fourier Transform•Fourier Series as

T→ ∞•Fourier Transform

•Fourier Transform

Examples

•Dirac Delta Function •Dirac Delta Function:

Scaling and Translation

•Dirac Delta Function:

Products and Integrals

•Periodic Signals •Duality •Time Shifting and Scaling •Gaussian Pulse •Summary E1.10 Fourier Series and Transforms (2014-5559)Fourier Transform: 6 - 2 / 12

Fourier Series:

u(t) =?∞n=-∞Unei2πnFt

The harmonic frequencies arenF?nand are spacedF=1

Tapart.

AsTgets larger, the harmonic spacing becomes smaller. e.g.T= 1s?F= 1Hz

T= 1day?F= 11.57μHz

IfT→ ∞then the harmonic spacing becomes zero, the sum becomes an integral and we get the

Fourier Transform

u(t) =?+∞ f=-∞U(f)ei2πftdf

Here,U(f), is the

spectral density ofu(t).

•U(f)is a

continuous function off.

Fourier Series asT→ ∞

6: Fourier Transform•Fourier Series as

T→ ∞•Fourier Transform

•Fourier Transform

Examples

•Dirac Delta Function •Dirac Delta Function:

Scaling and Translation

•Dirac Delta Function:

Products and Integrals

•Periodic Signals •Duality •Time Shifting and Scaling •Gaussian Pulse •Summary E1.10 Fourier Series and Transforms (2014-5559)Fourier Transform: 6 - 2 / 12

Fourier Series:

u(t) =?∞n=-∞Unei2πnFt

The harmonic frequencies arenF?nand are spacedF=1

Tapart.

AsTgets larger, the harmonic spacing becomes smaller. e.g.T= 1s?F= 1Hz

T= 1day?F= 11.57μHz

IfT→ ∞then the harmonic spacing becomes zero, the sum becomes an integral and we get the

Fourier Transform

u(t) =?+∞ f=-∞U(f)ei2πftdf

Here,U(f), is the

spectral density ofu(t).

•U(f)is a

continuous function off.

•U(f)is

complex-valued

Fourier Series asT→ ∞

6: Fourier Transform•Fourier Series as

T→ ∞•Fourier Transform

•Fourier Transform

Examples

•Dirac Delta Function •Dirac Delta Function:

Scaling and Translation

•Dirac Delta Function:

Products and Integrals

•Periodic Signals •Duality •Time Shifting and Scaling •Gaussian Pulse •Summary E1.10 Fourier Series and Transforms (2014-5559)Fourier Transform: 6 - 2 / 12

Fourier Series:

u(t) =?∞n=-∞Unei2πnFt

The harmonic frequencies arenF?nand are spacedF=1

Tapart.

AsTgets larger, the harmonic spacing becomes smaller. e.g.T= 1s?F= 1Hz

T= 1day?F= 11.57μHz

IfT→ ∞then the harmonic spacing becomes zero, the sum becomes an integral and we get the

Fourier Transform

u(t) =?+∞ f=-∞U(f)ei2πftdf

Here,U(f), is the

spectral density ofu(t).

•U(f)is a

continuous function off.

•U(f)is

complex-valued

•u(t)real?U(f)is

conjugate symmetric ?U(-f) =U(f)?.

Fourier Series asT→ ∞

6: Fourier Transform•Fourier Series as

T→ ∞•Fourier Transform

•Fourier Transform

Examples

•Dirac Delta Function •Dirac Delta Function:

Scaling and Translation

•Dirac Delta Function:

Products and Integrals

•Periodic Signals •Duality •Time Shifting and Scaling •Gaussian Pulse •Summary E1.10 Fourier Series and Transforms (2014-5559)Fourier Transform: 6 - 2 / 12

Fourier Series:

u(t) =?∞n=-∞Unei2πnFt

The harmonic frequencies arenF?nand are spacedF=1

Tapart.

AsTgets larger, the harmonic spacing becomes smaller. e.g.T= 1s?F= 1Hz

T= 1day?F= 11.57μHz

IfT→ ∞then the harmonic spacing becomes zero, the sum becomes an integral and we get the

Fourier Transform

u(t) =?+∞ f=-∞U(f)ei2πftdf

Here,U(f), is the

spectral density ofu(t).

•U(f)is a

continuous function off.

•U(f)is

complex-valued

•u(t)real?U(f)is

conjugate symmetric ?U(-f) =U(f)?.

Units:

ifu(t)is in volts, thenU(f)dfmust also be in volts ?U(f)is in volts/Hz (hence " spectral density

Fourier Transform

6: Fourier Transform

•Fourier Series as T→ ∞•Fourier Transform•Fourier Transform

Examples

•Dirac Delta Function •Dirac Delta Function:

Scaling and Translation

•Dirac Delta Function:

Products and Integrals

•Periodic Signals •Duality •Time Shifting and Scaling •Gaussian Pulse •Summary E1.10 Fourier Series and Transforms (2014-5559)Fourier Transform: 6 - 3 / 12

Fourier Series:

u(t) =?∞n=-∞Unei2πnFt

Fourier Transform

6: Fourier Transform

•Fourier Series as T→ ∞•Fourier Transform•Fourier Transform

Examples

•Dirac Delta Function •Dirac Delta Function:

Scaling and Translation

•Dirac Delta Function:

Products and Integrals

•Periodic Signals •Duality •Time Shifting and Scaling •Gaussian Pulse •Summary E1.10 Fourier Series and Transforms (2014-5559)Fourier Transform: 6 - 3 / 12

Fourier Series:

u(t) =?∞n=-∞Unei2πnFt The summation is over a set of equally spaced frequencies f n=nFwhere the spacing between them isΔf=F=1 T.

Fourier Transform

6: Fourier Transform

•Fourier Series as T→ ∞•Fourier Transform•Fourier Transform

Examples

•Dirac Delta Function •Dirac Delta Function:

Scaling and Translation

•Dirac Delta Function:

Products and Integrals

•Periodic Signals •Duality •Time Shifting and Scaling •Gaussian Pulse •Summary E1.10 Fourier Series and Transforms (2014-5559)Fourier Transform: 6 - 3 / 12

Fourier Series:

u(t) =?∞n=-∞Unei2πnFt The summation is over a set of equally spaced frequencies f n=nFwhere the spacing between them isΔf=F=1 T. U n=?u(t)e-i2πnFt?

Fourier Transform

6: Fourier Transform

•Fourier Series as T→ ∞•Fourier Transform•Fourier Transform

Examples

•Dirac Delta Function •Dirac Delta Function:

Scaling and Translation

•Dirac Delta Function:

Products and Integrals

•Periodic Signals •Duality •Time Shifting and Scaling •Gaussian Pulse •Summary E1.10 Fourier Series and Transforms (2014-5559)Fourier Transform: 6 - 3 / 12

Fourier Series:

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