I. Previously on IET

23
I. Previously on IET

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I. Previously on IET. Complex Exponential Function. Im-Axis. ω. Re-Axis. Complex Exponential Function. Im-Axis. ω. Re-Axis. Complex Exponential Function. Im-Axis. ω. Re-Axis. Complex Exponential Function. Im-Axis. ω. Re-Axis. Complex Exponential Function. Im-Axis. ω. Re-Axis. - PowerPoint PPT Presentation

Transcript of I. Previously on IET

Page 1: I. Previously on IET

I. Previously on IET

Page 2: I. Previously on IET

© Tallal Elshabrawy 2

Complex Exponential Function

( ) ( )tωsinj+tωcos=e tωj

Re-Axis

Im-Axis

ω

Page 3: I. Previously on IET

© Tallal Elshabrawy 3

Complex Exponential Function

( ) ( )tωsinj+tωcos=e tωj

Re-Axis

Im-Axis

ω

Page 4: I. Previously on IET

© Tallal Elshabrawy 4

Complex Exponential Function

( ) ( )tωsinj+tωcos=e tωj

Re-Axis

Im-Axis

ω

Page 5: I. Previously on IET

© Tallal Elshabrawy 5

Complex Exponential Function

( ) ( )tωsinj+tωcos=e tωj

Re-Axis

Im-Axis

ω

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© Tallal Elshabrawy 6

Complex Exponential Function

( ) ( )tωsinj+tωcos=e tωj

Re-Axis

Im-Axis

ω

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© Tallal Elshabrawy 7

Complex Exponential Function

( ) ( )tωsinj+tωcos=e tωj

Re-Axis

Im-Axis

ω

Page 8: I. Previously on IET

© Tallal Elshabrawy 8

Complex Exponential Function

( ) ( )tωsinj+tωcos=e tωj

Re-Axis

Im-Axis

ω

Page 9: I. Previously on IET

© Tallal Elshabrawy 9

Complex Exponential Function

( ) ( )tωsinj+tωcos=e tωj

Re-Axis

Im-Axis

ω

Page 10: I. Previously on IET

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Complex Exponential Function

( ) ( )tωsinj+tωcos=e tωj

Re-Axis

Im-Axis

ω

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Complex Exponential Function

( ) ( )tωsinj+tωcos=e tωj

Re-Axis

Im-Axis

ω

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The Fourier TransformRepresenting functions in terms of complex exponentials with different frequencies

jωt j2πftG f g t e dt g t e dt

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The Fourier Transform (Cosine Function)

Re-Axis

Im-Axis

-ωRe-Axis

Im-Axis

ω +

jωte cos ωt jsin ωt jωte cos ωt jsin ωt

jωt jωte e 2cos ωt

+

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The Fourier Transform (Cosine Function)

Re-Axis

Im-Axis

Re-Axis

Im-Axis

+

+

ω-ω

jωte cos ωt jsin ωt jωte cos ωt jsin ωt

jωt jωte e 2cos ωt

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The Fourier Transform (Cosine Function)

Re-Axis

Im-Axis

Re-Axis

Im-Axis

+

+

ω-ω

jωte cos ωt jsin ωt jωte cos ωt jsin ωt

jωt jωte e 2cos ωt

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The Fourier Transform (Cosine Function)

Re-Axis

Im-Axis

Re-Axis

Im-Axis

+

+

ω-ω

jωte cos ωt jsin ωt jωte cos ωt jsin ωt

jωt jωte e 2cos ωt

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The Fourier Transform (Cosine Function)

Re-Axis

Im-Axis

Re-Axis

Im-Axis

+

+

ω-ω

jωte cos ωt jsin ωt jωte cos ωt jsin ωt

jωt jωte e 2cos ωt

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The Fourier Transform (Sine Function)

Re-Axis

Im-Axis

Re-Axis

Im-Axis

- jωt jωte e 2jsin ωt

jωte cos ωt jsin ωt jωte cos ωt jsin ωt

ω-ω

-

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The Fourier Transform (Sine Function)

Re-Axis

Im-Axis

Re-Axis

Im-Axis

+ jωt jωte e 2jsin ωt

jωte cos ωt jsin ωt

ω-ω

+

jωte cos ωt jsin ωt

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The Fourier Transform (Sine Function)

Re-Axis

Im-Axis

Re-Axis

Im-Axis

- jωt jωtj e e 2sin ωt

jωtje jcos ωt sin ωt jωtje jcos ωt sin ωt

ω -ω

-

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The Fourier Transform (Sine Function)

Re-AxisRe-Axis

Im-Axis

+ jωt jωtj e e 2sin ωt

jωtje jcos ωt sin ωt jωtje jcos ωt sin ωt

ω

+

Im-Axis

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Fourier Transform of Sinusoids

Notes A real value for the coefficients in the frequency domain means that the starting

point for rotation is on the real axis An Imaginary value for the coefficients in the frequency domain means that the

starting point for rotation is on the imaginary axis

jωt jωte ecos ωt

2

jωt jωte e

sin ωt j2

0 ω-ω

1/21/2

0 ω-ω-j(1/2)

j(1/2)

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© Tallal Elshabrawy 23

Fourier Transform of Real Valued Functions

A real-valued function in time implies thatG(-f) = G*(f)

Re-Axis

Im-Axisω1

Re-Axis

Im-Axis

ω2

Re-Axis

Im-Axisωn

Re-Axis

Im-Axis

-ω1

Re-Axis

Im-Axis-ω2

Re-Axis

Im-Axis

-ωn