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Lecture 24: CT Fourier Transform

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Symmetry Property: Example

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Property of Duality

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Illustration of the duality property

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More Properties

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Example: Fourier Transform of a Step Signal

Lets calculate the Fourier transform X(j)of x(t) = u(t), making use of the knowledge that:

and noting that:

Taking Fourier transform of both sides

using the integration property. Since G(j) = 1:

We can also apply the differentiation property in reverse

1)()()( jGttgF

tdgtx )()(

)()0()(

)( Gj

jGjX

)(1

)(

j

jX

11)(

)(

j

jdt

tdut

F

Convolution Property

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Proof of Convolution Property

Taking Fourier transforms gives:

Interchanging the order of integration, we have

By the time shift property, the bracketed term is e-jH(j), so

dthxty )()()(

dtedthxjY tj )()()(

ddtethxjY tj)()()(

)()(

)()(

)()()(

jXjH

dexjH

djHexjY

j

j

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Convolution in the Frequency Domain

To solve for the differential/convolution equation using Fourier transforms:

1. Calculate Fourier transforms of x(t) and h(t)

2. Multiply H(j) by X(j) to obtain Y(j)

3. Calculate the inverse Fourier transform of Y(j)

Multiplication in the frequency domain corresponds to convolution in the time domain and vice versa.

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Example 1: Solving an ODE

Consider the LTI system time impulse response

to the input signal

Transforming these signals into the frequency domain

and the frequency response is

to convert this to the time domain, express as partial fractions:

Therefore, the time domain response is:

0)()( btueth bt

0)()( atuetx at

jajX

jbjH

1)(,

1)(

))((

1)(

jajbjY

)(

1

)(

11)(

jbjaabjY ba

)()()( 1 tuetuety btatab

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Example 2: Designing a Low Pass Filter

Lets design a low pass filter:

The impulse response of this filter is the inverse Fourier transform

which is an ideal low pass filter– Non-causal (how to build)– The time-domain oscillations may be undesirable

How to approximate the frequency selection characteristics?

Consider the system with impulse response:

Causal and non-oscillatory time domain response and performs a degree of low pass filtering

c

cjH

||0

||1)(

H(j)

cc

t

tdeth ctjc

c

)sin()( 2

1

jatue

Fat

1)(

Modulation property

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Lecture 27: Summary

The Fourier transform is widely used for designing filters. You can design systems which reject high frequency noise and just retain the low frequency components. This is natural to describe in the frequency domain.

Important properties of the Fourier transform are:

1. Linearity and time shifts

2. Differentiation

3. Convolution

Some operations are simplified in the frequency domain, but there are a number of signals for which the Fourier transform do not exist – this leads naturally onto Laplace transforms

)()( jXj

dt

tdx F

)()()()(*)()( jXjHjYtxthtyF

)()()()( jbYjaXtbytaxF