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Lecture 9 Fourier Transform Properties and

Examples

- 3. Basis functions (3 lectures) Concept of basis

function. Fourier series representation of time

functions. Fourier transform and its properties.

Examples, transform of simple time functions. - Specific objectives for today
- Properties of a Fourier transform
- Linearity
- Time shifts
- Differentiation and integration
- Convolution in the frequency domain

Lecture 9 Resources

- Core material
- SaS, OW, C4.3, C4.4
- Background material
- MIT Lectures 8 and 9.

Reminder Fourier Transform

- A signal x(t) and its Fourier transform X(jw) are

related by - This is denoted by
- For example (1)
- Remember that the Fourier transform is a density

function, you must integrate it, rather than

summing up the discrete Fourier series components

Linearity of the Fourier Transform

- If
- and
- Then
- This follows directly from the definition of the

Fourier transform (as the integral operator is

linear). It is easily extended to a linear

combination of an arbitrary number of signals

Time Shifting

- If
- Then
- Proof
- Now replacing t by t-t0
- Recognising this as
- A signal which is shifted in time does not have

its Fourier transform magnitude altered, only a

shift in phase.

Example Linearity Time Shift

- Consider the signal (linear sum of two time

shifted steps) - where x1(t) is of width 1, x2(t) is of width 3,

centred on zero. - Using the rectangular pulse example
- Then using the linearity and time shift Fourier

transform properties

Differentiation Integration

- By differentiating both sides of the Fourier

transform synthesis equation - Therefore
- This is important, because it replaces

differentiation in the time domain with

multiplication in the frequency domain. - Integration is similar
- The impulse term represents the dc or average

value that can result from integration

Example Fourier Transform of a Step Signal

- Lets calculate the Fourier transform X(jw)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(jw) 1
- We can also apply the differentiation property in

reverse

Convolution in the Frequency Domain

- With a bit of work (next) it can show that
- Therefore, to apply convolution in the frequency

domain, we just have to multiply the two

functions. - To solve for the differential/convolution

equation using Fourier transforms - Calculate Fourier transforms of x(t) and h(t)
- Multiply H(jw) by X(jw) to obtain Y(jw)
- Calculate the inverse Fourier transform of Y(jw)
- Multiplication in the frequency domain

corresponds to convolution in the time domain and

vice versa.

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-jwtH(jw), so

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

b?a

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

Lecture 9 Summary

- The Fourier transform is widely used for

designing filters. You can design systems with

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

Lecture 9 Exercises

- Theory
- SaS, OW,
- Q4.6
- Q4.13
- Q3.20, 4.20
- Q4.26
- Q4.31
- Q4.32
- Q4.33

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