3F3 2 Brief Review of Fourier Analysis

8/7/2019 3F3 2 Brief Review of Fourier Analysis

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Brief Review of Fourier Analysis

Elena Punskayawww-sigproc.eng.cam.ac.uk/~op205

Some material adapted from courses byProf. Simon Godsill, Dr. Arnaud Doucet,

Dr. Malcolm Macleod and Prof. Peter Rayner


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Time domain

Example: speech recognition

tiny segment

sound /a/as in father

sound /i/as in see

difficult to differentiate

between different sounds

in time domain


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How do we hear?

www.uptodate.com

Inner Ear

Cochlea spiral of tissue with liquidand thousands of tiny hairs thatgradually get smaller

Each hair is connected to the nerve

The longer hair resonate with lowerfrequencies, the shorter hairresonate with higher frequencies

Thus the time-domain air pressuresignal is transformed into frequencyspectrum, which is then processedby the brain

Our ear is a Natural Fourier Transform Analyser!


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Fouriers Discovery

Jean Baptiste Fourier showed that anysignal could be made up by adding together

a series of pure tones (sine wave) of

appropriate amplitude and phase

(Recall from 1A Maths)

Fourier Series

for periodic

square wave

infinitely large numberof sine waves is

required


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Prism Analogy

Analogy:

a prism which splitswhite light into aspectrum of colors

white light consists of allfrequencies mixedtogether

the prism breaks themapart so we can see theseparate frequencies

Whitelight

Spectrumof colours

FourierTransform

Signal Spectrum


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Signal Spectrum

Every signal has a frequency spectrum.

the signal defines the spectrumthe spectrum defines the signal

We can move back and forth between

the time domain and the frequency

domain without losing information


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Time domain / Frequency domain

Some signals are easier to visualise in thefrequency domain Some signals are easier to visualise in the

time domain

Some signals are easier to define in the timedomain (amount of information needed)

Some signals are easier to define in thefrequency domain (amount of informationneeded)

Fourier Transform is most usefultool for DSP


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Fourier Transforms Examples

peaks correspond tothe resonances ofthe vocal tract shape

they can be used todifferentiate betweensounds

in logarithmis units of dB

sound /i/as in see

signal spectrum

cosine

added higher

frequencycomponent

sound /a/

as in father

in logarithmis units of dB

t

t

t

t

Back to our sound recognition problem:


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Discrete Time Fourier Transform (DTFT)

What about sampled signal?

The DTFT is defined as the Fourier transform of the sampled

signal. Define the sampled signal in the usual way:

Take Fourier transform directly

using the sifting property of the -function to reach the last line


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Discrete Time Fourier Transform Signal Samples

Notethat this expression known as DTFT is a periodic function ofthe frequency usually written as

The signal sample values may be expressed in terms of DTFT bynoting that the equation above has the form of Fourier series (as a

function of) and hence the sampled signal can be obtained directly

as

[You can show this for yourself by first noting that (*) is a complex Fourier

series with coefficients however it is also covered in one of Part IBExamples Papers]


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Computing DTFT on Digital Computer

The DTFT

expresses the spectrum of a sampled signal in terms ofthe signal samples but is not computable on a digital

computer for two reasons:

1. The frequency variable is continuous.2. The summation involves an infinite number of

samples.


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Overcoming problems with computing DTFT

The problems with computing DTFT on a digitalcomputer can be overcome by:

Step 1. Evaluating the DTFT at a finitecollection of discrete frequencies.

no undesirable consequences, anyfrequency of interest can always beincluded in the collection

Step 2. Performing the summation over afinite number of data points

does have consequences sincesignals are generally not of finite duration


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The Discrete Fourier Transform (DFT)

The discrete set of frequencies chosen is arbitrary. However, since the

DTFT is periodic we generally choose a uniformly spaced grid of Nfrequencies covering the range Tfrom 0 to 2. If the summation is then

truncated to justNdata points we get the DFT

The inverse DFT can be used to obtain the sampled signal values from the

DFT: multiply each side by and sum overp=0 toN-1

Orthogonality property of complex exponentials

isNifn=q and 0 otherwise


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The Discrete Fourier Transform Pair


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is periodic, for eachp

is periodic, for each n

for real data

[You should check that you can show these results from first principles]

Properties of the Discrete Fourier Transform (DFT)


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DTFT Normalised Frequency

Please also note the DTFT and IDTFT pair is often written as:

The assumption here is that is a normalized frequency

We will adopt this notation for majority of the slides.

=2f = 2(f/fs) - normalized frequency(rad/sample)

f - cycles per second

fs - samples per second

f/fs - cycles per sample

3F3 2 Brief Review of Fourier Analysis

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