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Brief Review of Fourier Analysis

Elena Punskaya www-sigproc.eng.cam.ac.uk/~op205

Some material adapted from courses by Prof. 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 liquid and thousands of tiny hairs that gradually get smaller

Each hair is connected to the nerve

The longer hair resonate with lower frequencies, the shorter hair resonate with higher frequencies

Thus the time-domain air pressure signal is transformed into frequency spectrum, which is then processed by the brain

Our ear is a Natural Fourier Transform Analyser!

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Fourier’s Discovery

Jean Baptiste Fourier showed that any signal 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 number of sine waves is required

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Fourier Transform

The Fourier transform is an equation to calculate the frequency, amplitude and phase of each sine wave needed to make up any given signal :

(recall from 1B Signal and Data Analysis)

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Prism Analogy

Analogy:

a prism which splits white light into a spectrum of colors

white light consists of all frequencies mixed together

the prism breaks them apart so we can see the separate frequencies

White light

Spectrum of colours

Fourier Transform

Signal Spectrum

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Signal Spectrum

Every signal has a frequency spectrum. • the signal defines the spectrum • the 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 the frequency domain

•  Some signals are easier to visualise in the time domain

•  Some signals are easier to define in the time domain (amount of information needed)

•  Some signals are easier to define in the frequency domain (amount of information needed)

Fourier Transform is most useful tool for DSP

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Fourier Transforms Examples

peaks correspond to the resonances of the vocal tract shape

they can be used to differentiate between sounds

in logarithmis units of dB

sound /i/ as in see

signal spectrum

cosine

added higher frequency component

sound /a/ as in father

in logarithmis units of dB

t

t

t

t

ω

Back to our sound recognition problem:

ω

ω

ω

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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 δ-function to reach the last line

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Discrete Time Fourier Transform – Signal Samples

Note that this expression known as DTFT is a periodic function of the frequency usually written as

The signal sample values may be expressed in terms of DTFT by noting that the equation above has the form of Fourier series (as a function of ω) 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 IB Examples Papers]

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Computing DTFT on Digital Computer

The DTFT

expresses the spectrum of a sampled signal in terms of the signal samples but is not computable on a digital computer for two reasons:

1.  The frequency variable ω 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 digital computer can be overcome by:

Step 1. Evaluating the DTFT at a finite collection of discrete frequencies.

no undesirable consequences, any frequency of interest can always be included in the collection

Step 2. Performing the summation over a finite number of data points

does have consequences since signals 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 N frequencies covering the range ωT from 0 to 2π. If the summation is then truncated to just N data 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 over p=0 to N-1

Orthogonality property of complex exponentials

is N if n=q and 0 otherwise

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The Discrete Fourier Transform Pair

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•  is periodic, for each p

•  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 ω is a normalized frequency

We will adopt this notation for majority of the slides.

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

f - cycles per second

fs - samples per second f/fs - cycles per sample