Brief Review of Fourier Analysis
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Transcript of Brief Review of Fourier Analysis
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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