Lab 7: Fourier analysis and synthesis

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Lab 7 Fourier analysis and synthesis

• Fourier series (periodic phenomena)
• Fourier transform (aperiodic phenomena)
• Fast Fourier transform (FFT)

A powerful analytic tool that has many
applications.
The Fourier Transform and its Applications Brad
G. OsgoodStanford http//see.stanford.edu/see/co
urseinfo.aspx?coll84d174c2-d74f-493d-92ae-c3f45c0
ee091
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Applications of Fourier analysis

• Periodic phenomena (in space and time)
• Physics
• harmonic oscillation
• waves (sounds, lights, and etc.)
• Acoustics
• Image processing
• Crystallography
• Astronomy and earth science.

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Fourier series (formula)
A periodic function f(t) with period T
Any periodic function f(t) with period T can be
mathematically expressed as a sum of harmonics.
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Fourier series (animation)
By Lucas V. Barbosa
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Fourier analysis and synthesis
Fourier analysis (Fourier transform)
Given a periodic function f(t), calculate An and
Bn
In practice, f(t) is a waveform from measurement
Fourier synthesis
Given An and Bn, reconstruct function
f(t). Synthesize (arbitrary) periodic waveforms.
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Symmetry of trigonometric functions
odd
even
if f(t) is an even function
if f(t) is an odd function,
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Fourier analysis (an example)
Square wave
Since f(t) is an odd function
Only need to calculate
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Fourier analysis (another example)
Triangular wave
Since f(t) is an even function
Only need to calculate
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Fourier synthesis
Sum of Fourier components (e.g. square wave)
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135
1357
13579
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Fourier synthesis square wave
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Lab 7 B Sawtooth
Sawtooth wave
Odd function ? An0

• Complete the derivation in your lab report!

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Using complex exponential functions
Using Eular formula, we could rewrite Sine and
Cosine functions as complex exponential
functions, which greatly simply the notation and
algebra.
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Reciprocal relationship and the conjugate
variables
Angular freq.
frequency
Frequency domain
Time domain
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Fourier Transform (FT)
A generalization of Fourier series to analyze
aperiodic functions. E.g. random noise.
Roughly speaking, the fundamental frequency tends
to zero, i.e. ?? 0, or T??. Therefore, we need
All frequencies (i.e. continuum limit) to
describe aperiodic functions. In other words, the
Fourier coefficients become a continuous
function, and the Fourier sum becomes an
integral, i.e. Fourier integral (inverse
FT). Note FT is also applicable for periodic
functions.
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FT and inverse FT
Fourier Transform
Inverse FT
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Application of Fourier transform find out
periodic signal in a noisy background
0.333 Hz
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How does a spectrum analyzer perform Fourier
transform?
Sampling Fast Fourier Transform (FFT)
Most physical signals are continuous function of
time. However, computers can only process
discrete signal. To utilize the powerful
computation capacity of modern computer, we need
to covert a continuous signal to a digital
signal, i.e. sampling.
The hardware for sampling a voltage signal
analog-to-digital convertor (ADC)
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Digitizing continuous signal (sampling)
In signal processing, sampling is the reduction
of a continuous signal to a discrete signal. A
common example is the conversion of a sound wave
(a continuous signal) to a sequence of samples (a
discrete-time signal).
http//en.wikipedia.org/wiki/Signal_processing
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An example cell phone/radio
http//en.wikipedia.org/wiki/Signal_processing
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Fast Fourier Transform (FFT)
FFT was invented by Cooley and Tukey in 1965.
It is a revolutionary numerical method that
allows rapid and accurate Fourier analysis of
discrete (digitized) signals. It is the
fundamental mechanism behind any spectrum
analyzer and many other digital (numerical)
processing.
FFT
Discrete FT
FFT
ADC
N points
N/2 points
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Frequency span and resolution
Spectrum analyzer can ONLY record finite number
of data points.
Same for the FFT spectrum (SR760 records 400
points for the whole frequency span. The
frequency bin size (resolution) Freq Span/400,
which is inversely proportional to total
acquisition time (length of signal).
Reciprocal relationship
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Spectrum analyzer (SR760)
Do not apply voltages greater than 1 volt
amplitude.
http//www.thinksrs.com/products/SR760770.htm
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Key pad
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Unit dBV
By definition, dBV is a measure of voltage in log
scale.