Main Title Slide
Frequency Domain Signals
Fourier theory
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Frequency scale
You may have noticed that the DFT / FFT doesn’t refer to an actual frequency, they are normalised in the sense that they do not care what the sampling frequency is. k is just the number of cycles that fit into the array (f is k/N).
To get a real frequency value we need to use the time scale information we have that corresponds to the array of data being transformed.
Note: F will be 1+N/2 points long this matches the range of the unique output of DFT/FFT.
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What controls the spacing of points in the frequency domain?
What could we do to improve this?
Note: F will be 1+N/2 points long this matches range of the unique output of DFT/FFT.
Frequency scale
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Zero padding
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Fourier Examples
Time Domain x(t)
Frequency Domain X(f)
Delta function
Comb function
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Fourier Examples
Time Domain x(t)
Frequency Domain X(f)
cos function
sin function
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Fourier Examples
Rectangular
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Fourier Examples
Exponential
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Example: Take the example waveform below:
What do we expect it’s Fourier transform to look like?
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We can make this by convolving a pulse train which is repeating every 0.05 seconds with a Gaussian which has a FWHM of 0.0118.
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This means we should multiple their respective Fourier transforms.
FT of Gaussian is another Gaussian with a modified width , the FT of the pulse train is another pulse train with spacing 1/T
(in ω, where ω=2πf)
Pulses every 20Hz (1/0.05)
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Without needing to do the integration we can get the form of the spectrum.
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Aside: Integrating digital signals
You’ll note I have switched back and forth between integral symbols (for continuous data) and summation symbols for discrete data.
When we integrate a set of discrete points what do we need to do?
This depends on how well your data is sampled for example:
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Aside: Integrating digital signals
Summing the points is the same as assuming each sample represents a rectangle under that point of the true curve. This can introduce errors if you have few samples on the waveform.
In these cases you might want to use other methods (trapezium, simpsons, or higher order approximations)
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Aside: Integrating digital signals
If your data is well sampled though just summing is fine.
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