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DFT/FFT and Wavelets
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Additive Synthesis demonstration (wave addition)
Standard Definitions
Computing the DFT and FFT
●Sine and cosine wave multiplication and integration
●Input form
●Windowing
Interpreting the results
●Thresholds
Limitations of the FFT
●Frequency resolution
●Artifacts
Wavelets
Summary of Demonstration
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Complex waveforms are a summation of simple waves at differing
frequencies
Each simple wave has two coefficients
●Amplitude
●Phase
Examining the time-domain waveform does not provide any real
information about the coefficients:
●Small changes in amplitude and phase can produce very different
results in the time domain waveform
The DFT/FFT is a method that is designed to recover the simple
waveform coefficients from a complex wave
The DFT/FFT makes a lot of assumptions
Definition
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DFT: Discrete fourier transform
FFT: Optimized DFT.
Time-Domain waveform: A simple or complex waveform which is
plotted wrt to time (x-axis)
Frequency domain: The data which represents the amplitude and
phase of the series of simple waves which, when summed, produce a
given complex waveform. Also called the “Spectrum”.
Amplitude: The peak value of a wave (either positive or negative)
Phase: The relation of a periodic waveform to its initial value expressed
in factorial parts of the complete cycle. Usuall expressed as an angular
measurement (0-360 degrees or 0-2*pi)
Stationary signals: Signals which maintain the same parameters over
time. The FFT does not work well on non-stationary signals.
Computing the DFT
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Given a sine wave:
Integrate the sine wave over 1 cycle
●Result will be zero due to the symmetric nature of sine.
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Take the same sine wave and multiple by itself. (ie. Squared)
The resulting waveform is no longer symmetric wrt the baseline
●Integration will now yield a non-zero result
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One can isolate a frequency component from any complex waveform by
multipling the complex waveform by a simple waveform and then
integrating.
●If the integration yields a small result, we say that the frequency of
the simple waveform is not a major component of the complex
waveform
●If the integration yields a large result, we say that the frequency of
the simple waveform is a major component of the complex waveform
The FFT
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The DFT involves many calculations involving complex numbers
2
●N algorithm
●Where N is the number of samples in the waveform
The FFT uses the “divide and conquer” approach
●Involves breaking the N samples down into two N/2 sequences
●Smaller sequences involve less computation and the recombination
adds less overhead
Note: The FFT assumes that samples are equally spaced in time.
Because of the divide and conquer approach, N must be a power of 2.
If your waveform does not have 2a samples, the waveform can be
“padded” with zeros to fill up to 2a samples.
Sampling the waveform
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Sine is a continuous function. The DFT/FFT works on discrete data
If you wish to perform an FFT on a continuous function, it is often most
optimal to represent the continuous function in a discrete manner.
●This process is called Sampling
As the function progresses in time, we can measure the distance
between the function and some arbitrary baseline. (example on board)
●This process yields a series of numbers which approximate the
waveform.
Sampling can introduce artifacts/errors (example on board)
●Sampling rate too small (Nyquist limit)
●Not enough discrete levels
Real and Complex numbers
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The definition of the DFT involves multiplying a complex signal by a
sine wave.
●If a major component of the complex waveform is equivalent the
multiplied sine, the result is sin(x)2
●We need to take the square root of the integration, but the
integration might yield a negative value
The result is that we need to use complex numbers to perform the
computation.
●Both the input and the output of the FFT include real an imaginary
components.
Input to the FFT
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Input to the FFT is an array of complex numbers which represents the
input waveform
●The complex portion is set to zero because the waveform exists in
“real” space
0
1
2
3
...
2N
Real
Imaginary
Real
Imaginary
...
Sample 1
Sample 2
...
Sample N
Output from the FFT
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The output of the FFT is an array of complex numbers which represents
the spectrum
From each complex number we can compute:
●Amplitude information
●Phase information
0
1
2
3
...
Real
Imaginary
Real
Imaginary
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1st Harmonic (Positive Frequencies)
2nd Harmonic
...
N/2 Harmonic (postive)
N/2 Harmonic (negative)
...
2nd Harmonic
2N
1st Harmonic (Negative Frequencies)
Computing Amplitude and Phase
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Plot the real and imaginary components on a plane (where one axis is
the real component and the other is the imaginary component).
From the right angle triangle:
Imaginary Axis
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The amplitude is the hypotenuse
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The phase is the angle (a)
Real
Amplitude
a
Imaginary
Real Axis
Time localization
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The FFT assumes that the signals are stationary.
●The frequency components are present throughout the entire wave
(from negative to positive infinity)
●The phase components are present throughout the entire wave
(from negative to positive infinity)
However, what happens when the wave is NOT stationary?
●The FFT can tell you that the frequency is present, but it cannot tell
you where the frequency exists in time within the wave (website pic)
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Very few waveforms that exist within the real world are stationary.
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If you do not care where in time the frequency exists, no problem.
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If you do care where in time the frequency exists, you have to adjust
how you use the FFT.
Windowing
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Rather than analyse the whole waveform at once, we break the
waveform down into discrete pieces.
This is called “Windowing”
●Define a window which can hold N samples where N is a power of 2
●Copy the samples from time T within the original waveform into the
window
●Perform the FFT on the window
●Slide the window over D samples and repeat the process (window
slide)
The FFT will show which frequencies are present within the waveform
from time T to (T+N samples)
We cannot localize within the window, but we can localize the window
within the original waveform.
Unfortunately, windowing introduces artifacts
●Solution: Use a windowing function: Hamming, Hanning, KaiserBessel, Blackman, etc.
Windowing
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When we apply a windowing function, we are (generally) trying to
reduce high frequency artifacts which are introduced because of
windowing.
In doing so, we are de-emphasizing the beginning and the end of the
window.
●When we slide the window, if we make the slide too large, we will
lose information about the waveform.
The value for window slide is a power of 2 that is smaller than the
window size.
●ie. if we had a window size of 256 samples we would choose a slide
value which is less than 128 samples
●64, 32, or 16 (typically)
Interpreting the results of the FFT
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If we have a single window, we can simply plot the spectrum on a
graph
●The X axis is frequency
●The Y axis is amplitude
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Amplitude
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Frequency
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Interpreting the results of the FFT
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If multiple FFTs have been performed, then the result is a 3 dimensional
graph.
This graph is usually projected to 2 dimensional space where
●The X axis is time
●The Y axis is frequency
●The intensity of the point is the amplitude
Some examples exist on the following site:
http://pages.cpsc.ucalgary.ca/~hill/papers/synthesizer/index.html
Limitations of the FFT
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The FFT produces approximate results. There are always artifacts
introduced throughout the process
●These artifacts manifest as noise within the spectrum
●Filter out the noise using thresholds
The FFT does not deal well with discontinuities
●Discontinuities manifest (typically) as high frequency noise in the
spectrum
Because the FFT uses a single window size, the resolution is different
for different frequencies.
●You need a short window to capture high frequency components
●You need a longer window to capture low frequency components
Wavelets
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There are many similarities between wavelets and the FFT
●They both attempt to factor a time-domain function into its spectrum
●They both use translation functions to perform the factoring
Where wavelets differ from the FFT is precisely where the FFT has it's
limitations
●Wavelets are localized in space (ie. The wavelet translation function
is localized) and thus work better with non-stationary signals
●Wavelets have a scaling factor which provides better resolutions for
differing frequencies
The Continuous Wavelet Transform
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(Web Page) Formula
From the formula, we can see that there is a translation function (tao)
and a scaling factor (s)
Tau defines the “mother wavelet”.
●It is a local function
●There are many different “mothers” to choose from.
●Each mother tends to emphasize particular parts of the
spectrum and de-emphasize other parts of the spectrum
●The user needs to research which mother function is suitable to
his/her application
The Scaling Factor
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(Web Page) Diagram
When an FFT is performed, the window size remains constant.
●This is why the FFT has different resolutions for different frequencies
Wavelets are applied with a changing window size.
●The size of the window is called the “scale”
The variable “s” in the CWT definition causes the transform function to
be “scaled” to differing sizes.
To obtain a clear definition of low frequency components, a large
window is required
To obtain a clear definition of high frequency components, a small
window is required
●(web page diagram)