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DFT/FFT and Wavelets ● 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

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Summary of Demonstration ● 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

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Definition ● 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.

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Computing the DFT ● Given a sine wave: ● Integrate the sine wave over 1 cycle ● Result will be zero due to the symmetric nature of sine. ● 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 ● 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

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The FFT ● The DFT involves many calculations involving complex numbers ● N 2 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 2 a samples, the waveform can be “padded” with zeros to fill up to 2 a samples.

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Sampling the waveform ● 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

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Real and Complex numbers ● 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.

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Input to the FFT ● 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 Real Imaginary Real Imaginary Sample 1 Sample N Sample N

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Output from the FFT ● 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 Real Imaginary Real Imaginary st Harmonic (Positive Frequencies) 2 nd Harmonic... N/2 Harmonic (postive) N/2 Harmonic (negative)... 2N 2 nd Harmonic 1 st Harmonic (Negative Frequencies)

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Computing Amplitude and Phase ● 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: Real Axis Imaginary Axis ● The amplitude is the hypotenuse ● The phase is the angle (a) Real Imaginary Amplitude a

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Time localization ● 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) ● Very few waveforms that exist within the real world are stationary. ● If you do not care where in time the frequency exists, no problem. ● If you do care where in time the frequency exists, you have to adjust how you use the FFT.

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Windowing ● 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, Kaiser- Bessel, Blackman, etc.

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Windowing ● 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)

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Interpreting the results of the FFT ● 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 Frequency Amplitude

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Interpreting the results of the FFT ● 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:

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Limitations of the FFT ● 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

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Wavelets ● 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

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The Continuous Wavelet Transform ● (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

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The Scaling Factor ● (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)

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