DFT/FFT and Wavelets

1. 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

2. 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

3. 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.

4. 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

5. The FFT • The DFT involves many calculations involving complex numbers • N2 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.

6. 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

7. 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.

8. 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 0 1 2 3 ... Real Imaginary Real Imaginary ... Sample 1 Sample 2 ... Sample N 2N

9. 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 0 1 2 3 ... Real Imaginary Real Imaginary ... 1st Harmonic (Positive Frequencies) 2nd Harmonic ... N/2 Harmonic (postive) N/2 Harmonic (negative) ... 2nd Harmonic 1st Harmonic (Negative Frequencies) 2N

10. 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: Imaginary Axis • The amplitude is the hypotenuse • The phase is the angle (a) Real Imaginary Amplitude a Real Axis

11. 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.

12. 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.

13. 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)

14. 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 50 40 Amplitude 30 20 10 100 200 300 Frequency

15. 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: http://pages.cpsc.ucalgary.ca/~hill/papers/synthesizer/index.html

16. 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

17. 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

18. 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

19. 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)