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Transform Techniques 1 Mark Stamp. Intro  Signal can be viewed in… o Time domain  usual view, raw signal o Frequency domain  transformed view  Many.

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Presentation on theme: "Transform Techniques 1 Mark Stamp. Intro  Signal can be viewed in… o Time domain  usual view, raw signal o Frequency domain  transformed view  Many."— Presentation transcript:

1 Transform Techniques 1 Mark Stamp

2 Intro  Signal can be viewed in… o Time domain  usual view, raw signal o Frequency domain  transformed view  Many types of transformations o Fourier transform  most well-known o Wavelet transform  some advantages Transform Techniques 2

3 Intro  Fourier and wavelet transforms are reversible o From time domain representation to frequency domain, and vice versa  Fourier transform is in terms of functions sin(nx) and cos(nx)  Wavelet can use a wide variety of different “basis” functions Transform Techniques 3

4 Fourier Series  Generally, we can write f(x) in terms of series of sin(nx) and cos(nx) o Exact, but generally need infinite series o Finite sum usually just an approximation  Coefficients on sin(nx) and cos(nx) tell us “how much” of that frequency o May not be obvious from function o Can be very useful information Transform Techniques 4

5 Fourier Series  For example, consider sawtooth function: s(x) = x / π  The graph is… Transform Techniques 5

6 Sawtooth Function 1 term of Fourier series 2 terms Transform Techniques 6 3 terms 4 terms 5 terms

7 Fourier Transform  A function f(x) is usually viewed in the “time domain”  Transform allows us to also view it in “frequency domain”  What does this mean? o See next slide…  Why might this be useful? o Again, reveals non-obvious structure Transform Techniques 7

8 Time vs Frequency  Function f(x) written as sums of functions a n sin(nx) and b n cos(nx)  Coefficients (amplitudes) a n and b n o Tell us “how much” of each frequency Transform Techniques 8

9 Time vs Frequency  Frequency domain view gives us info about the function o More complicated the signal, less obvious the frequency perspective may be Transform Techniques 9 Time domain Frequency domain

10 Time vs Frequency  Time domain in red o Frequency domain in blue  What does blue tell us? o Dominant low frequency o Some high frequencies  Note that blue tells us nothing about time… o I.e., we do not know where frequencies occur Transform Techniques 10

11 Speech Example  Frequency domain info used to extract important characteristics Transform Techniques 11  Time domain signal  Sonogram

12 Fourier Transform  Many different transforms exist  So, why is Fourier so popular? o Fast, efficient algorithms o Fast Fourier Transform (FFT)  Apply transform to entire function? o May not be too informative, since we lose track of where frequencies occur o Usually, want to understand local behavior Transform Techniques 12

13 Global vs Local  Function can change a lot over time…  Global frequency info not so useful  Local frequency info is much better Transform Techniques 13

14 Global vs Local  Use Short Time Fourier Transform (STFT) for each window o Note that windows can overlap Transform Techniques 14

15 Window Size  How big should the window be? o Small? May not have enough freq info o Big? May not have useful time info Transform Techniques 15 about right too smalltoo big

16 Window Size  Looks like ideal case would be windows that match frequency o Bigger windows for low frequency areas o Smaller windows for high frequency  The bottom line? o Too big of window gives good frequency resolution, but poor time resolution o Too small of window gives good time resolution, but poor frequency resolution Transform Techniques 16

17 Uncertainty Principle  Cannot have accurate frequency and time resolution simultaneously o Form of Heisenberg Uncertainty Principle  So, this is something we must deal with o Since it’s the law! (of physics…)  Is there any alternative to STFT? o Yes, “multiresolution analysis”  What the … ? Transform Techniques 17

18 Windowing Revisited  Window in STFT is really a function o Selects f(x) within current window o “Window function” is essentially 1 within current window, 0 outside of it  For wavelets, “windows” much fancier o Like Windows 95 vs Windows 7… o Effect is to filter based on frequencies o Can mitigate some of the problems inherent in the uncertainty principle Transform Techniques 18

19 Fourier Transform  In Fourier transform, frequency resolution, but no time resolution Transform Techniques 19 frequency time

20 Short Time Fourier Trans.  In STFT, time resolution via windowing Transform Techniques 20 frequency time

21 Wavelet Transform  Time resolution based on frequency Transform Techniques 21 frequency time

22 Wavelet  Recall that Fourier analysis is based on sin(nx) and cos(nx) functions  Wavelet analysis based on wavelets o Duh!  But, what is a wavelet? o A small wave, of course… o “Wave”, so it oscillates (integrates to 0) o “Small”, meaning acts like finite window Transform Techniques 22

23 Wavelets  Many different wavelet functions to choose from o Select a “mother” wavelet or basis o Form translations and dilations of basis  Examples include o Haar wavelets (piecewise constant) o Daubechies wavelets o …and many others Transform Techniques 23

24 Haar: Translation & Dilation Transform Techniques

25 Advantages of Wavelets  Wavelet basis is local o Unlike Fourier basis of sine and cosine o Local, implies better time resolution  Basis functions all mutually orthogonal o Makes computations fast o Fourier basis also orthogonal, but requires “extreme cancellation” outside window o In effect, “windowing” built in to wavelet basis  Wavelets faster to compute than FFT o A recursive paradise… Transform Techniques 25

26 Disadvantage of Wavelets  Approximation with Haar functions… o For example, sine function is trivial in Fourier analysis, not so easy with Haar Transform Techniques 26

27 Wavelets: Bottom Line  Fourier ideal wrt frequency resolution o But sine/cosine bad wrt time resolution  Wavelets excels at time resolution o Since basis functions finite (compact) support, and employ translation/dilation o In effect, filters by frequency and time  Complicated mathematics o But fairly easy to implement and use Transform Techniques 27

28 Discrete Transforms  In practice, apply transforms to discrete time series, a 0,a 1,a 2,… o We assume a i = f(x i ) for unknown f(x)  Discrete transforms are very fast o FFT is O(n log n) o Fast wavelet transform is O(n)  Discrete transforms based on some fancy linear algebra Transform Techniques 28

29 Transform Uses  Speech processing o Construct sonogram (spectrogram) o Speech recognition  Image/video processing o Remove noise, sharpen images, etc., etc.  Compression  And many, many more… Transform Techniques 29

30 What About Malware Detection?  We measure some characteristic of a exe file to obtain series a 0,a 1,a 2,…  Compute wavelet transform and… o Filter out high frequency “noise” (i.e., insignificant variations) o And segment file based on where the significant changes occur  Ironically, transform used to pinpoint significant changes wrt time Transform Techniques 30

31 Malware Detection Example 1. Compute entropy measurement using a i = entropy(B i ) for i = 0,1,2,…,n o Where B i is block of i consecutive bytes o Computed on (overlapping) “windows” o “Window” here not same as in transform 2. Apply discrete transform to a 0,a 1,… 3. Find significant changes in entropy 4. Use resulting sequence for scoring Transform Techniques 31

32 References  R. Polikar, The wavelet tutorialThe wavelet tutorial  A. J. Jerri, Introduction to WaveletsIntroduction to Wavelets  G. Strang, Wavelet transforms versus Fourier transforms, Bulletin of the American Mathematical Society, 28: , 1993Wavelet transforms versus Fourier transforms Transform Techniques 32


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