Information and Communications Security: Encryption and Information Hiding

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1 Short Course on Information and Communications Security: Encryption and Information Hiding Tuesday, 10 March Friday, 13 March, 2015 Lecture 5: Signal Analysis

2 Contents The complex exponential The complex Fourier series The Fourier transform The delta function Fourier transform of important functions Important properties The convolution theorem The sampling theorem

3 Contents (continued) The DFT Time-frequency sampling relation The Fast Fourier Transform (FFT) Data windowing Spectral leakage Example windows Summary

4 The complex exponential Fundamental result Euler s equation: i to the power of i:

5 The complex Fourier series (CFS) Let a function f(t) with period 2T be given by Then

6 The CFS (continued) Evaluation of the RHS integral gives Complex coefficients given by

7 The Fourier transform For a signal with period 2T the CFS is given by (with )

8 Fourier transform (continued) Let Then and

9 Fourier transform (continued) In the limit as function i.e. for a non-periodic

10 Lord Kelvin The Fourier transform (Fourier s theorem) is not only one of the most beautiful results of modern analysis, but it may be said to furnish an indispensable instrument in the treatment of nearly every recondite question in modern physics

The (Dirac) delta function First introduced by Paul Dirac in the 1930s for use in quantum mechanics Dirac referred to it as an improper function, i.e. a strictly hypothetical (abstract) function or

11 The (Dirac) delta function First introduced by Paul Dirac in the 1930s for use in quantum mechanics Dirac referred to it as an improper function, i.e. a strictly hypothetical (abstract) function or generalisation Dirac also first introduced the hypothetical idea of anti-matter (a positively charged electron or positron) and won the Nobel prize for Physics in 1933 after experimental verification of his idea

12 What is d(x) d(x) is not a function of x according to the usual definition, of a function, which requires a function to have a definite value for each point in its domain, but is something more general, which we may call an improper function. Paul Dirac ( ) in The Principles of Quantum Mechanics

13 Basic definition

14 Integral representation

15 Sampling property

16 The Green s function Suppose a system can be modelled in terms of the linear differential equation The Green s function is the solution of

17 Green s function solution The Green s function provides a general solution (transformation) of the form The Green s function is the Impulse Response Function (IRF) of the system characterised by the differential operator, e.g. the surface waves produced on a pond when a small stone disturbs it, producing an impulse

18 George Green: A scientific enigma

19 The comb function Defines a sequence of delta functions and is used in proving the sampling theorem, i.e. the sample rate required for A-to-D without loss of information

20 The Kronecker delta The discrete equivalent of the delta function with the following analogous properties:

21 The Fourier operator Basic definition Real space and Fourier space

22 Physical interpretation The Fourier transform provides a quantitative statement on the frequency content of a function. The variable w has dimensions that are reciprocal to those of the variable t. If t is time in seconds then w is the temporal (angular) frequency in cycles per second or Hertz (Hz). N.B. w =2pf where f is the frequency proper

23 Spatial frequency x is length and k is the wave-number given by where l is the wavelength and c is the wave speed

24 DC value The DC (Direct Current) value is the value of the Fourier transform at zero frequency

25 Differentiation

26 Integration

27 Fourier transform of important functions The Tophat or Square wave function

28 Fourier transform of important functions (continued) The cosine and sine functions

29 Fourier transform of important functions (continued) The Gaussian function

30 Fourier transform of important functions (continued) The sign or sgn function and related functions

31 Important properties Addition theorem Similarity theorem Shift theorem

32 Parseval s theorem Important properties (continued) Rayleigh s (the energy) theorem

33 Band-limited functions A band-limited function is one whose Fourier transform is of limited extent, i.e. of compact support : A band-limited function is given by

34 The convolution theorem One of the single most important theorems of Fourier analysis Used routinely to process signals in Fourier space The convolution of two function in real space is equivalent to the product of their Fourier transforms in Fourier (frequency) space

35 The convolution theorem expressed mathematically

36 The product theorem The product of two function in real space is equivalent to the convolution of their Fourier transforms in Fourier (frequency) space

37 The correlation and autocorrelation theorems Correlation theorem Autocorrelation theorem

38 The sampling theorem Used to determine the rate at which an analogue signals need to be sampled into digital form without loss of information. Underpins all A-to-D (& D-to-A) conversion

39 The comb function Consider a comb function of period T Let a digital function g(t) be written in terms of an analogue function f(t) as

40 The sampling problem

41 CFS of the comb function The comb function is a periodic function and can therefore be written as Coefficients are given by

42 Fourier transform of comb(t) CFS of a comb function is Fourier transform is therefore

43 Fundamental result

44 Basic result Sampling a signal at regular interval dt: From the product theorem

45 Interpretation Sampling a function f(t) creates a new spectrum G(w) which is a periodic replica of the spectrum F(w) spaced at regular intervals If f(t) has a bandwidth W then the total width of the spectrum is W-(-W)=2W and the replicated spectra will overlap if (aliasing)

46 Sampling intervals To ensure that replicated spectra do not overlap causing aliasing we require that This is equivalent to a sampling rate of

47 Aliasing

48 The Nyquist frequency A signal that is sampled according to the condition is known as a Nyquist sampled signal The Nyquist frequency is given by

49 Nyquist sampling

50 Derivation of the DFT The CFS is Consider f(t) to be uniformly sampled by Dt so that

51 Derivation of the DFT (continued) Let Then

52 DFT and the Fourier transform DFT pair is Discretizing the Fourier transform pair:

53 Time-frequency sampling relation Let Then, by inspection, i.e. comparing the DFT with the discretized Fourier transform

54 Example of the discrete time-frequency relation Consider a digital signal composed of 1000 element and a sampling interval of 0.001s Frequency sampling interval is then 2p The more precisely time is determined the less precisely the frequency is known

55 Discrete spatial frequency relationship x is length and k = w/c = 2p/l is the wave-number where l is the wavelength and c is the wave-speed The more precisely position is determined the less precisely the spatial frequency is known

56 Standard and Optical versions of the DFT The DFT is usually written in standard form with n=0,1,2,,n-1 where the DC term occurs at n=0, i.e. The optical form sets the DC term in the middle of the array and is compatible with the Fourier transform, i.e.

57 The Fast Fourier Transform (FFT) Consider the DFT (in standard form) writing it in the form In matrix form

58 Computational issues Written in matrix form, the DFT is computed by multiplying an N-point vector by a matrix of complex elements This requires N x N multiplications To compute the DFT of a 1000 point digital signal requires multiplications!

59 Basic idea

60 Fundamental property Basic result is With e & o representing even and odd component respectively, we have Computation of arrays is now over N/2 elements and not N elements.

61 Successive doubling We can repeat the trick to obtain 4 arrays Can continue sub-divding the data into odd and even component until we get to the DFT of just 2 points

62 Base 2 condition Because the data is subdivided into odd and even components, we require to start with a array size of (with k=1,2,3, ) Computing the DFT in this way reduces the number of multiplication to the order of

63 Example Consider the 4-point array FFT is

64 Array order Consider 8-point array Decomposition in to odd and even components gives

65 Bit reversal To obtain output in correct order, original array must be input as Reordering by bit reversing the index:

66 Computational efficiency FFT reduces number of multiplication from to Can consider decomposition into 3, 4, arrays instead of 2 arrays but reordering of I/O is more complex than bit reversal

67 MATAB FFT Based on the function FFT output_array=fft(input_array) The inverse transform is IFFT output_array=ifft(input_array) FFT is based on standard form (DC occurs at the first element of the array) MATLAB function for producing output in optical form is FFTSHIFT output_array=fftshift(input_array) DC component occurs at 1+N/2 for array size of N

68 Data windowing Unlike the Fourier transform the DFT operates on a discrete array of finite length Computing the DFT of a signal consisting of N samples is equivalent to computing the DFT of an infinite run of samples multiplied by a square window function (tophat function) This result allows us to evaluate the effect of operating on a array of finite size

69 Effect of windowing If then

70 Effect of windowing (continued) Using the product theorem The discrete spectrum F m is not given by F(w m ) but by F(w m ) convolved with a sinc function Note that

71 Spectral leakage Each computed sample F m depends on the influence of the sinc function associated with one sample bin on the next The sinc function leaks from one bin to the next producing errors in the values of neighbouring spectral components This is due to the discontinuous nature of the window function

72 Window functions The larger the size of the array, the less effect spectral leakage has on the output For small arrays, spectral leakage can be reduced by application of a window that approaches zero at the end of the array Many window have been invented for this purpose. They are based on trade-offs between the narrowness and peakedness of the spectral leakage function

73 Some example window functions Parzan window Welch window Hanning window (cosine taper)

74 Summary The Fourier transform pair:

75 Summary (continued) The convolution theorem The product theorem

76 Summary (continued) Sampling theorem (fundamental result)

77 Summary (continued) DFT pair Time frequency sampling relation

78 Summary (continued) Principle of the (base-2) FFT Bit reversal: Reversal of the binary number representation of the position of an element in an array which is used to reorder the data before repeated application of the principle above

79 Summary (continued) MATLAB FFT function y=fft(x); Inverse FFT function y=ifft(x); Shifting function (to compute spectrum in optical form ) y=fftshift(x);

80 Summary (continued) DFT approximation to F(w m ) Data windows: Function with edge tapers that reduce the spectral leakage generated by the sinc function in the equation above

SEISMIC WAVE PROPAGATION. Lecture 2: Fourier Analysis

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