DCSP-2: Fourier Transform

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1 DCSP-2: Fourier Transform Jianfeng Feng Department of Computer Science Warwick Univ., UK

2 Data transmission Channel characteristics, Signalling methods (ADC) Interference and noise Fourier transform Data compression and encryption

3 Bandwidth The range of frequencies occupied by the signal is called its bandwidth. Power 0 B Frequency

4 Nyquist-Shannon Theorem

5 The ADC process is governed by an important la Nyquist-Shannon Theorem (will be discussed in Chapter 3) An analogue signal of bandwidth B can be completely recreated from its sampled form provided its sampled at a rate equal to at least twice it bandwidth. That is S > 2 B

6 Example I will guess that B = 1 Hz Sample at 2B = 2 Hz: x[n] = [ ] Intuitively, I would say it will not work

7 Example I will guess that B = 1 Hz Sample at 2B < 4 Hz: x[n] = [ ] According to N-S Thm, we can fully recover the original signal

Example I will guess that B = 1 Hz Sample at 2B < 4 Hz: x[n] = [ 1 0-1 0 1 0-1 0 ] According to N-S Thm, we

8 Example I will guess that B = 1 Hz Sample at 2B < 4 Hz: x[n] = [ ] According to N-S Thm, we can fully recover the original signal Well, the blue line has the identical frequency, and x[n]. What is wrong?

9 Noise in a channel

10 Noise in a channel Attenuation

11 Noise in a channel

12 Noise in a channel

13 Noise in a channel

14 SNR Noise therefore places a limit on the channel at which we can transfer information Obviously, what really matters is the signal to noise ratio (SNR). This is defined by the ratio signal power S to noise power N, and is often expressed in decibels (db): SNR=10 log 10 (S/N) db

15 Noise sources Input noise is common in low frequency circuits and arises from electric fields generated by electrical switching. It appears as bursts at the receiver, and when present can have a catastrophic effect due to its large power. Other peoples signals can generate noise: cross-talk is the term give to the pick-up of radiated signals from adjacent cabling.

16 Noise sources When radio links are used, interference from other transmitters can be problematic. Thermal noise is always present. It is due to the random motion of electric charges present in all media. It can be generated externally, or internally at the receiver. How to tell signal from noise?

17 Communication Techniques I Time frequency Fourier Transform bandwidth noise power

18 Communication Techniques I Time frequency Fourier Transform bandwidth noise power

19 Communication Techniques II Time, frequency and bandwidth We can describe a signal in two ways. One way is to describe its evolution in time domain, as we usually do. The other way is to describe its frequency content, in frequency domain: what we will learn The

20 Your heartbeat Ingredients: a frequency ω (units: radians) an initial phase φ (units: radians) an amplitude A (units depending on underlying measurement) a trigonometric function e.g. x[n]= A cos(ωn+φ) cosine wave, x(t), has a single frequency, w =2 p/t where T is the period i.e. x(t+t)=x(t).

21 What do we expect? Power Time 1 Hz Fre

22 What do we expect? Power Time 1 Hz Fre

23 What do we expect? Power Time 1 Hz Fre

24 What do we expect? Power Time 1 Hz Fre

25 What do we expect? Power Time 1 Hz Fre

26 Fourier Transform I This representation is quite general. In fact we have the following theorem due to Fourier. Any signal x(t) of period T can be represented as the sum of a set of cosinusoidal and sinusoidal waves of different frequencies and phases

27 The term Fourier transform can refer to either the frequency domain representation of a function or to the process/formula that "transforms" one function into the other. Fourier Transform II In mathematics, the continuous Fourier transform is one of the specific forms of Fourier analysis. As such, it transforms one function into another, which is called the frequency domain representation of the original function (which is often a function in the timedomain). In this specific case, both domains are continuous and unbounded.

28 Fourier Transform III

29 Fourier Transform IV Continuous time (analogous signals): FT (Fourier transform) it is in theory (in Warwick, we need it) Discrete time: DTFT (infinity digital signals) it is in theory (discrete version) DFT: Discrete Fourier transform (finite digital signals what we can use, one line in Matlab (fft))

30 History of FT I Gauss computes trigonometric series efficiently in 1805 Fourier invents Fourier series in 1807 People start computing Fourier series, and develop tricks Good comes up with an algorithm in 1958 Cooley and Tukey (re)-discover the fast Fourier transform algorithm in 1965 for N a power of a prime Winograd combined all methods to give the most efficient FFTs

31 History of FT II Gauss

32 History of FT III Fourier

33 History of FT IV Jianfeng Feng

34 History of FT V Prof Feng

35 Complex Numbers

36 Euler Formular Exp(j a) = cos a + j sin a

37 The complex eponential the trigonometric function of choice in DSP is the complex exponential: x[n] = Aexp(j(ωn+φ)) = A[cos(ωn + φ) + j sin(ωn + φ)]

38 The complex eponential

39 Most beautiful Math Formula exp ( j π ) + 1 = 0 Where e is Euler's number J is the imaginary unit

40 Fourier's Song Integrate your function times a complex exponential It's really not so hard you can do it with your pencil And when you're done with this calculation You've got a brand new function - the Fourier Transformation What a prism does to sunlight, what the ear does to sound Fourier does to signals, it's the coolest trick around Now filtering is easy, you don't need to convolve All you do is multiply in order to solve. From time into frequency - from frequency to time Every operation in the time domain Has a Fourier analog - that's what I claim Think of a delay, a simple shift in time It becomes a phase rotation - now that's truly sublime! And to differentiate, here's a simple trick Just multiply by J omega, ain't that slick? Integration is the inverse, what you gonna do? Divide instead of multiply - you can do it too. Or make the pulse wide, and the sinc grows dense, The uncertainty principle is just common sense. From time into frequency - from frequency to time Let's do some examples... consider a sine It's mapped to a delta, in frequency - not time Now take that same delta as a function of time Mapped into frequency - of course - it's a sine! Sine x on x is handy, let's call it a sinc. Its Fourier Transform is simpler than you think. You get a pulse that's shaped just like a top hat... Squeeze the pulse thin, and the sinc grows fat.

41 Example Frequency-space k1 IFT Image space y k2 FT x

42 Fun: Decoding dream (Horikawa et al. Science, 2013)

43 DCSP-3: Fourier Transform II Jianfeng Feng Department of Computer Science Warwick Univ., UK

44 This week s summary Introduce FT in layman s language (yesterday) How to calculate it (today)

45 This week s summary Introduce FT in layman s language (yesterday) How to calculate it (today) You might find the world is different!

46 100m-for-medical-imaging-tech/

47 This week s summary Introduce FT in layman s language (yesterday) Talk about continuous FT since it is clean and simple Come back to it on how to numerically calculate it later on Intuition Fourier theorem Examples Bandwidth

48 Fourier Theorem This representation is quite general. In fact we have the following theorem due to Fourier. Any signal x(t) of period T can be represented as the sum of a set of cosinusoidal and sinusoidal waves of different frequencies

50 Intuition of FT Two dimensional space (all points) (a, b) = a (1,0) + b (0,1) Signal space (all functions of t) x (t) = a sin(ω t) + b sin (2 ω t)

51 Intuition of FT Two a point in it (a, b) = a (1,0) + b (0,1)

52 Intuition of FT Two a point in it (a, b) = a (1,0) + b (0,1) Signal = coef basis coef basis

53 Intuition of FT Two a point in it (a, b) = a (1,0) + b (0,1) x(t) =? Signal = coef basis coef basis

54 Intuition of FT Two a point in it (a, b) = a (1,0) + b (0,1) x(t) = a F1(t) +b F2(t) Signal = coef basis coef basis

55 Intuition of FT It is a branch of mathematics called functional analysis: treat each function as a point in a functional space It is the starting pointe of modern mathematics It is also the actual power of mathematics Branch space, Hilbert space etc.

56 Intuition of FT How to calculate these coefficients? o (a, b) = a (1,0) + b (0,1) Functional x (t) = a cos(ω t) + b cos (2 ω t) w = 2 pi /T

57 Intuition of FT 1. Coefficient is obtained via Inner product <(x, y), (m,n)> = xm+yn 2. All bases are orthogonal <(0, 1), (1,0)> = 0, (0,1) (1, 0) (a, b) = a (1,0) + b (0,1) Therefore <(a, b), (1,0)> = a <(1,0), (1,0)> + b <(0,1), (1,0)> = a <(a, b), (0,1)> = a <(1,0), (0,1)> + b <(0,1), (0,1)> = b

58 Intuition of FT Two dimensi coefficient is obtained via Inner product (a, b) = a (1,0) + b (0,1) <(a, b), (1,0)> = a <(1,0), (1,0)> + b <(0,1), (1,0)> = a Functional coefficient is obtained via inner product x (t) = a cos(ω t) + b cos (2 ω t) <x (t), cos(ω t) > = <a cos(ω t), cos(ω t) > + <b cos (2 ω t), cos(ω t) > = a (?)

59 Intuition of FT Two dimension orthogonal basis (a, b) = a (1,0) + b (0,1) Functional space basis x (t) = a cos(ω t) + b cos (2 ω t)

60 Intuition of FT It turns out that we can define the inner product in the signal space x(t), y ( t ) 2 T T / 2 T / 2 x ( t ) y ( t) dt 2 cos( wt ),cos(2wt ) cos( wt)cos(2wt ) dt T T T

61 Intuition of FT Cos ( ω t) Cos ( 2* ω t) Cos ( 2 ω t) * Cos ( 2* ω t) 2 cos( wt ),cos(2wt ) cos( wt)cos(2wt ) dt T T T

62 they form orthogonal bases Intuition of FT Cos ( 2 ω t) Cos ( 2* ω t) Cos ( 2 ω t) * Cos ( 2* ω t) { cos (n ω t ), n=1,2 } are orthogonal

63 Intuition of FT It turns out that we can define the inner product in the signal space cos( mwt),cos( nwt) In particular, we have 2 / T, n 0, n m m x (t) = a cos(ω t) + b cos (2 ω t) a = <X(t), cos(ω t)> = b =? 2 T T /2 ò -T /2 x(t) cos( wt)dt

64 Intuition of FT X X 1 X 6 ( ) X 2 X 3 x 4 X5 ( ) ( ) ( ) ( ) ( ) In an n-dim Euclidean space, we decompose any vector X in terms of orthogonal bases Coefficient is obtained by inner product between a basis and X We sometime call x i weight X = x 1 ( ) + x 2 ( ) + + x 6 ( )

65 Intuition of FT X(t) A 1 A 6. cos (ω t ) A 2 A 3 A 4 A5 x( t) A1 cos( wt) more

66 Intuition of FT X(t) A 1 A 6. cos (ω t ) A 2 A 3 A 4 A5 cos (2 ω t ) x t) A cos( wt) A cos(2 t) more ( 1 2 w

67 Intuition of FT X(t) A 1 A 6. cos (ω t ) A 2 A 3 A 4 A5 cos (3 ω t ) cos (2 ω t ) x t) A cos( wt) A cos(2wt ) A cos(3 t) more ( w

68 Intuition of FT X(t) A 1 A 6. cos (ω t ) A 2 A 3 A 4 A5 cos (3 ω t ) cos (4 ω t ) cos (2 ω t ) x t) A cos( wt) A cos(2wt ) A cos(3wt ) A cos(4 t) more ( w

69 Intuition of FT X(t) A 1 A 6. cos (ω t ) A 2 A 3 A 4 0 cos (3 ω t ) cos (4 ω t ) cos (2 ω t ) x t) A cos( wt) A cos(2wt ) A cos(3wt ) A cos(4 t) more ( w

70 Intuition of FT X(t) A 1 A 6 cos (ω t ) A 2 A 3 A 4 0 cos (3 ω t ) cos (4 ω t ) cos (2 ω t ) x t) A cos( wt) A cos(2wt ) A cos(3wt ) A cos(4wt) A cos(6w ) ( t

71 Intuition of FT X(t) A 1 A 6. cos (ω t ) A 2 A 3 A 4 0 cos (3 ω t ) cos (4 ω t ) cos (2 ω t ) In an n-dim Euclidean space, we decompose any vector X in terms of orthogonal bases Coefficient is obtained by inner product between a basis and X We sometime call x i weight

72 FT: Fourier Thm in terms of Sin and Cos simply a generalization of common knowledge of the Euclidean space { 1, cos (nωt), sin(nwt), n=1,2 } are orthogonal and complete they form orthogonal bases

73 FT: Fourier Thm in terms of Sin and Cos simply a generalization of common knowledge of the Euclidean space { 1, cos (nωt), sin(nwt), n=1,2 } are orthogonal and complete they form orthogonal bases

74 FT: Fourier Thm in terms of Sin and Cos simply a generalization of common knowledge of the Euclidean space { 1, cos (nωt), sin(nwt), n=1,2 } are orthogonal and complete they form orthogonal bases x(t) = A 0 + å A n cos(nwt)+ n=1 n=1 ì ï A 0 =<1, x(t) >= 1 ïï T ï í ï ïï ï ï î T /2 -T /2 B n sin(nwt) where A 0 is the d.c. term, and T is the period of the waveform. ò å x(t)dt A n =< cos(nwt), x(t) >= 2 T B n =< sin(nwt), x(t) >= 2 T w = 2p T T /2 ò -T /2 T /2 ò -T /2 x(t) cos(nwt) dt x(t)sin(nwt) dt

75 FT: Fourier Thm in terms of Sin and Cos simply a generalization of common knowledge of the Euclidean space { 1, cos (nωt), sin(nwt), n=1,2 } are orthogonal and complete they form orthogonal bases x(t) = A 0 + å A n cos(nwt)+ n=1 n=1 ì ï A 0 =<1, x(t) >= 1 ïï T ï í ï ïï ï ï î T /2 -T /2 B n sin(nwt) where A 0 is the d.c. term, and T is the period of the waveform. ò å x(t)dt A n =< cos(nwt), x(t) >= 2 T B n =< sin(nwt), x(t) >= 2 T w = 2p T T /2 ò -T /2 T /2 ò -T /2 x(t) cos(nwt) dt x(t)sin(nwt) dt

76 FT: Fourier Thm in terms of Sin and Cos simply a generalization of common knowledge of the Euclidean space { 1, cos (nωt), sin(nwt), n=1,2 } are orthogonal and complete they form orthogonal bases x(t) = A 0 + å A n cos(nwt)+ n=1 n=1 ì ï A 0 =<1, x(t) >= 1 ïï T ï í ï ïï ï ï î T /2 -T /2 B n sin(nwt) where A 0 is the d.c. term, and T is the period of the waveform. ò å x(t)dt A n =< cos(nwt), x(t) >= 2 T B n =< sin(nwt), x(t) >= 2 T w = 2p T T /2 ò -T /2 T /2 ò -T /2 x(t) cos(nwt) dt x(t)sin(nwt) dt

77 FT: Fourier Thm in terms of Sin and Cos simply a generalization of common knowledge of the Euclidean space { 1, cos (nωt), sin(nwt), n=1,2 } are orthogonal and complete they form orthogonal bases x(t) = A 0 + å A n cos(nwt)+ n=1 n=1 ì ï A 0 =<1, x(t) >= 1 ïï T ï í ï ïï ï ï î T /2 -T /2 B n sin(nwt) where A 0 is the d.c. term, and T is the period of the waveform. ò å x(t)dt A n =< cos(nwt), x(t) >= 2 T B n =< sin(nwt), x(t) >= 2 T w = 2p T T /2 ò -T /2 T /2 ò -T /2 x(t) cos(nwt) dt x(t)sin(nwt) dt

79 Example I x(t)=1, 0< t < p, 2p < t < 3p, 0 otherwise Hence x(t) is a signal with a period of 2p p 2p 3p 4p

80 Example I ì A 0 = 1 p ï ò dt = 1 ï 2p 0 2 A n = 2 p ï í ò cos(nt)dt = 1 sin(np ) = 0,n =1,2,... ï 2p 0 np ï B n = 2 p ï ò sin(nt)dt = 1 (1- cos(np )), n =1, 2,... îï 2p np 0 Finally, we have x(t) = p [sin(t)+ 1 3 sin(3t)+ 1 5 sin(5t)+...]

Example I X(F) 2/p Frequency domain x(t) = 1 2 + 2 p [sin(t)+ 1 3 sin(3t)+ 1 5 sin(5t)+.

81 Example I X(F) 2/p Frequency domain x(t) = p [sin(t)+ 1 3 sin(3t)+ 1 5 sin(5t)+...] The description of a signal in terms of its constituent frequencies is called its frequency (power) spectrum.

82 Example I X(F) 2/p = Time domain Frequency domain

83 Example I A periodic signal is uniquely determined by its coefficients {A n, B n }. If we truncated the series into finite term, the signal can be approximated by a finite sines as shown below (compression, MP3, MP4, JPG, ) one term Two terms Three terms Five terms

84 Example II (understanding music)

85 Example II a. Pure tone: Script1_1.m This confirms our earlier believe that it is a signal with a finite bandwidth (N-S sampling Thm)

86 Example II b. Different waveforms Script1_2.m This confirms our earlier believe that it is a signal without bandlimit (N-S sampling Thm)

87 Bandwidth can be properly defined Bandwidth is the difference between the upper and lower frequencies in a set of frequencies. It is typically measured in hertz

88 Bandwidth can be properly defined Power 0 B Frequency Bandwidth is the difference between the upper and lower frequencies in a continuous set of frequencies. It is typically measured in hertz

89 Example II C. Approximation (compression) Script1_3.m Without doing much, we can compress the original data now, as in Example I.

DCSP-5: Fourier Transform II

DCSP-5: Fourier Transform II Jianfeng Feng Department of Computer Science Warwick Univ., UK Jianfeng.feng@warwick.ac.uk http://www.dcs.warwick.ac.uk/~feng/dcsp.html Assignment:2018 Q1: you should be able