The Fourier Transform

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1 The Fourier Transorm Fourier Series Fourier Transorm The Basic Theorems and Applications Sampling Bracewell, R. The Fourier Transorm and Its Applications, 3rd ed. New York: McGraw-Hill, 2. Eric W. Weisstein. "Fourier Transorm.

2 Fourier Series 2 Fourier series is an epansion (German: Entwicklung) o a periodic unction () (period 2L, angular requency /L) in terms o a sum o sine and cosine unctions with angular requencies that are integer multiples o /L Any piecewise continuous unction () in the interval [-L,L] may be approimated with the mean square error converging to zero

3 Fourier Series 3 Sine and cosine unctions orm a complete orthogonal system over [-,] Basic relations (m,n ): mn : Kronecker delta mn = or m n, mn = or m = n sin n, cosn sin cos msinnd mn mcosnd mn mcosnd sin md sin cos md

4 () Fourier Series: Sawtooth 4 Eample: Sawtooth wave / (2L)

5 Fourier Series 5 Fourier series is given by 2 n n n a a cosn b sinn n where a n b n a d cosnd sinnd I the unction () has a inite number o discontinuities and a inite number o etrema (Dirichlet conditions): The Fourier series converges to the original unction at points o continuity or to the average o the two limits at points o discontinuity

6 6 Fourier Series For a unction () periodic on an interval [-L,L] instead o [-,] change o variables can be used to transorm the interval o integration Then and L ' L d d ' L n b L n a a n n n n ' sin ' cos 2 ' ' cos ' d L n L a L L n ' ' sin ' d L n L b L L n ' ' d L a L L

7 Fourier Series 7 For a unction () periodic on an interval [,2L] instead o [-,]: a n 2 L L ' d' a n 2 L L ' n' cos d' L b n 2 L L ' n' sin d' L

8 () Fourier Series: Sawtooth 8 Eample: Sawtooth wave a a 2 L n d L 2L L L 2 L 2 n cos d L 2n cosn sinn sin n 2 2 n.8 2L n sin 2 n n L Finite Fourier series b n L 2 L L 2 n sin d L 2n cos 2 2 2n n 2n sin2n / (2L)

9 () Fourier Series: Sawtooth 9 Eample: Sawtooth wave Fourier series: g 2L 2 m m, n sin n n L / (2L)

10 Fourier Series: Sawtooth / (2L) 2 3 Amplitude a /2, b n n : Frequency w / (/L) / (2L)

11 Eample: Approimation by Fourier-Series m= m m n n n a a cosn b sinn n

12 Approimation by Fourier-Series m= m m n n n a a cosn b sinn n

13 Approimation by Fourier-Series m= m m n n n a a cosn b sinn n

14 Fourier Series: Properties 4 Around points o discontinuity, a "ringing, called Gibbs phenomenon occurs I a unction () is even, n is odd and thus Even unction: b n = or all n I a unction () is odd, n is odd and thus Odd unction: a n = or all n sin sinnd cos cosnd

15 Comple Fourier Series 5 Fourier series with comple coeicients n A n e in Calculation o coeicients: A n 2 e in d Coeicients may be epressed in terms o those in the real Fourier series A n 2 cos n isinn d a 2 a 2 a 2 n n ib ib n n or n < or n = or n >

16 Fourier Transorm 6 Generalization o the comple Fourier series or ininite L and continuous variable k L, n/l k, A n F(k)dk F k Forward transorm or e 2ik d Fk F k Inverse transorm F k e 2ik dk F k F k Symbolic notation Fk

17 Forward and Inverse Transorm 7 I. d eists 2. () has a inite number o discontinuities 3. The variation o the unction is bounded Forward and inverse transorm: For () continuous at For () discontinuous at F F 2ik 2ik e e d dk k F k F 2

18 Fourier Cosine Transorm and Fourier Sine Transorm 8 Any unction may be split into an even and an odd unction 2 2 E O Fourier transorm may be epressed in terms o the Fourier cosine transorm and Fourier sine transorm F k Ecos2 kd i Osin2k d

19 Real Even Function 9 I a real unction () is even, O() = F k Ecos2 kd cos2 kd 2 cos2 k I a unction is real and even, the Fourier transorm is also real and even d () F(k) k Bracewell, R. The Fourier Transorm and Its Applications, 3rd ed. New York: McGraw-Hill, 2.

20 Real Odd Function 2 I a real unction () is odd, E() = sin(..-k..) = -sin(..k..) F k i Osin2kd i sin2kd 2i sin2k d I a unction is real and odd, the Fourier transorm is imaginary and odd () F(k) Bracewell, R. The Fourier Transorm and Its Applications, 3rd ed. New York: McGraw-Hill, 2.

21 Symmetry Properties 2 () Real and even Real and odd Imaginary and even Imaginary and odd Real asymmetrical Imaginary asymmetrical Even Odd F(k) Real and even Imaginary and odd Imaginary and even Real and odd Comple hermitian Comple antihermitian Even Odd Hermitian: () = * (-) Antihermitian: () = -*(-)

22 Symmetry Properties 22 () F(k) Bracewell, R. The Fourier Transorm and Its Applications, 3rd ed. New York: McGraw-Hill, 2.

23 23 Fourier Transorm: Properties and Basic Theorems Linearity a bg af k bgk Shit theorem Modulation theorem Derivative 2i e k Fk cos2 k Fk k Fk k 2 i2kfk

24 24 Fourier Transorm: Properties and Basic Theorems Parseval's / Rayleigh s theorem 2 d F k 2 dk Fourier Transorm o the comple conjugated * * F k Similarity (a : real constant) Special case (a = -): a a F k a F k

25 Similarity Theorem 25 Fk 2 k F 2 2 k 5 k k F 5 5 k

26 Convolution 26 Convolution describes an action o an observing instrument (linear instrument) when it takes a weighted mean o a physical quantity over a narrow range o a variable Eamples: Photo camera: Measured photo is described by real image convolved with a unction describing the apparatus Spectrometer: Measured spectrum is given by real spectrum convolved with a unction describing limited resolution o the spectrometer Other terms or convolution: Folding (German: Faltung), composition product System theory: Linear time (or space) invariant systems are described by convolution

27 Convolution 27 Deinition o convolution y g g d Eric W. Weisstein. Convolution. wolram.com/convolution.html JAVA Applets:

28 Properties o the Convolution 28 Properties o the convolution g ( g h) g gh commutativity associativity ( g h) g h distributivity a g a g ag d d g d d g dg d The integral o a convolution is the product o integrals o the actors gd d g d

29 The Basic Theorems: Convolution 29 The Fourier transorm o the convolution o () and g() is given by the product o the individual transorms F(k) and G(k) g Fk Gk g e 2ik ' g ' d' d 2ik' 2ik ' e ' d' e g ' e 2 '' ' d' e g' ' d' ' 2ik' ik d Fk Gk Convolution in the spectral domain g Fk Gk

30 Cross-Correlation 3 Deinition o cross-correlation r g * t ( t) gt g t d ( t) g t * t gt g g! Fourier transorm: ( t) g t * t gt * F * k, (see FT rules) t ( t) g t F * k Gk

31 Signal sent by radar antenna Eample: Cross-Correlation Radar (Low Noise) Signal received by radar antenna 3 Time Time Cross correlation o the signals Time

32 Signal sent by radar antenna Eample: Cross-Correlation Radar Signal received by radar antenna 32 Time Time Cross correlation o the signals Time

33 Signal sent by ultrasonic transducer Eample: Cross-Correlation Ultrasonic Distance Measurement 33 Signal received by ultrasonic transducer Time Time Cross correlation o the two signals Pulse Timer Start Stop Transmitter Object Receiver d vt light 2 () t light Time d

34 Autocorrelation Theorem 34 Autocorrelation Autocorrelation theorem: Proo ( t) ( t) t * t t ( t), * t t d t * F * k t Fk 2 ( t) t F k F k F k * 2

35 Signal Eample: Autocorrelation Detection o a Signal in the Presence o Noise 35 Signal: Noise Time Autocorrelation unction: Peak at t = : Noise r Time

36 Signal Eample: Autocorrelation Detection o a Signal in the Presence o Noise 36 Signal: Sinusoid plus noise Time Autocorrelation unction: Peak at t = : Noise Period o r reveals that signal is periodic r Time

37 37

38 The Delta Function 38 Function or the description o point sources, point masses, point charges, surace charges etc. Also called impulse unction: Brie (unit area) impulse so that all measuring equipment is unable to resolve pulse Important attribute is not value at a given time (or position) but the integral Deinition o the delta unction (distribution, also called Dirac unction). or 2. d

39 The Delta Function 39 The siting (sampling) property o the delta unction d Sampling o () at = a a d a Fourier transorm: 2ik 2ik e d e

40 Fourier Transorm Pairs 4 Cosine unction cos II II k Sine unction sin I I i I I k

41 Fourier Transorm Pairs 4 Gaussian unction 2 2 e e k k P Rectangle unction o unit height and base P() sinck sinc sin k

42 Fourier Transorm Pairs 42 Triangle unction o unit height and area 2 sinc k k Constant unction (unit height) (k) k

43 Sampling 43 Sampling: Noting a unction at inite intervals Bracewell, R. The Fourier Transorm and Its Applications, 3rd ed. New York: McGraw-Hill, 2.

44 Sampling o a Signal 44 Signal Samples taken at equal intervals o time Sampled signal

45 Sampling 45 Signal + Signal 2 Signal 2 (very short) Samples taken at equal intervals o time Sampled signal does not contain the additional Signal

46 Aliasing 46 Sampling with a low sampling requency Sampled signal Apparent signal

47 Sampling 47 Sampling unction (sampling comb) III() Shah III III n n n n a a a Multiplication o () with III() describes sampling at unit intervals III n n n Sampling at intervals : III n n n Bracewell, R. The Fourier Transorm and Its Applications, 3rd ed. New York: McGraw-Hill, 2.

48 Sampling 48 Fourier transorm o the sampling unction III IIIk III III k III IIIk 2 III 2III2k here: = 2 Bracewell, R. The Fourier Transorm and Its Applications, 3rd ed. New York: McGraw-Hill, 2.

49 Sampling: Band limited signal 49 Band limited signal: F(k) = or k >k c () F(k) k c k Bracewell, R. The Fourier Transorm and Its Applications, 3rd ed. New York: McGraw-Hill, 2.

50 Sampling in the Two Domains 5 () F(k) III IIIk k c k - = = III IIIk Fk k c k c Bracewell, R. The Fourier Transorm and Its Applications, 3rd ed. New York: McGraw-Hill, 2.

51 Reconstruction o the Signal 5 III IIIk Fk k c k Convolution Multiplication with rectangular unction in order to remove additional signal components = = () Fourier Transorm F(k) k c Original signal in requency domain k k

52 Critical Sampling and Undersampling 52 () F(k) k c k 2k c III 2k c k 2k 2k III Fk c c Undersampling 2k c k c k c Bracewell, R. The Fourier Transorm and Its Applications, 3rd ed. New York: McGraw-Hill, 2.

53 Sampling 53 Sampling theorem A unction whose Fourier transorm is zero (F(k) = ) or k >k c is ully speciied by values spaced at equal intervals < /(2k c ). Alternative (simpliied) statement: The sampling requency must be higher than twice the highest requency in the signal Remark: This does not imply that a reconstruction is impossible or all cases i the sampling requency is lower. I additional knowledge about the signal is available (e.g. lower limit o Fourier transorm, single requency signal), reconstruction may be possible. Description o sampling at intervals : Multiplication o () with III(/ ) Reconstruction o the signal: Multiplication o the Fourier transorm III k F k with P and inverse transormation Or: Convolution o the sampled unction with a sinc-unction k 2 k c

ENSC327 Communications Systems 2: Fourier Representations. School of Engineering Science Simon Fraser University

ENSC327 Communications Systems 2: Fourier Representations. School of Engineering Science Simon Fraser University ENSC37 Communications Systems : Fourier Representations School o Engineering Science Simon Fraser University Outline Chap..5: Signal Classiications Fourier Transorm Dirac Delta Function Unit Impulse Fourier