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 expansion o a periodic unction (x) (period 2L, angular requency π/l) in terms o a sum o sine and cosine unctions with angular requencies that are integer multiples o π/l Any piecewise continuous unction (x) in the interval [-L,L] may be approximated with the mean square error converging to zero

3 Fourier Series 3 Sine and cosine unctions orm a complete orthogonal system over [-π,π] Basic relations (m,n ): δ mn : Kronecker delta δ mn = or m n, δ mn = or m = n { sin( nx), cos( nx) } π π π π sin cos π ( mx) sin( nx) dx = πδmn ( mx) cos( nx) dx = πδ mn ( mx) cos( nx) dx sin = π π π ( mx) dx sin = π π ( mx) dx cos =

4 Fourier Series: Sawtooth 4 Example: Sawtooth wave.8 (x) x / (2L)

5 Fourier Series 5 Fourier series is given by 2 n n= n= ( x) = a + a cos( nx) + b sin( nx) n where a n b n a = π = π = π π π π π π ( x)dx ( x) cos( nx)dx ( x) sin( nx)dx I the unction (x) has a inite number o discontinuities and a inite number o extrema (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 Fourier Series 6 For a unction (x) periodic on an interval [-L,L] instead o [-π,π] change o variables can be used to transorm the interval o integration Then and 2 x πx' L dx nπx' + L πdx' L ( x) a + a cos b sin = n n n= n= L nπx' L L a = dx L ( x' ) ' a n L = L L nπx' L ( x' ) cos dx' b n L = L L nπx' L ( x' ) sin dx'

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

8 Fourier Series: Sawtooth 8 Example: Sawtooth wave ( x) a a 2 L = n x = dx L 2L L 2 x nπx = dx L cos 2L L = = [ 2nπ cos( nπ ) sin( nπ )] sin( nπ ) 2 2 n π =.8 x 2L nπx = 2 π n= n L ( x) sin Finite Fourier series b n L 2 = L 2 x L nπx sin dx L 2nπ cos = 2 2 2n π = nπ ( 2nπ ) + sin( 2nπ ) (x) x / (2L)

9 Fourier Series: Sawtooth 9 Example: Sawtooth wave Fourier series: g ( x) = x 2L = 2 π m ( m, x) n= n sin nπx L.8.6 (x) x / (2L)

10 Fourier Series: Sawtooth x / (2L) 2 3 Amplitude a /2, b n n : Frequency ω / (π/l) x / (2L)

11 Fourier Series: Properties Around points o discontinuity, a "ringing, called Gibbs phenomenon occurs ( ) ( ) I a unction (x) is even, x nx is odd and thus Even unction: b n = or all n I a unction (x) is odd, x nx is odd and thus Odd unction: a n = or all n sin ( x) sin ( nx) dx = ( ) ( ) cos ( x) cos ( nx) dx = π π π π

12 Complex Fourier Series 2 Fourier series with complex coeicients ( x inx ) = A e n n= Calculation o coeicients: A n = 2π π π ( x) e inx dx Coeicients may be expressed in terms o those in the real Fourier series A n = 2π π π ( x) [ cos( nx) i sin( nx) ] dx = 2 a 2 2 ( a + ib ) n ( a ib ) n n n or n < or n = or n >

13 Fourier Transorm 3 Generalization o the complex Fourier series or ininite L and continuous variable k L, n/l k, A n F(k)dk F πikx = e dx ( k) ( x) Forward transorm or 2 F ( k ) = F ( ( x ) )( k) x Inverse transorm ( x) F( k) 2πikx = e dk ( x) = F ( F( k) )( x) k Symbolic notation ( x) F( k)

14 Forward and Inverse Transorm 4 I ( ). x dx exists 2. (x) has a inite number o discontinuities 3. The variation o the unction is bounded Forward and inverse transorm: For (x) continuous at x For (x) discontinuous at x = k x 2πikx 2πikx ( x) F ( F ( x) )( x) = e ( x) e dx dk F ( ( x) )( x) = ( ( x ) + ( x )) k Fx 2 +

15 Fourier Cosine Transorm and Fourier Sine Transorm 5 Any unction may be split into an even and an odd unction ( x) = [ ( x) + ( x) ] + [ ( x) ( x) ] = E( x) O( x) Fourier transorm may be expressed in terms o the Fourier cosine transorm and Fourier sine transorm F ( k) E( x) cos( 2πkx) dx i O( x) sin( 2πkx) = dx

16 Real Even Function 6 I a real unction (x) is even, O(x) = F ( k) = E( x) cos( 2πkx) dx = ( x) cos( 2πkx) dx = 2 ( x) cos( 2πkx) I a unction is real and even, the Fourier transorm is also real and even dx (x) F(k) x k Bracewell, R. The Fourier Transorm and Its Applications, 3rd ed. New York: McGraw-Hill, 2.

17 Real Odd Function 7 I a real unction (x) is odd, E(x) = F ( k) = i O( x) sin( 2πkx) dx = i ( x) sin( 2πkx) dx = 2i ( x) sin( 2πkx) dx I a unction is real and odd, the Fourier transorm is imaginary and odd (x) F(k) Bracewell, R. The Fourier Transorm and Its Applications, 3rd ed. New York: McGraw-Hill, 2.

18 Symmetry Properties 8 (x) 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 Complex hermitian Complex antihermitian Even Odd Hermitian: (x) = * (-x) Antihermitian: (x) = -*(-x)

19 Symmetry Properties 9 (x) F(k) Bracewell, R. The Fourier Transorm and Its Applications, 3rd ed. New York: McGraw-Hill, 2.

20 2 Fourier Transorm: Properties and Basic Theorems Linearity ( x) bg( x) a + ( k) bg( k) af + Shit theorem Modulation theorem Derivative ( x ) e 2πix k F( k) x ( x) cos( 2π k x) ( F ( k k ) + F( k + k )) 2 ( x) i2πkf ( k)

21 2 Fourier Transorm: Properties and Basic Theorems Parseval's / Rayleigh s theorem 2 ( x) dx F( k) 2 = dk Fourier Transorm o the complex conjugated * * ( x) F ( k) Similarity (a : real constant) ( ax) Special case (a = -): a F k a ( x) F( k)

22 Similarity Theorem 22 ( x) F( k) ( 2x) x k F 2 2 k ( 5x) x k k F 5 5 x k

23 Convolution 23 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 Examples: 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

24 Convolution 24 Deinition o convolution y ( x) g = ( τ ) g( x τ ) dτ = Eric W. Weisstein. Convolution. wolram.com/convolution.html JAVA Applets:

25 Properties o the Convolution 25 Properties o the convolution g = g commutativity ( g h) = ( g) h associativity ( g + h) = ( g) + ( h) distributivity a ( g) = ( a ) g = ( ag) d dx ( g) = d dx g = dg dx The integral o a convolution is the product o integrals o the actors ( g) dx = ( x) dx g( x) dx

26 The Basic Theorems: Convolution 26 The Fourier transorm o the convolution o (x) and g(x) is given by the product o the individual transorms F(k) and G(k) ( x) g( x) F( k) G( k) ( x) g( x) e 2πikx ( x' ) g( x x' ) dx' dx 2πikx' 2πik ( x x' ) [ e ( x' ) dx' ] e g( x x' ) [ ] = dx 2 ' = e 2 '' ( x' ) dx' e g( x' ') dx' ' πikx πikx = F( k) G( k) Convolution in the spectral domain ( x) g( x) F( k) G( k)

27 Cross-Correlation 27 Deinition o cross-correlation * ( t) ( t) g( t) = ( τ ) g( t τ ) dτ r g = + ( t) g ( t) = * ( t) g( t) g g! Fourier transorm: ( t) g ( t) = * ( t) g( t) * ( ) F *( k ), (see FT rules) t ( t) g ( t) F * ( k) G( k)

28 Signal sent by radar antenna Example: Cross-Correlation Radar (Low Noise) Signal received by radar antenna 28 Time Time Cross correlation o the signals Time

29 Signal sent by radar antenna Example: Cross-Correlation Radar Signal received by radar antenna 29 Time Time Cross correlation o the signals Time

30 Signal sent by ultrasonic transducer Example: Cross-Correlation Ultrasonic Distance Measurement 3 Signal received by ultrasonic transducer Time Time Cross correlation o the two signals Pulse Timer Start Stop Transmitter Θ Object Receiver d = vt light 2 (Θ ) t light Time d

31 Autocorrelation Theorem 3 Autocorrelation = * ( t) ( τ ) ( t τ ) dτ ( t) + Autocorrelation theorem: Proo ( t) ( t) = * ( t) ( t) ( t), ( t) F ( k ) 2 * ( t) F *( k ) ( t) ( t) F k F k = F k * ( ) ( ) ( ) 2

32 Example: Autocorrelation Detection o a Signal in the Presence o Noise 32 Signal: Noise Signal Time Autocorrelation unction: Peak at t = : Noise r Time

33 Example: Autocorrelation Detection o a Signal in the Presence o Noise 33 Signal: Sinusoid plus noise Signal Time Autocorrelation unction: Peak at t = : Noise Period o r reveals that signal is periodic r Time

34 34

35 The Delta Function 35 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). 2. ( x) = δ x δ or ( x) dx =

36 The Delta Function 36 The siting (sampling) property o the delta unction δ Sampling o (x) at x = a δ ( x ) ( x) dx = ( ) ( x a) ( x) dx = ( a) Fourier transorm: ( ) x x = 2πikx 2πikx δ δ ( x x ) e dx e

37 Fourier Transorm Pairs 37 Cosine unction cos( πx) II ( x) = δ x + δ x II( x) x k Sine unction sin( πx) I I ( x) = δ x + δ x i I I ( x) x k

38 Fourier Transorm Pairs 38 Gaussian unction 2 2 e πx e πk x k Π ( x) Rectangle unction o unit height and base Π(x) = 2 x x x < 2 = 2 > 2 sinc( k) ( x) sinc = sinπx πx x k

39 Fourier Transorm Pairs 39 Λ ( x) Triangle unction o unit height and area x x = x > 2 sinc ( k) x k Constant unction (unit height) δ(k) x k

40 Sampling 4 Sampling: Noting a unction at inite intervals Bracewell, R. The Fourier Transorm and Its Applications, 3rd ed. New York: McGraw-Hill, 2.

41 Sampling o a Signal 4 Signal Samples taken at equal intervals o time Sampled signal

42 Sampling 42 Signal + Signal 2 Signal 2 (very short) Samples taken at equal intervals o time Sampled signal does not contain the additional Signal

43 Aliasing 43 Sampling with a low sampling requency Sampled signal Apparent signal

44 Sampling 44 Sampling unction (sampling comb) III(x) Shah III III ( x) = δ ( x n) n= ( ) ax = a n= δ x Multiplication o (x) with III(x) describes sampling at unit intervals III n a ( x) ( x) = ( n) δ ( x n) n= Sampling at intervals τ: x III τ ( x) = τ ( nτ ) δ ( x nτ ) n= Bracewell, R. The Fourier Transorm and Its Applications, 3rd ed. New York: McGraw-Hill, 2.

45 Sampling 45 Fourier transorm o the sampling unction III ( x) III( k ) x III τ τ III( τk ) III( x) III( k) τ 2 III x 2 III( 2k) τ here: τ = 2 Bracewell, R. The Fourier Transorm and Its Applications, 3rd ed. New York: McGraw-Hill, 2.

46 Sampling: Band limited signal 46 Band limited signal: F(k) = or k >k c (x) F(k) x k c k Bracewell, R. The Fourier Transorm and Its Applications, 3rd ed. New York: McGraw-Hill, 2.

47 Sampling in the Two Domains 47 (x) F(k) τ x k c x III τ τ III( τk ) τ - k = = x III ( x) τ III( τk ) F( k) τ k c k c Bracewell, R. The Fourier Transorm and Its Applications, 3rd ed. New York: McGraw-Hill, 2.

48 Reconstruction o the Signal 48 x III ( x) τ III( τk ) F( k) τ k c k Convolution Multiplication with rectangular unction in order to remove additional signal components = = (x) Fourier x Transorm F(k) k c Original signal in requency domain k k

49 Critical Sampling and Undersampling 49 (x) F(k) x k c k 2k c III ( 2k c x) ( x) k 2k ( 2k ) III F( k ) c c Undersampling τ > 2k c k c k c Bracewell, R. The Fourier Transorm and Its Applications, 3rd ed. New York: McGraw-Hill, 2.

50 Summary: Sampling 5 Sampling theorem A unction whose Fourier transorm is zero (F(k) = ) or k >k c is ully speciied by values spaced at equal intervals τ < /(2k c ). Alternative (simpliied) statement: The sampling requency must be higher than twice the highest requency in the signal Description o sampling at intervals τ: Multiplication o (x) with III(x/ τ) Reconstruction o the signal: Multiplication o the Fourier ( ( ) ( )) transorm τ III τk F k with Π and inverse transormation Or: Convolution o the sampled unction with a sinc-unction k 2 k c

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