FOURIER TRANSFORMATION FOR PEDESTRIANS
CONTENTS INTRODUCTION 1 1 FOURIER SERIES 3 FOURIER SERIES 3 EVEN AND ODD FUNCTIONS 3 DE?NITION OF THE FOURIER SERIES 4 CALCULATION OF THE FOURIER COE?CIENTS 6 FOURIER SERIES IN COMPLEX NOTATION 11 THEOREMS AND RULES 13 LINEARITY THEOREM 13 THE FIRST SHIFTING RULE 14 THE SECOND SHIFTING RULE 17 SCALING THEOREM 21 PARTIAL SUMS, BESSEL’S INEQUALITY, PARSEVAL’S EQUATION 21 GIBBS’ PHENOMENON 24 DIRICHLET’S INTEGRAL KERNEL 24 INTEGRAL NOTATION OF PARTIAL SUMS 26 GIBBS’ OVERSHOOT 27 PLAYGROUND 30 2 CONTINUOUS FOURIER TRANSFORMATION 33 CONTINUOUS FOURIER TRANSFORMATION 33 EVEN AND ODD FUNCTIONS 33 THE ?-FUNCTION 34 FORWARD AND INVERSE TRANSFORMATION 35 POLAR REPRESENTATION OF THE FOURIER TRANSFORM 40 THEOREMS AND RULES 42 LINEARITY THEOREM 42 THE FIRST SHIFTING RULE 42 THE SECOND SHIFTING RULE 43 SCALING THEOREM 44 CONVOLUTION, CROSS CORRELATION, AUTOCORRELATION, PARSEVAL’S THEOREM 46 CONVOLUTION 46 CROSS CORRELATION 55 AUTOCORRELATION 56 PARSEVAL’S THEOREM 57 FOURIER TRANSFORMATION OF DERIVATIVES 58 PITFALLS 60 “TURN 1 INTO 3” 60 TRUNCATION ERROR 63 PLAYGROUND 66 3 WINDOW FUNCTIONS 69 THE RECTANGULAR WINDOW 69 ZEROS 70 INTENSITY AT THE CENTRAL PEAK 70 SIDELOBE SUPPRESSION 71 3 DB-BANDWIDTH 72 ASYMPTOTIC BEHAVIOUR OF SIDELOBES 73 THE TRIANGULAR WINDOW (FEJER WINDOW) 73 THE COSINE WINDOW 74 THE COS2-WINDOW (HANNING) 75 THE HAMMING WINDOW 77 THE TRIPLET WINDOW 78 THE GAUSS WINDOW 79 THE KAISER–BESSEL WINDOW 80 THE BLACKMAN–HARRIS WINDOW 81 84 87 PLAYGROUND 88 4 DISCRETE FOURIER TRANSFORMATION 89 DISCRETE FOURIER TRANSFORMATION 89 EVEN AND ODD SERIES AND WRAP-AROUND 89 THE KRONECKER SYMBOL OR THE “DISCRETE ?-FUNCTION” 90 DE?NITION OF THE DISCRETE FOURIER TRANSFORMATION 92 THEOREMS AND RULES 96 LINEARITY THEOREM 96 THE FIRST SHIFTING RULE 96 THE SECOND SHIFTING RULE 97 SCALING RULE/NYQUIST FREQUENCY 98 CONVOLUTION, CROSS CORRELATION, AUTOCORRELATION, PARSEVAL’S THEOREM 99 CONVOLUTION CROSS CORRELATION AUTOCORRELATION PARSEVAL’S THEOREM THE SAMPLING THEOREM DATA MIRRORING ZERO-PADDING FAST FOURIER TRANSFORMATION (FFT) PLAYGROUND 5 FILTER E?ECT IN DIGITAL DATA PROCESSING TRANSFER FUNCTION LOW-PASS, HIGH-PASS, BAND-PASS, NOTCH FILTER SHIFTING DATA DATA COMPRESSION DI?ERENTIATION OF DISCRETE DATA INTEGRATION OF DISCRETE DATA REFERENCES