Fourier Series Hlya Yaln 1 Hlya Yaln 2

Fourier Series Hülya Yalçın © 1

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Nice Java Applet http: //www. falstad. com/fourier/ Fourier series java applet Hülya Yalçın © 12

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Well, there also -Fourier Transform -Discrete Fourier Transform - etc How all these things are related? Hülya Yalçın © 29

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Going from Fourier Series to Fourier Transform Hülya Yalçın © 39

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Famous Fourier Transforms Sine wave Delta function

Famous Fourier Transforms Gaussian

Famous Fourier Transforms Sinc function Square wave

Famous Fourier Transforms Sinc function Square wave

Famous Fourier Transforms Exponential Lorentzian

FFT of FID

FFT of FID

FFT of FID

Effect of changing sample rate

Effect of changing sample rate

Effect of changing sample rate Lowering the sample rate: Reduces the Nyquist frequency, which Reduces the maximum measurable frequency Does not affect the frequency resolution

Effect of changing sampling duration

Effect of changing sampling duration

Effect of changing sampling duration Reducing the sampling duration: Lowers the frequency resolution Does not affect the range of frequencies you can measure

Effect of changing sampling duration

Effect of changing sampling duration

Measuring multiple frequencies

Measuring multiple frequencies

Trigonometric Fourier series: example Hülya Yalçın © 75

Complex Exponential Fourier Series Hülya Yalçın © 76

Complex Exponential Fourier Series Hülya Yalçın © 77

Symmetry Properties for real signals Hülya Yalçın © 78

Amplitude and phase spectra Hülya Yalçın © 79

Fourier Transform Hülya Yalçın © 80

Recap: Fourier Series Hülya Yalçın © 81

Spectral content of aperiodic signals: the Fourier transform Hülya Yalçın © 82

Example: rectangular pulse Hülya Yalçın © 83

Inverse Fourier Transform Hülya Yalçın © 84

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Properties of Fourier Transform Hülya Yalçın © 86

Properties of Fourier Transform: Linearity Hülya Yalçın © 87

Properties of Fourier Transform: Time-shift Hülya Yalçın © 88

Multiplication by a complex exponential Multiplication by a cosine Hülya Yalçın © 89

Convolution in time-domain Hülya Yalçın © 90

Parseval’s Theorem Hülya Yalçın © 91

Duality Example Hülya Yalçın © 92

Generalized Fourier Transform Hülya Yalçın © 93

Fourier Transform of cosine Hülya Yalçın © 94

Fourier transform of a periodic signal Hülya Yalçın © 95

Bandlimited and timelimited signals Hülya Yalçın © 96

Frequency response of LTI systems Hülya Yalçın © 97

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Some useful links http: //www. falstad. com/fourier/ Fourier series java applet http: //www. jhu. edu/~signals/ Collection of demonstrations about digital signal processing http: //www. ni. com/events/tutorials/campus. htm FFT tutorial from National Instruments http: //www. cf. ac. uk/psych/Culling. J/dictionary. html Dictionary of DSP terms http: //jchemed. chem. wisc. edu/JCEWWW/Features/Mcad. In. Chem/mcad 0 08/FT 4 Free. Ind. Decay. pdf Mathcad tutorial for exploring Fourier transforms of free-induction decay Hülya Yalçın © 106

Review: vectors Hülya Yalçın © 107

Vector Spaces Hülya Yalçın © 108

Signal Spaces Hülya Yalçın © 109

The space of periodic signals Hülya Yalçın © 110

Complete orthonormal sets of functions Hülya Yalçın © 111

Fourier Series Hülya Yalçın © 112

Fourier Coefficients Hülya Yalçın © 113

Convergence of Fourier Series Hülya Yalçın © 114

Trigonometric Fourier Series Hülya Yalçın © 115

Trigonometric Fourier Series Hülya Yalçın © 116

Trigonometric Fourier Series Hülya Yalçın © 117

Symmetry Properties Hülya Yalçın © 118

Trigonometric Fourier series: example Hülya Yalçın © 119