A New Twist to Fourier Transforms

Transcription

1 Hamish D. Meikle A New Twist to Fourier Transforms WILEY VCH WILEY-VCH Verlag GmbH &, Co. KGaA

2 Table ofcontents 1 The Fourier Transform and the Helix Fourier Transform Conventions Fourier Transforms in Physics Fourier Transform in Electrical Engineering Fourier Transform in Statistics The FourierTransform and the Helical Functions Radar and Sonar Echo Signals Colour Television Signals Modulation and Demodulation Communications Circularly PolarisedWaves Noise Other Forms ofthe Fourier Transform 8 2 Spiral and Helical Functions Complex Arithmetic Unary Operations Vector Addition and Subtraction Vector Multiplication Division Powers of Vectors Unbalanced Polyphase Voltages and Currents 18 3 Fourier Transforms From Fourier Series to Fourier Transform Fourier Series Period ofintegration for a Fourier Series Fourier Transform Inverse Transform Three ofthe Conventions for FourierTransforms Fourier Transforms and Spatial Spirals Properties offourier Transforms Addition, Subtraction, and Scaling- Linearity 34 A New 7Mst to rourier 7iansform. Hamish D. Meilde Copyright 2004 WILLYVCH Verlag GmbH &Co.KGM, Weinheim ISBN : ,4

3 Vlll I Table ofcontents Multiplication oftransforms Division Differentiation Moments Special Functions used for Fourier Transforms Summary offouriertransform Properties Examples of Fourier Transforms Cosine and Sine Waveforms Rectangular Pulse Triangular Pulse Ramp Pulse Gaussian Pulse Unequally Spaced Samples 60 4 Continuous, Finite, and Discrete Fourier Transforms Finite Fourier Transforms -Limited in Time or Space Discrete Fourier Transforms Cyclic Nature of Discrete Transforms Other Forms ofthe Discrete FourierTransform Summary of Properties Sampling Sampling Errors Sampling of Polyphase Voltages Examples of Discrete Fourier Transforms Finite Impulse Response Filters and Antennae The z-transform S Inverse Fourier Transforms, a Lowpass Filter inverse Fourier Transforms, a Highpass Filter Inverse Fourier Transforms, Bandpass and Bandstop Filters Arrays of Sensors, Linear Antennae Pattern Synthesis, the Woodward-Levinson Sampling Method Conversion of Analogue Signals to Digital Words 93 4,5.1 Dynamic Range Dynamic Range in Vector Systems 93 4,5.3 Quantisation Noise 95 4,5.4 Conversion Errors 95 4,5.5 Image Frequency or Negative Phase Sequence Component Power 97 5 Tapering Functions Conventions and Normalisation Parameters used with Tapering Functions Parameters A and C Efficiency Parameter n Noise Width Half-power Width 102

4 Table ofcontents I IX Parameters D and G Root Mean Square (rms) Width in Terms ofp' Root Mean Square (rms) Width in Terms of q' First Sidelobe or Sideband Height Fall-off Scalloping and Worst Case Processing Losses 104 5,2.11 Spectral Leakage or Two-tone Characteristics Other names for tapering functions Tapering Functions Trapezoidal Tapering (1-- 4p'2)n Tapering Cosine to the Power tt Tapering Cosine on a Pedestal Tapering Hamming, Blackman, and Blackman-Harris Tapering Truncated Gaussian tapering Dolph-Chebyshev Tapering with 10 Discrete Samples Taylor Tapering FourierTransforms in Statistics Basic Statistics Higher Moments Moment Generating Functions Properties of Characteristic Functions Linearity--Addition, Subtraction, and Scaling Multiplication Characteristic Functions ofsome Continuous Distributions Uniform Distribution Gaussian or Normal Distribution Family Cauchy Distribution Gamma Family ,5 Beta Family Weibull Distribution Characteristic Functions ofsome Discrete Distributions Binomial Distribution ,2 Hypergeometric Distribution Pascal, Negative Binomial, and Geometric Distributions ,4 Poisson Distribution An Example from Operational Research Noise and Pseudo-random Signals The Fourier Transform of Noise Optimum Filtering of Signals and Noise Auto- and Cross-correlation Functions Pseudo-random Signals Example of the Convolution of Radar Echo Signals and Noise 195

5 X I Table of Contents False Alarm probability with NNoise Samples Probability o Detection with Signal Plus Noise 197 Appendix A Glossary 203 Appendix B Maple Graphical Expressions 209 B.1 Fourier Transforms 209 B.2 Plotting Expressions Time Plot Frequency Plot, Spectrum, or Characteristic Function combined Plots 213 B.2.4 Exporting Plots 213 B.2.5 Stereographic Pairs 213 B.3 Other Types ofplots in Three Dimensions 214 Index 217