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DFT LINEARITY

The DFT has a very important property known as linearity. This property states that the DFT of the sum of two signals is equal to the sum of the transforms of each signal; that is, if an input sequence x1(n) has a DFT X1(m) and another input sequence x2(n) has a DFT X2(m), then the DFT of the sum of these sequences xsum(n) = x1(n) + x2(n) is

This is certainly easy enough to prove. If we plug xsum(n) into Eq. (3-2) to get Xsum(m), then

Without this property of linearity, the DFT would be useless as an analytical tool because we could transform only those input signals that contain a single sinewave. The real-world signals that we want to analyze are much more complicated than a single sinewave.

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Chapter One. Discrete Sequences and Systems

Chapter Two. Periodic Sampling

Chapter Three. The Discrete Fourier Transform

Chapter Four. The Fast Fourier Transform

Chapter Five. Finite Impulse Response Filters

Chapter Six. Infinite Impulse Response Filters

Chapter Seven. Specialized Lowpass FIR Filters

Chapter Eight. Quadrature Signals

Chapter Nine. The Discrete Hilbert Transform

Chapter Ten. Sample Rate Conversion

Chapter Eleven. Signal Averaging

Chapter Twelve. Digital Data Formats and Their Effects

Chapter Thirteen. Digital Signal Processing Tricks

Appendix A. The Arithmetic of Complex Numbers

Appendix B. Closed Form of a Geometric Series

Appendix C. Time Reversal and the DFT

Appendix D. Mean, Variance, and Standard Deviation

Appendix E. Decibels (dB and dBm)

Appendix F. Digital Filter Terminology

Appendix G. Frequency Sampling Filter Derivations

Appendix H. Frequency Sampling Filter Design Tables

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Understanding Digital Signal Processing
Understanding Digital Signal Processing (2nd Edition)
ISBN: 0131089897
EAN: 2147483647
Year: 2004
Pages: 183
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