Although the DFT is the major topic of this chapter, it's appropriate, now, to introduce the inverse discrete Fourier transform (IDFT). Typically we think of the DFT as transforming time-domain data into a frequency-domain representation. Well, we can reverse this process and obtain the original time- domain signal by performing the IDFT on the X(m) frequency-domain values. The standard expressions for the IDFT are

and equally,

Equation 3-23'

Remember the statement we made in Section 3.1 that a discrete time- domain signal can be considered the sum of various sinusoidal analytical frequencies and that the X(m) outputs of the DFT are a set of N complex values indicating the magnitude and phase of each analysis frequency comprising that sum. Equations (3-23) and (3-23') are the mathematical expressions of that statement. It's very important for the reader to understand this concept. If we perform the IDFT by plugging our results from DFT Example 1 into Eq. (3-23), we'll go from the frequency-domain back to the time-domain and get our original real Eq. (3-11') x(n) sample values of

Notice that Eq. (3-23)'s IDFT expression differs from the DFT's Eq. (3-2) only by a 1/N scale factor and a change in the sign of the exponent. Other than the magnitude of the results, every characteristic that we've covered thus far regarding the DFT also applies to the IDFT.

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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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