The frequency axis m of the DFT result in Figure 3-4 deserves our attention once again. Suppose we hadn't previously seen our DFT Example 1, were given the eight input sample values, from Eq. (3-11'), and asked to perform an 8-point DFT on them. We'd grind through Eq. (3-2) and get the X(m) values shown in Figure 3-4. Next we ask, "What's the frequency of the highest magnitude component in X(m) in Hz?" The answer is not "1." The answer depends on the original sample rate fs. Without prior knowledge, we have no idea over what time interval the samples were taken, so we don't know the absolute scale of the X(m) frequency axis. The correct answer to the question is to take fs and plug it into Eq. (3-5) with m = 1. Thus, if fs = 8000 samples/s, then the frequency associated with the largest DFT magnitude term is

If we said the sample rate fs was 75 samples/s, we'd know, from Eq. (3-5), that the frequency associated with the largest magnitude term is now

OK, enough of this—just remember that the DFT's frequency spacing (resolution) is fs/N.

To recap what we've learned so far:

  • each DFT output term is the sum of the term-by-term products of an input time-domain sequence with sequences representing a sine and a cosine wave,
  • for real inputs, an N-point DFT's output provides only N/2+1 independent terms,
  • the DFT is a linear operation,
  • the magnitude of the DFT results are directly proportional to N, and
  • the DFT's frequency resolution is fs/N.

It's also important to realize, from Eq. (3-5), that X(N/2+1), when m = N/2+1, corresponds to half the sample rate, i.e., the folding (Nyquist) frequency fs/2.

Prev don't be afraid of buying books Next

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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