Scalloping is the name used to describe fluctuations in the overall magnitude response of an N-point DFT. Although we derive this fact in Section 3.16, for now we'll just say that when no input windowing function is used, the sin(x)/x shape of the sinc function's magnitude response applies to each DFT output bin. Figure 3-19(a) shows a DFT's aggregate magnitude response by superimposing several sin(x)/x bin magnitude responses.[] (Because the sinc function's sidelobes are not key to this discussion, we don't show them in Figure 3-19(a).) Notice from Figure 3-19(b) that the overall DFT frequency-domain response is indicated by the bold envelope curve. This rippled curve, also called the picket fence effect, illustrates the processing loss for input frequencies between the bin centers.

[] Perhaps Figure 3-19(a) is why individual DFT outputs are called "bins." Any signal energy under a sin(x)/x curve will show up in the enclosed storage compartment of that DFT's output sample.

Figure 3-19. DFT bin magnitude response curves: (a) individual sin(x)/x responses for each DFT bin; (b) equivalent overall DFT magnitude response.

From Figure 3-19(b), we can determine that the magnitude of the DFT response fluctuates from 1.0, at bin center, to 0.637 halfway between bin centers. If we're interested in DFT output power levels, this envelope ripple exhibits a scalloping loss of almost –4 dB halfway between bin centers. Figure 3-19 illustrates a DFT output when no window (i.e., a rectangular window) is used. Because nonrectangular window functions broaden the DFT's main lobe, their use results in a scalloping loss that will not be as severe as the rectangular window[10,12]. That is, their wider main lobes overlap more and fill in the valleys of the envelope curve in Figure 3-19(b). For example, the scalloping loss of a Hanning window is approximately 0.82, or –1.45 dB, halfway between bin centers. Scalloping loss is not, however, a severe problem in practice. Real-world signals normally have bandwidths that span many frequency bins so that DFT magnitude response ripples can go almost unnoticed. Let's look at a scheme called zero padding that's used to both alleviate scalloping loss effects and to improve the DFT's frequency resolution.

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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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