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real-world signal detection have learned that their overall frequency resolution and signal sensitivity are affected much more by the size and shape of their window function than the mere size of their DFTs. Figure 3-18 Increased signal detection sensitivity afforded using windowing: (a) 64-sample product of a Hanning window and the sum of a 3.4 cycles and a 7 cycles per sample interval sinewaves; (b) reduced leakage Hanning DFT output response versus rectangular window DFT output response. As we become more experienced using window functions on our DFT input data, we\u2019ll see how different window functions have their own individual advantages and disadvantages. Furthermore, regardless of the window function used, we\u2019ve decreased the leakage in our DFT output from that of the rectangular window. There are many different window functions described in the literature of digital signal processing\u2014so many, in fact, that they\u2019ve been named after just about everyone in the digital signal processing business. It\u2019s not that clear that there\u2019s a great deal of difference among many of these window functions. What we find is that window selection is a trade-off between main lobe widening, first sidelobe levels, and how fast the sidelobes decrease with increased frequency. The use of any particular window depends on the application [5], and there are many applications. Windows are used to improve DFT spectrum analysis accuracy[6], to design digital filters[7,8], to simulate antenna radiation patterns, and even in the hardware world to improve the performance of certain mechanical force to voltage conversion devices[9]. So there\u2019s plenty of window information available for those readers seeking further knowledge. (The mother of all technical papers on windows is that by Harris[10]. A useful paper by Nuttall corrected and extended some portions of Harris\u2019s paper[11].) Again, the best way to appreciate windowing effects is to have access to a computer software package that contains DFT, or FFT, routines and start analyzing windowed signals. (By the way, while we delayed their discussion until Section 5.3, there are two other commonly used window functions that can be used to reduce DFT leakage. They\u2019re the Chebyshev and Kaiser window functions, which have adjustable parameters, enabling us to strike a compromise between widening main lobe width and reducing sidelobe levels.) 3.10 DFT Scalloping Loss 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\u2019ll just say that when no input windowing function is used, the sin(x)/x shape of the sinc function\u2019s magnitude response applies to each DFT output bin. Figure 3-19(a) shows a DFT\u2019s aggregate magnitude response by superimposing several sin(x)/x bin magnitude responses.\u2020 (Because the sinc function\u2019s sidelobes are not key to this discussion, we don\u2019t 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. \u2020 Perhaps Figure 3-19(a) is why individual DFT outputs are called \u201cbins.\u201d Any signal energy under a sin(x)/x curve will show up in the enclosed storage compartment of that DFT\u2019s 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\u2019re interested in DFT output power levels, this envelope ripple exhibits a scalloping loss of almost \u22124 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\u2019s main lobe, their use results in a scalloping loss that will not be as severe as with 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 \u22121.45 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\u2019s look at a scheme called zero padding that\u2019s used to both alleviate scalloping loss effects and to improve the DFT\u2019s frequency granularity. 3.11 DFT Resolution, Zero Padding, and Frequency-Domain Sampling One popular method used to improve DFT spectral estimation is known as zero padding. This process involves the addition of zero-valued data samples to an original DFT input sequence to increase the total number of input data samples. Investigating this zero-padding technique illustrates the DFT\u2019s important property of frequency-domain sampling alluded to in the discussion on leakage. When we sample a continuous time- domain function, having a continuous Fourier transform (CFT), and take the DFT of those samples, the DFT results in a frequency-domain sampled approximation of the CFT. The more points in our DFT, the better our DFT output approximates the CFT. To illustrate this idea, suppose we want to approximate the CFT of the continuous f(t) function in Figure 3-20(a). This f(t) waveform extends to infinity in both directions but is nonzero only over the time interval of T seconds. If the nonzero portion of the time function is a sinewave of three cycles in T seconds, the magnitude of its CFT is shown in Figure 3-20(b). (Because the CFT is taken over an infinitely wide time interval, the CFT has infinitesimally small frequency resolution, resolution so fine-grained that it\u2019s continuous.) It\u2019s this CFT that we\u2019ll approximate with a DFT. Figure 3-20 Continuous Fourier transform: (a) continuous time-domain f(t) of a truncated sinusoid of frequency 3/T; (b) continuous Fourier transform of f(t). Suppose we want to use a 16-point DFT to approximate the CFT of f(t) in Figure 3-20(a). The 16 discrete samples of f(t), spanning the three periods of f(t)\u2019s sinusoid, are those shown on the left side of Figure 3-21(a). Applying those time samples to a 16-point DFT results in discrete frequency- domain samples, the positive frequencies of which are represented by the dots on the right side of Figure 3-21 (a). We can see that the DFT output comprises samples of Figure 3-20(b)\u2019s CFT. If we append (or zero-pad) 16 zeros to the input sequence and take a 32-point DFT, we get the output shown on the right side of Figure 3-21 (b), where we\u2019ve increased our DFT frequency sampling by a factor of two. Our DFT is sampling the input function\u2019s CFT more often now. Adding 32 more zeros and taking a 64-point DFT, we get the output shown on the right side of Figure 3-21(c). The 64-point DFT output now begins to show the true shape of the CFT. Adding 64 more zeros and taking a 128-point DFT, we get the output shown on the right side of Figure 3-21(d). The DFT frequency-domain sampling characteristic is obvious now, but notice that the bin index for the center of the main lobe is different for each of the DFT outputs in Figure 3-21. Figure 3-21 DFT frequency-domain sampling: (a) 16 input data samples and N = 16; (b) 16 input data samples, 16 padded zeros, and N = 32; (c) 16 input data samples, 48 padded zeros, and N = 64; (d) 16 input data samples, 112 padded zeros, and N = 128. Does this mean we have to redefine the DFT\u2019s frequency axis when using the zero-padding technique? Not really. If we perform zero padding on L nonzero input samples to get a total of N time samples for an N-point

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