Lyons, Richard G - Understanding Digital Signal Processing
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Lyons, Richard G - Understanding Digital Signal Processing

DisciplinaProcessamento Digital de Sinais1.103 materiais4.084 seguidores
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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.
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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