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software package) often label the frequency axis in terms of the normalized f/fs variable. 3.13.4.3 DFT Frequency Axis Using a Normalized Angle We can multiply the above normalized f/fs frequency variable by 2\u3c0 to create a normalized angular notation representing frequency. Doing so would result in a frequency variable expressed as \u3c9=2\u3c0(f/fs) radians/sample. Using this notation, each X(m) DFT sample is associated with a normalized frequency of 2\u3c0m/N radians/sample, and our highest frequency is \u3c0 radians/sample as shown in Figure 3-34(b). In this scenario the sample spacing of X(m) is 2\u3c0/N radians/sample, and the DFT repetition period is one radian/sample as shown by the expressions in brackets in Figure 3-34(b). Using the normalized angular \u3c9 frequency variable is very popular in the literature of DSP, and its characteristics are described in the last row of Table 3-1. Unfortunately having three different representations of the DFT\u2019s frequency axis may initially seem a bit puzzling to a DSP beginner, but don\u2019t worry. You\u2019ll soon become fluent in all three frequency notations. When reviewing the literature, the reader can learn to convert between these frequency axis notation schemes by reviewing Figure 3-34 and Table 3-1. 3.13.5 Alternate Form of the DFT of an All-Ones Rectangular Function Using the radians/sample frequency notation for the DFT axis from the bottom row of Table 3-1 leads to another prevalent form of the DFT of the all-ones rectangular function in Figure 3-31. Letting our normalized discrete frequency axis variable be \u3c9 = 2\u3c0m/N, then \u3c0m = N\u3c9/2. Substituting the term N\u3c9/2 for \u3c0m in Eq. (3-48), we obtain (3-51) Equation (3-51), taking the third form of Eq. (3-34) sometimes seen in the literature, also has the DFT magnitude shown in Figures 3-32(b) and 3-32(c). 3.14 Interpreting the DFT Using the Discrete-Time Fourier Transform Now that we\u2019ve learned about the DFT, it\u2019s appropriate to ensure we understand what the DFT actually represents and avoid a common misconception regarding its behavior. In the literature of DSP you\u2019ll encounter the topics of continuous Fourier transform, Fourier series, discrete-time Fourier transform, discrete Fourier transform, and periodic spectra. It takes effort to keep all those notions clear in your mind, especially when you read or hear someone say something like \u201cthe DFT assumes its input sequence is periodic in time.\u201d (You wonder how this can be true because it\u2019s easy to take the DFT of an aperiodic time sequence.) That remark is misleading at best because DFTs don\u2019t make assumptions. What follows is how I keep the time and frequency periodicity nature of discrete sequences straight in my mind. Consider an infinite-length continuous-time signal containing a single finite-width pulse shown in Figure 3-35(a). The magnitude of its continuous Fourier transform (CFT) is the continuous frequency-domain function X1(\u3c9). If the single pulse can be described algebraically (with an equation), then the CFT function X1 (\u3c9), also an equation, can be found using Fourier transform calculus. (Chances are very good that you actually did this as a homework, or test, problem sometime in the past.) The continuous frequency variable \u3c9 is radians per second. If the CFT was performed on the infinite-length signal of periodic pulses in Figure 3-35(b), the result would be the line spectra known as the Fourier series X2(\u3c9). Those spectral lines (impulses) are infinitely narrow and X2(\u3c9) is well defined in between those lines, because X2(\u3c9) is continuous. (A well-known example of this concept is the CFT of a continuous squarewave, which yields a Fourier series whose frequencies are all the odd multiples of the squarewave\u2019s fundamental frequency.) Figure 3-35 Time-domain signals and sequences, and the magnitudes of their transforms in the frequency domain. Figure 3-35(b) is an example of a continuous periodic function (in time) having a spectrum that\u2019s a series of individual spectral components. You\u2019re welcome to think of the X2(\u3c9) Fourier series as a sampled version of the continuous spectrum in Figure 3-35(a). The time-frequency relationship between x2(t) and X2(\u3c9) shows how a periodic function in one domain (time) leads to a function in the other domain (frequency) that is a series of discrete samples. Next, consider the infinite-length discrete time sequence x(n), containing several nonzero samples, in Figure 3- 35(c). We can perform a CFT of x(n) describing its spectrum as a continuous frequency-domain function X3(\u3c9). This continuous spectrum is called a discrete-time Fourier transform (DTFT) defined by (see page 48 of reference [15]) (3-52) where the \u3c9 frequency variable is measured in radians/sample. To illustrate the notion of the DTFT, let\u2019s say we had a time sequence defined as xo(n) = (0.75)n for n \u2265 0. Its DTFT would be (3-53) Equation (3-53) is a geometric series (see Appendix B) and can be evaluated as (3-53\u2032) Xo(\u3c9) is continuous and periodic with a period of 2\u3c0, whose magnitude is shown in Figure 3-36. This is an example of a sampled (or discrete) time-domain sequence having a periodic spectrum. For the curious reader, we can verify the 2\u3c0 periodicity of the DTFT using an integer k in the following (3-54) Figure 3-36 DTFT magnitude |Xo(\u3c9)|. because e\u2212j2\u3c0kn = 1 for integer values of k. X3(\u3c9) in Figure 3-35(c) also has a 2\u3c0 periodicity represented by \u3c9s = 2\u3c0fs, where the frequency fs is the reciprocal of the time period between the x(n) samples. The continuous periodic spectral function X3(\u3c9) is what we\u2019d like to be able to compute in our world of DSP, but we can\u2019t. We\u2019re using computers and, sadly, we can\u2019t perform continuous signal analysis with the discrete (binary number) nature of computers. All of our processing comprises discrete numbers stored in our computer\u2019s memory and, as such, all of our time-domain signals and all of our frequency-domain spectra are discrete sampled sequences. Consequently the CFT, or inverse CFT, of the sequences with which we work will all be periodic. The transforms indicated in Figures 3-35(a) through 3-35(c) are pencil-and-paper mathematics of calculus. In a computer, using only finite-length discrete sequences, we can only approximate the CFT (the DTFT) of the infinite-length x(n) time sequence in Figure 3-35(c). That approximation is called the discrete Fourier transform (DFT), and it\u2019s the only DSP Fourier transform tool we have available to us. Taking the DFT of x1 (n), where x1(n) is a finite-length portion of x(n), we obtain the discrete periodic X1(m) spectral samples in Figure 3-35(d). Notice how X1(m) is a sampled version of the continuous periodic X3(\u3c9). That sampling is represented by (3-55) We interpret Eq. (3-55) as follows: X3(\u3c9) is the continuous DTFT of the N\u2212sample time sequence x1(n). We can evaluate X3 (\u3c9) at the N frequencies of \u3c9 = 2\u3c0m/N, where integer m is 0 \u2264 m \u2264 N\u22121, covering a full period of X3(\u3c9). The result of those N evaluated values is a sequence equal to the X1(m) DFT of x1(n). However, and here\u2019s the crucial point, X1(m) is also exactly equal to the CFT of the periodic time sequence x2 (n) in Figure 3-35(d). So when people say \u201cthe DFT assumes its input sequence is periodic in time,\u201d what they really mean is the DFT is equal to the continuous Fourier transform (the DTFT) of a periodic time-domain discrete sequence. After all this rigmarole, the end of the story is this: if a function is periodic, its forward/inverse DTFT will be discrete; if a function is discrete, its forward/inverse DTFT will be periodic. In concluding this discussion of the DTFT, we mention that in the literature of DSP the reader may encounter the following expression (3-56) as an alternate definition of the DTFT. Eq. (3-56) can be used to evaluate a full period of the DTFT of an x(n) sequence by letting the frequency variable

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