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\section{The DFT in practice}
Now that we have illustrated the properties of the DFT as a mathematical object,
\subsection{Physical Interpretation}
see odds and ends
\subsection{Labeling the DFT axis}
When the discrete-time data signal is obtained by uniformly sampling a real-world signal, the DFT plot can be relabeled so that each DFT index corresponds to a real-world frequency in Hertz. Assume that the original signal is sampled at a frequency of $F_s$ samples per second; music, for instance, is often sampled using $F_s = 44100$.
The DFT bin number $k$, for $k \leq N/2$, corresponds to a sinusoidal generator that completes $k$ full counterclockwise rotations over $N$ samples; this is equivalent to $k$ cycles over $N/F_s$ seconds. For the remaining values of $k$, the associated generators complete $N-k$ clockwise rotations. The real-world frequency corresponding to each DFT bin is therefore
\begin{equation}
f_k = \begin{cases}
\displaystyle\frac{k}{N}F_s & 0 \leq k \leq N/2 \\ \\
\displaystyle\frac{k-N}{N}F_s & N/2 < k < N.
\end{cases}
\end{equation}
\subsection{Applications}
The Fourier transform, being a change of basis, represents a fundamental \textit{change in perspective:} the frequency domain allows us to look at the data from a different point of view and, if we are lucky, this new vantage point may highlight features that were not immediately visible in the time domain, or it may reveal some hidden structural pattern that we can exploit to analyze and process the information contained in the signal.
\itempar{Denoising} Consider the 1024-point signal plotted in Figure~\ref{fig:fa:mistery}; by a quick visual inspection we can easily detect an oscillatory component but it's not altogether clear if its period is constant and whether the changes in amplitude represent anything meaningful or if they are just noise. By looking at the same data in the frequency domain, we are presented with a much clearer picture. We can see in Figure~\ref{fig:fa:misteryDFT} that the real part of the DFT contains two peaks at indices $k = 64$ and $k = 960$; since there are no noticeable peaks at the corresponding locations in the imaginary part, we know from the discussion in Section~\ref{sec:fa:dft_elementary} that the signal will contain a sinusoidal component of the form
\[
A\cos\left(2\pi\frac{64}{1024}n \right) = A\cos\left(\frac{\pi}{8}n \right);
\]
since the amplitude of the Fourier coefficients at the peak are approximately equal to $1024/2 = 512$, we can assume $A=1$. Once we remove the peaks from the spectrum, we are left with a set of DFT coefficients that show no particular structure or pattern; this will be clearer once we study stochastic signal processing in Chapter~\ref{ch:rn}, but a random-looking, unstructured spectrum is the tell-tale sign of additive noise, as we would observe at the receiver's end of a communication channel over which a sinusoid was transmitted. Indeed, the original signal has been created by summing the cosine to a sequence $\sigma[n]$ where each sample is a random number uniformly distributed over the $[-2, 2]$ range so that
\[
x[n] = \cos\left(\frac{\pi}{8}n \right) + \sigma[n].
\]
In general, we are interested in removing the noise from a signal and, in this case, we are able to correctly identify the frequency of the sinusoid buried in noise by looking at the spectrum.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\center
\begin{dspPlot}[height=0.3\dspWidth,yticks=2,xout=true]{0,1024}{-3, 3}
\dspFunc[plotpoints=1024,linecolor=ColorDT]{x 1024 div 360 mul 64 mul cos rand 2147483647 div 0.5 sub 4 mul add}
\end{dspPlot}
\caption{Noisy signal in the time domain.}\label{fig:fa:mistery}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[b]
\center
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8.5184 -7.3253 32.2135 0.9185 -12.2617 37.4823 -29.6325 -6.3893 25.4368 -30.6156 9.7880 27.0488 43.7854 57.8213 -22.2382 36.2494 -29.5395 -20.0065 28.2966 15.1594 -22.7622 -42.0391 -4.4060 -36.0579 11.8772 -7.8731 -26.8984 25.0062 -8.1341 0.5083 -22.0461 0.0000 22.0461 -0.5083 8.1341 -25.0062 26.8984 7.8731 -11.8772 36.0579 4.4060 42.0391 22.7622 -15.1594 -28.2966 20.0065 29.5395 -36.2494 22.2382 -57.8213 -43.7854 -27.0488 -9.7880 30.6156 -25.4368 6.3893 29.6325 -37.4823 12.2617 -0.9185 -32.2135 7.3253 -8.5184 12.5919 19.7644 18.0886 -11.2458 17.3080 31.3749 7.6709 33.9984 3.4764 -23.7571 -16.4444 25.9115 3.3041 -20.5523 8.2151 37.3566 28.8989 11.0174 -28.5538 20.7331 -17.7010 -34.1950 -14.0108 12.0380 -12.1473 -10.9795 37.3001 63.2439 3.2556 -0.4680 -11.8822 16.7077 4.3255 10.6168 -14.1407 10.9963 3.5175 -27.6497 -15.5839 -28.9191 -22.9411 5.0867 28.1526 -43.7659 4.0255 7.8398 7.3461 15.5093 -5.9273 -13.9764 -30.3254 -32.0142 9.1783 4.0823 -34.0937 -10.0524 -12.5305 -2.2950 -20.5589 -6.1123 12.0799 -3.3974 -31.1600 2.3515 -15.2353 40.5576 8.6059 -4.8609 33.1129 -15.5122 -11.7729 26.6368 63.1508 -1.2763 -12.4180 15.1234 35.5033 17.3918 38.3182 -0.9102 9.8518 0.3749 8.1948 -41.2214 -35.5551 6.6833 51.5492 -5.0038 24.2865 -9.5901 -30.8376 -17.7035 -10.1660 -64.9064 14.0644 -1.7034 10.2208 -35.0884 29.0838 57.3552 -23.6084 33.1488 -15.7169 -25.0657 -41.7051 -4.9086 14.4071 -6.7180 0.7979 16.5875 6.0835 -53.1350 23.2243 -19.7793 -31.2407 14.0993 29.7887 -13.5245 68.5029 25.0333 41.5809 30.4146 13.8282 9.8172 -59.1258 -13.3751 15.5935 12.0196 -20.4238 -28.8955 10.0139 -38.4705 -10.4057 19.9051 -27.7483 -24.7847 24.0768 38.6479 17.1756 -18.3925 -21.3410 -3.4674 -21.1944 32.5922 18.5973 24.9932 -31.5238 2.3995 1.0166 29.1097 -19.6614 -0.7415 34.5680 34.7193 18.9402 13.6320 -0.2763 10.5979 -20.4123 -18.2011 -36.6584 -25.8490 -2.1541 -0.0348 10.1492 -41.4065 56.0860 -3.9283 15.1060 -3.5816 -6.5596 28.7996 23.6047 -30.5305 8.8250 -53.4462 11.6822 37.9561 -0.8036 -11.3600 -1.1976 4.6581 -14.0453 -15.4802 29.7667 -33.3379 -19.8856 -14.5631 54.2468 -1.2491 -39.7367 32.3050 -45.8054 -31.3553 21.5844 -26.8967 0.2956 -36.9062 -20.3810 38.1330 -9.9579 56.2703 14.3501 -2.1245 -16.1132 15.2390 6.7283 57.2200 5.6434 0.4633 -42.5532 -10.2172 50.0265 -49.1820 -37.0716 -7.9533 31.4079 17.9685 -11.3359 -9.1533 -12.2429 18.9037 -11.8931 -46.2012 -21.4831 -4.7908 -5.3757 0.3082 2.9280 -57.3504 1.5419 -32.8454 6.5470 -8.9597 17.7463 -26.1387 27.2176 -52.5012 -5.4941 1.6355 -34.7502 -26.5220 -33.5738 12.0468 16.6511 41.0443 -34.1140 -36.4058 -28.4501 2.8659 -2.3284 -9.0654 -1.6417 22.2922 27.2093 -10.1001 -8.7572 -31.3104 -36.0248 -34.4140 -10.0621 36.8702 32.5673 -28.7139 17.1142 26.9485 -6.6516 -7.4067 54.4694 68.2551 22.2593 45.3212 47.1539 -7.5279 5.1913 64.7273 -27.8904 21.2534 -17.9118 14.7628 -45.5446 -9.2772 -28.2431 -8.3021 68.7156 -1.8498 10.8989 -24.7510 -49.9905 1.1920 15.0506 -23.9909 -26.8919 -35.3457 14.1412 -8.5928 -16.3760 19.7002 8.9539 17.8820 34.4359 51.2600 -4.0823 1.8712 -25.2538 -4.3745 -3.4124 36.6624 11.7040 29.2360 -6.2425 11.9515 11.3453 33.2840 -18.5012 -26.1757 -8.2436 35.2984 0.1131 -26.4934 41.2068 -71.2839 8.4924 -41.4721 6.0543 -61.6112 46.1464 -31.9483 -39.3606 42.3604 -12.9450 7.0387 40.9283 -0.7189 -3.6236 45.6358 -34.1526 -18.1363 44.4881 15.3352 53.5478 -11.0049 6.4674 1.2552 13.6345 18.3111 18.5418 22.4265 15.3441 -17.6642 50.5312 -17.2781 2.6448 -16.4643 21.9495 -20.1078 21.1313 -2.4587 -2.7112 35.4158 -38.9406 -22.8851 29.2118 64.8528 -21.1498 -13.7386 -5.5878 -15.3519 4.2601 -24.7027 42.5357 -3.7960 -4.9297 -52.9568 39.9732 -7.5184 44.9211 -3.6851 -4.8691 8.9130 53.0143 -19.5522 14.7468 30.3041 34.8342 13.7251 -34.5211 15.0315 12.9881 18.1701 18.9793 4.9275 -16.7810 20.1705 52.4315 -38.3375 -16.9387 -2.4628 -44.7773 -53.0838 -8.5939 2.7924 -3.1910 -14.0272 -12.0399 -6.9726 5.2977 -43.2811 -14.3409 44.3269 -18.8259 -31.9437 -22.1225 27.5517 5.1457 12.4269 26.2181 11.8919 1.1018 2.2491 -25.1450 33.9462 8.5584 22.1950 -5.9418 -22.7845 2.5183 0.6453 43.6089 39.1061 21.0137 9.6406 17.3876 -6.9525 10.0981 -40.0794 -6.5178 31.8304 17.9756 -8.2193 15.3736 7.8735 -27.6148 16.5173 15.9402 2.8838 10.7081 2.6690 -8.4072 24.1858 -25.0391 -12.5130 -29.8414 -11.4370 16.4645 -12.9397 6.7983 -13.5887 7.0969 20.1369 35.8306 20.2117 -14.1198 -0.9530 32.4853 -6.7419 -17.6172 -8.5050 32.4758 -19.3499 -48.4952 -29.4615 9.1220 -26.7151 -10.9252 -52.5232 14.1742 0.8879 52.6496 -18.9229}
\end{dspPlot}
\caption{Real and imaginary parts of the DFT of the signal in Figure~\ref{fig:fa:mistery}.}\label{fig:fa:misteryDFT}
\end{figure}
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\clearpage
\itempar{Analysis of musical sounds.} The spectrum of musical sounds generally possesses a \textit{harmonic} structure: the frequency content is concentrated at increasing multiples of a \textit{fundamental} frequency, so that the Fourier transform has the typical comb-like pattern shown in Figure~\ref{fig:fa:cello}. The fundamental frequency determines the perceived \textit{pitch} of a musical note; subsequent spectral peaks are referred to as the \textit{partials} and the way that the overall energy is distributed across partials determines the perceived \textit{timbre} of a sound.
\clearpage
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