4.4. DISTRIBUTION OF ENERGY IN FREQUENCIES
67
Note that (4.30) holds only for t in [− L, L]. For the same signal x(t), its Fourier transform is ZL x(t)e− jωt dt (4.32) X (ω) = t=− L
Comparing (4.31) and (4.32) yields cm = X (mω0 )/2L
(4.33)
for all m. In other words, the Fourier coefficients are simply the samples of the Fourier transform divided by 2L or X (ω)/2L. On the other hand, the Fourier transform X (ω) can be computed from the Fourier coefficients as ∞ X
X (ω) =
cm m=− ∞
2 sin[(ω − mω0
(4.34)
)L] ω − mω0
for all ω. See Problem 4.18. Thus for a signal of a finite duration, there is a one-to-one correspondence between its Fourier series and Fourier transform. For a signal of infinite duration, the Fourier series is applicable only if the signal is periodic. However the Fourier transform is applicable whether the signal is periodic or not. Thus the Fourier transform is more general than the Fourier series. 2. There is a unique relationship between a time function x(t) for all t in (− ∞, ∞) and its Fourier transform X (ω) for all ω in (− ∞, ∞). If x(t) is identically zero for some time interval, the information will be imbedded in X (ω) and the inverse Fourier transform of X (ω) will yield zero in the same time interval. For example, the Fourier transforms of a periodic signal of one period, two periods and ten periods are all different and their inverse Fourier transforms will yield zero outside one period, two periods, or ten periods. The Fourier series of one period, two periods, or ten periods are all the same. Its applicable time interval is specified separately. 3. The Fourier transform is a function of ω alone (does not contain explicitly t) and describes fully a time signal. Thus it is the frequency-domain description of the time signal. The Fourier coefficients cannot be used alone, they make sense only in the expression in (4.14) which is still a function of t. Thus the Fourier series is, strictly speaking, not a frequency-domain description. However, many texts consider the Fourier series to be a frequency-domain description and call its coefficients frequency spectrum. 4. The basic result of the Fourier series is that a periodic signal consists of fundamental and harmonics. Unfortunately, the expression remains the same no matter how long the periodic signal is. The Fourier transform of a periodic signal will also exhibit fundamental and harmonics. Moreover, the longer the periodic signal, the more prominent the harmonics. See Problem 5.14 and Figure 5.19 of the next chapter. In conclusion, the Fourier transform is a better description than the Fourier series in describing both periodic and aperiodic signals. Thus we downplay the Fourier series. The only result in the Fourier series that we will use in this text is the Fourier series of the sampling function discussed in Example 4.2.3.
4.4
Distribution of energy in frequencies
Every real-world signal satisfies, as discussed in Subsections 3.5.1 and 2.5.2 the Dirichlet conditions, thus its frequency spectrum is well defined. Furthermore the spectrum is bounded and continuous. Even so, its analytical computation is generally not possible. For example, there is no way to compute analytically the spectra of the signals in Figures 1.1 and 1.2. However they can be easily computed numerically. See the next chapter. Thus we will not
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