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fourier_transform

# fourier_transform - The Fourier Transform Do not distribute...

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The Fourier Transform Do not distribute these notes 189 Prof. R.T. M’Closkey, UCLA

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Background Consider a continuous-time signal u on the interval ( -∞ , ) . This signal has finite energy if the limit lim τ →∞ Z τ - τ | u ( t ) | 2 dt, (69) exists. We will call signals that satisfy (69) “energy signals". For example, the impulse response of any stable linear system is an energy signal. Other examples inlcude any bounded signal that is non-zero for only a finite interval of time. Note that periodic signals do not have finite energy on the interval ( -∞ , ) because (69) does not exist. The periodic signals studied in this course, however, have finite energy in one period. A power signal is one which satisfies 0 < lim τ →∞ 1 2 τ Z τ - τ | u ( t ) | 2 dt < , (70) which is consistent with the notion that power is defined as energy per unit time . If u is periodic with period T then lim τ →∞ 1 2 τ Z τ - τ | u ( t ) | 2 dt = 1 T Z T | u ( t ) | 2 dt = 1 T k u k E . Thus, u is a power signal if it is periodic. Finally, note that energy signals have zero power. This section is concerned with energy signals and their Fourier transforms . To motivate the Fourier transform let’s return to periodic signals. Let u be a periodic signal with period T . The Fourier series coefficients are given by Z T u ( t ) e - jkω 0 t dt = Z T/ 2 - T/ 2 u ( t ) e - jkω 0 t dt, where ω 0 = 2 π/T . Imagine taking one period of u and then inserting zeros for some duration and repeating this process in order to make a periodic signal with longer period. For example, a periodic signal is shown in Fig. 1 and a modified version of the signal is shown in Fig. 2. The interesting fact is that the Fourier series coefficients for each case lie on the envelope function that we associate with the original periodic signal in Fig. 1. Note that in Fig. 2 the Do not distribute these notes 190 Prof. R.T. M’Closkey, UCLA
effective period is larger so that frequency spacing of the Fourier series coefficients along the envelope function is actually smaller . This is evident when comparing the amplitude spectrum plots in Figs. 1 and 2. If we insert more zeros as shown in Figs. 3 and 4, the spacing between Fourier series coefficients is reduced even further. Finally, in the limit as T → ∞ , we are left with a single pulse and its associated Fourier coefficients which have in fact “converged" to the envelope function. We are not submitting to a formal proof of this limit function but rather that this argument seems reasonable. Thus, the Fourier transform of an energy signal u , denoted ˆ u , is defined to be ˆ u ( ω ) = Z -∞ u ( t ) e - jωt dt. Previously in these notes we denoted this as “ χ ( ω ) ". We already showed that the frequency response function of an asymptotically stable linear system is actually the Fourier transform of its impulse response. The Fourier transform can be applied to any energy signal, though, not just those associated with the impulse response of a stable system. An energy signal can also be computed from its Fourier transform just like a periodic signal can be

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fourier_transform - The Fourier Transform Do not distribute...

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