By Amy Dahan-Dalmedico, Jeanne Peiffer
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Use the zoom option in order to examine closely spaced results. Use the log option when the behavior for small values of the DFT is to be checked. 10 Objective To understand engineering units for one- and two-sided spectra, according to the signal class. Reminder The DFT of an N point sequence x(n) is X (k ) ϭ N Ϫ1 2π ∑ x(n) exp Ϫj N nk nϭ0 Fourier Methods 37 with X (Ϫk) ϭ X(N Ϫ k) Negative frequencies are associated with k Ͼ N/2 for a two-sided transform X(k). A translation of the section comprising the last N/2 … N transform values to the negative frequency region, to give the more classical symmetrical representation, with N/2 the Nyquist frequency.
Reminder ؕ x (t ) ϭ ∫ X( f ) exp( j2πf )df Ϫؕ ∫ X ( f ) ϭ TX k ϭ x (t ) exp(Ϫ j2π ft)dt T Aperiodic functions can be synthesized and decomposed by harmonic functions, spanning a continuous range of frequencies. Description Two types of transient signals to be synthesized can be chosen, as well as the number of synthesizing components. A dynamic display is updated as each additional component is added, until the maximum number is used. 5 shows the desired final signal, and the sum of components at each step.
10. The FFT, twoand one-sided spectral representations are shown in the left, middle and right lower plots. Three signal types can be chosen. The relevant EUs appear in the lower plots. 10 38 Discover Signal Processing: An Interactive Guide for Engineers Instructions/Tasks For all signal types, note the EU of the spectra. Next check the quantitative aspects of these, as related to the DFT pair x(n) and X(k). Finally observe the differences of the one- and two-sided representations, checking those of end points as well.