L14-15

The discrete sequence x dened by the xn values is

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Unformatted text preview: e sequence x[n] = cos(2π f t) |t=nTs = cos(2π f Ts n), (14.1) where Ts = f1s . The discrete sequence, X defined by the x[n] values, is said to have a discrete frequency, Ω, defined as Ω = 2π f Ts = 2π f . fs We can therefore relate the continuous frequency f to its discrete equivalent Ω, in terms of the sampling frequency used to discretize the continuous waveform, fs . Furthermore, it suffices to consider discrete frequencies Ω in the range [−π , π ). The reason is that for any frequency outside this range, there is an equivalent frequency within this range. To see why, suppose x[n] = ej (π+φ)n and φ ∈ (0, 2π ) , which defines a sequence with frequency outside the range. Because ej πn = e−j πn , x[n] = ej (φ−π)n , which is a frequency in the range [−π , π ). If φ > 2π or φ < 0, we can simply substract out the largest integer multiple of 2π and apply the same argument, because ej 2πn = 1 for all integers n. ￿ 14.3 Periodicity and the Fourier Series The Fourier series can represent any periodic sequence, that is, any sequence for which there is some finite N such that x[n + N ] = x[n], ∀n ∈ (−∞, ∞). (14.2) The assumption of periodicity is not as limiting as it seems. One can make a periodic sequence out of a any finite-length sequence, just by repeating the sequence, as shown for N = 400 in Figure 14-2. In addition, we can assume that N is even (if N were odd, we can just double it to produce an even N for which Eq. (14.2) holds). 4 LECTURE 14. FREQUENCY-DOMAIN SHARING AND FOURIER SERIES Figure 14-2: Making a finite sequence periodic. ￿ 14.3.1 Discretized Frequencies There are some limitations imposed by assuming periodicity. In particular, the frequencies of sines and cosines are restricted to discrete values. We already know that for cosine and sine sequences with frequency Ω, we can limit consideration to −...
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This document was uploaded on 02/26/2014 for the course CS 6.02 at MIT.

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