L14-15

# There are only certain values of for which ej n 1 we

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Unformatted text preview: ≤ Ω &lt; π . If the sequence is now also periodic with period N , Ω must satisfy an additional constraint. To see this, consider ej Ω(n+N ) = ej Ωn ej ΩN , ∀n ∈ (−∞, ∞) (14.3) and therefore ej ΩN = 1. There are only certain values of Ω for which ej ΩN = 1. We can now say that an N -periodic sine or cosine must have a frequency in the set ￿ ￿ 2 2π N 2π Ω ∈ {0, ± , ±2 , . . . , ± −1 , ± π }. (14.4) N N 2 N ￿ 14.3.2 The Fourier Series Any periodic function can be exactly represented with a Fourier series, a statement we will not prove here (the proof is given in the annotated slides of lecture 14). Instead, we will state the Fourier Series Theorem in the form we will ﬁnd most useful. For any periodic sequence, X , with period N , there exists a representation of that sequence as a sum of complex exponentials. That is, x [ n] = K −1 ￿ 2π N , 2 (14.5) 2π k, N (14.6) X [k ]ej N kn K = k =− K or to simplify notation, x [ n] = K −1 ￿ k = −K X [ k ] e j Ωk n Ωk = where the complex number, X [k ], is the referred to as the Fourier coefﬁcient associated with Ωk . Note that the Fourier series has 2K = N complex coefﬁcients, X [k ], − N ≤ K ≤ N − 1, but 2 2 5 SECTION 14.4. MODULATION AND DEMODULATION there are only N unique real values for X , x[0], ..., x[N − 1]. It may seem like there are too many Fourier coefﬁcients (because the Fourier coefﬁcients have a real and an imaginary part), but that is not the case. As shown in the Lecture 14 annotated notes, the real part of the Fourier coefﬁcients are an even function of k , and the imaginary part of the Fourier coefﬁcients is an odd function of k , real(X [k ]) = real(X [−k ]) imag (X [k ]) = −imag (X [−k ]). ￿ 14.4 (14.7) Modulation and Demodulation Modulating is deﬁned as multiplying our input X (whose nth value is x[n]) by a cosine (or sine) sequence. If we multiply x[n] by a cosine of frequency...
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## This document was uploaded on 02/26/2014 for the course CS 6.02 at MIT.

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