L14-15

8 which can be simplied to k 1 k 1 1 1 x k e j k

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Unformatted text preview: m , we can observe what happens to the Fourier coefficients. The product can be represented by two copies of the Fourier series representation for x[n], one shifted up by Ωm and one shifted down by Ωm , and each scaled by 1 . Mathematically, 2 x[n] cos Ωm [n] = ￿ K −1 ￿ k =− K X [k ]e j Ωk n ￿￿ ￿ 1 j Ωm n 1 −j Ωm n e +e , 2 2 (14.8) which can be simplified to = K −1 K −1 1￿ 1￿ X [ k ] e j (Ω k + Ω m )n + X [ k ] e j (Ω k − Ω m ) n . 2 2 k = −K (14.9) k =− K Equation 14.9 shows that the Fourier coefficients of the product are exactly the Fourier coefficients of x[n] (X [K ]), scaled and at the new frequencies of Ωk +Ωm and Ωk -Ωm . If the channel is ideal, so that Y = X , then demodulation by multiplying by cos Ωm n is given by ￿ ￿￿ ￿ K −1 K −1 1￿ 1￿ 1 j Ωm n 1 −j Ωm n j (Ω k + Ω m ) n j (Ω k − Ω m )n X [k ]e + X [k ]e e +e , 2 2 2 2 k =− K (14.10) k = −K which can be simplified to K −1 K −1 K −1 1￿ 1￿ 1￿ j (Ωk +2Ωm )n j (Ω k − 2 Ω m )n X [k ]e + X [k ]e + X [ k ] e j (Ω k )n . 4 4 2 k = −K k =− K (14.11) k =− K As is clear from (14.11), the process of multiplying by a cosine to modulate, and then a cosine to demodulate, results in a version of the original Fourier series for X , scaled by 1 2 , and two copies of the Fourier series representation for X , one shifted up by 2Ωm and one shifted down by 2Ωm , with each scaled by 1 . If X is bandlimited, so that X [k ] = 0 4 whenever |Ωk | ≥ |Ωm |, then the three sums in (14.11) have no overlapping terms (note: it must also be true that 3 ∗ |Ωm | ≤ π to avoid “wrap-around”). Then, X can be recovered with a low-pass filter. 6 LECTURE 14. FREQUENCY-DOMAIN SHARING AND FOURIER SERIES If demodulation is performed by multiplying by sin Ωm n, then ￿ ￿￿ ￿ K −1 K −1 1￿ 1￿ j j Ωm n j −j Ωm n j (Ω k + Ω m )n j (Ω k − Ω m )n X [k ]e + X [k ]e −e +e , 2 2 2 2 k = −K (14.12) k =− K which can be simplified to K −1 K −1 −j ￿ j￿ X [k ]ej (Ωk +2Ωm )n + X [ k ] e j (Ω k − 2 Ω m )n 4 4 k =− K k =− K and there is no unshifted version of the Fourier series of X to low-pass filter. (14.13)...
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This document was uploaded on 02/26/2014 for the course CS 6.02 at MIT.

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