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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 “wraparound”). Then, X can be recovered
with a lowpass ﬁlter. 6 LECTURE 14. FREQUENCYDOMAIN 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 lowpass ﬁlter. (14.13)...
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This document was uploaded on 02/26/2014 for the course CS 6.02 at MIT.
 Fall '13
 HariBalakrishnan

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