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8.8. Consider the modulation system shown in Figure P8.8. The input signal $x(t)$ has a Fourier transform $X(j\omega )$ that is zero for $|\omega |>{\omega }^M$. Assuming that ${\omega }^c>{\omega }^M$, answer the following questions: (a) Is $y(t)$ guaranteed to be real if $x(t)$ is real? (b) Can $x(t)$ be recovered from $y(t)$ ? 

8.22. In Figure P8.22(a), a system is shown with input signal x(t) and output signal y() The input signal has the Fourier transform X(j) shown in Figure P8.22(b). Deter- mine and sketch Y(jo), the spectrum of y(1). x(t) y(t) -5w-3w 3w 5w -3w 3w ω cos(5wt) cos(3wt) Figure P8.22 632 Communication Systems Chap8 X(jo) -2w 2w Figure P8.22 Continued

8.41. In Problem 8.40, we introduced the concept of quadrature multiplexing, whereby two signals are summed after each has been modulated with carrier signals of iden- tical frequency, but with a phase difference of 90°. The corresponding discrete-time multiplexer and demultiplexer are shown in Figure P8.41. The signals xı[n] and x2[n] are both assumed to be band limited with maximum frequency wm, so that X;(ejw) = x2(ej“) = 0 for WM <w < 271 - wm. = = coswet H(jo) y(t) r(t) Demultiplexed outputs H(jw) y(t) sinct H(jw) -WM WM (b) Figure P8.40 Continued Coswon X,[n] r[n]=multiplexed signal X2[n] sinon (a) COS Hels) y, [n] r[n] Demultiplexed outputs H() y In] sinwon (b) Figure P8.41 040 (a) Determine the range of values for Wc so that xi[n] and x2[n] can be recovered from r[n]. (b) With we satisfying the conditions in part (a), determine H(ejw) so that yı[n] xi[n] and y2[n] x2[n].