Lower side-band suppressed carrier (lsb-sc) modulation i: in lecture we studied how to modulate a (band-limited) signal x(t) such that we can remove the lsb from a signal r(t) and keep only the upper side-band (usb); we will assume that the bandwidth w of r(t) is less than the carrier frequency wc. note that for this we first produced the hilbert transform of the signal r(t), which we call (t), by passing it through an lti system with frf h(w) j sgn(w), where is the sign function. then we constructed yusb(t)t) cos(wct) - (t) sin(wct), which is the modulated signal with the lsb eschewed. answer the following with minimal calculations but using properties of fourier transforms (a) show that ylsb(t)-r(t) cos(wct) + (t) sin(wct) (b) show that you can decode using synchronous decoding just as with either the dsb or usb modulation schemes, i. e., by multiplying by cos(wct) and then using a (ideal) low-pass filter. hint: work exclusively in the fourier domain. determine the fourier transform of the hilbert transform x(t) using the fourier transform of x(t), then apply the consequences of modulation. use generic shapes for the fourier transform of x(t). the following trigonometric identities maybe useful too cos(a) cos(b) sin(a) cos(b) - sin( a) sin(b) cos(a+ b) +cos(a- b) sin(a + b) +sin(a - b) cos(a b) - cos(a b) 2