For the single-sided spectra, write the signal in terms of cosines: x(t) = 10cos(4πt + π/8) + 6 s... more For the single-sided spectra, write the signal in terms of cosines: x(t) = 10cos(4πt + π/8) + 6 sin(8πt + 3π/4) = 10cos(4πt + π/8) + 6 cos(8πt + 3π/4 − π/2) = 10cos(4πt + π/8) + 6 cos(8πt + π/4) For the double-sided spectra, write the signal in terms of complex exponentials using Euler's theorem: x(t) = 5exp[(4πt + π/8)] + 5 exp[−j(4πt + π/8)] +3 exp[j(8πt + 3π/4)] + 3 exp[−j(8πt + 3π/4)] The two sets of spectra are plotted in Figures 2.1 and 2.2. Problem 2.2 The result is x(t) = 4e j(8πt+π/2) + 4e −j(8πt+π/2) + 2e j(4πt−π/4) + 2e −j(4πt−π/4) = 8cos(8πt + π/2) + 4 cos (4πt − π/4) = −8 sin (8πt) + 4 cos (4πt − π/4)
For the single-sided spectra, write the signal in terms of cosines: x(t) = 10cos(4πt + π/8) + 6 s... more For the single-sided spectra, write the signal in terms of cosines: x(t) = 10cos(4πt + π/8) + 6 sin(8πt + 3π/4) = 10cos(4πt + π/8) + 6 cos(8πt + 3π/4 − π/2) = 10cos(4πt + π/8) + 6 cos(8πt + π/4) For the double-sided spectra, write the signal in terms of complex exponentials using Euler's theorem: x(t) = 5exp[(4πt + π/8)] + 5 exp[−j(4πt + π/8)] +3 exp[j(8πt + 3π/4)] + 3 exp[−j(8πt + 3π/4)] The two sets of spectra are plotted in Figures 2.1 and 2.2. Problem 2.2 The result is x(t) = 4e j(8πt+π/2) + 4e −j(8πt+π/2) + 2e j(4πt−π/4) + 2e −j(4πt−π/4) = 8cos(8πt + π/2) + 4 cos (4πt − π/4) = −8 sin (8πt) + 4 cos (4πt − π/4)
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