x(t) = 1+x, (t) + x,(?) + x,(t) = 1+cos (6zt -+ cns (8r t) + cns ( 14z t +). x, (t), x,(t) and x, (t) are delayed by differeat amounts of time (from the origin). Calculale these delays. a) b) How can you show that x(t) is a periodic signal. Find the fundamental frequency and period T, of x(t). Mathematically explain which harmonics exist and which are nonexistent in x(t). c)

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D E F

x(t) = 1+x, (t) + x,(t) + x,(t) = 1+cos ( fiz t
cns ( 87 t) + cns (147t +).
x, (t), x2(t) and x, (t) are delayed by different amounts of time (from the origin).
a)
Calculate these delays.
How can you show that x(t) is a periodic signal. Find the fundamental frequency and
period T, uf x(t).
Mathematically explain which harmonics exist and which are nonexistent in x(t).
We know that the Fourier-series expansion of any periodic signal can be given in the
following three fumms:
b)
c)
d)
f(t) = a, +> (ae cos(kl,t) + b, sin(kant)}
k-1
= ao + E, vat + b {cos(klot – o)} = a, + E, 2|c4l?{cos(kh,t – )}
= E-C peikilat
By comparing x(t) with f(t) find a, h, Pk and c. Fill in the blanks in the table
below (including k), showing every slep of your caleulations.
k
klo
br
CK
e)
Fomulate x(t) in the above three forms.
Draw the magnitude and phase frequency spectrums of x(t).
Compute the average and RMS values of x(t).
g)
Transcribed Image Text:x(t) = 1+x, (t) + x,(t) + x,(t) = 1+cos ( fiz t cns ( 87 t) + cns (147t +). x, (t), x2(t) and x, (t) are delayed by different amounts of time (from the origin). a) Calculate these delays. How can you show that x(t) is a periodic signal. Find the fundamental frequency and period T, uf x(t). Mathematically explain which harmonics exist and which are nonexistent in x(t). We know that the Fourier-series expansion of any periodic signal can be given in the following three fumms: b) c) d) f(t) = a, +> (ae cos(kl,t) + b, sin(kant)} k-1 = ao + E, vat + b {cos(klot – o)} = a, + E, 2|c4l?{cos(kh,t – )} = E-C peikilat By comparing x(t) with f(t) find a, h, Pk and c. Fill in the blanks in the table below (including k), showing every slep of your caleulations. k klo br CK e) Fomulate x(t) in the above three forms. Draw the magnitude and phase frequency spectrums of x(t). Compute the average and RMS values of x(t). g)
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