S 3.1 (P 1.19)
______________
(i) We know that w = W*T_s, i.e.,
w = 150*pi*(3/1000) = 0.45*pi
(ii) w is a rational multiple of pi, therefore x[n]
is periodic.
The period N is determined by expressing
w as
(r/N)*2*pi
where r and N have no common factors.
In this case,
w = (9/40)*2*pi
so N=40.
(iii) x[n] is constant for all n if
w = 2*k*pi
i.e.,
T_s = (2*k*pi)/(150*pi) = k*(13.3333.
..) ms
or
f_s = 1/T_s = 75/k samples/sec
for some integer k.
x[n] alternates in value between cos(q) and cos(q) if
w = pi + 2*k*pi
i.e.,
T_s = (0.5 + k)*(13.3333.
..) ms
or
f_s = 1/T_s = 75/(k+0.5) samples/sec
for some integer k.
S 3.2
_____
(i) By inspection, the period T is given by 0.5 seconds. Thus
W = 2*pi/T = 4*pi. Again by inspection, A=3.0, and since
x(0) = 3*sqrt(2)/2, it follows that
3*cos(0*t + q) = 3*sqrt(2)/2 = sqrt(2)/2
i.e., cos(q) = pi/4 or pi/4.
Since the waveform is
increasing at t=0, it follows that q = pi/4.
(ii) x[n] = 3*cos(W*T_s*n + q)
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 Spring '08
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