Question

Discrete-time signals with different frequencies can be equal; this is one manifestation of aliasing and folded aliases. Determine which of the following discrete-time cosine signals are equal (i.e., plots of xi[n]versus n will be identical):

x_1[n] = cos (0.9pi n + 0.1pi)
x_2[n] = cos (-2.9pi n + 0.1pi)
x_3[n] = cos (5.1pi n - 0.1pi)
x_4[n] = cos (6.9pi n - 0.1pi)
x_5[n] = cos (-7.1pi n - 0.1pi)

Answer 1

Answer:

$x_4[n]\:\:and\:\:x_5[n]$

Explanation:

Firstly, x_1 and x_2 must be reviewed together. Then, x_3, x_4 and x_5 must be reviewed together. Since the phase differences (+0.1π and -0.1π) are different.

1)

$x_1[n]=\cos(0.9\pi n+0.1\pi)$

$x_2[n]=\cos(-2.9\pi n+0.1\pi)=\cos(-2.9\pi n+0.1\pi+4\pi n)=\cos(1.1\pi n+0.1\pi)$

$x_1[n]\:\:and\:\:x_2[n]\:\:are\:\:not\:\:equal$

2)

$x_3[n] = \cos (5.1\pi n - 0.1\pi)=\cos (5.1\pi n - 0.1\pi-4\pi n)=\cos (1.1\pi n - 0.1\pi)$

$x_4[n] = \cos (6.9\pi n - 0.1\pi)=\cos (6.9\pi n - 0.1\pi-6\pi n)=\cos (0.9\pi n - 0.1\pi)$

$x_5[n] =\cos (-7.1\pi n - 0.1\pi)=\cos (-7.1\pi n - 0.1\pi+8\pi n) =\cos (0.9\pi n - 0.1\pi)$

$x_4[n]\:\:and\:\:x_5[n]\:\:are\:\:equal\:\:,and\:\:x_3[n]\:\:is\:\:not\:\:equal\:\:to\:\:them$

Although the magnitude plot of $x_1[n]$ is equal to the magnitude plots of $x_4[n]\:\:and\:\:x_5[n]$, their phase plot of $x_1[n]$  is not equal to the phase plots of the $x_4[n]\:\:and\:\:x_5[n]$ .

Moreover, although the magnitude plots of $x_2[n]\:\:and\:\:x_3[n]$ are equal, their phase plot are not equal.