Aliasing
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Two discrete-time signals can have different frequencies but be equal at every sample time n; for instance,
\sin(0.9\pi n) = \sin(0.9\pi n + 2\pi n) = \sin(2.9\pi n)
or
\cos(0.9\pi n) = \cos(0.9\pi n - 2\pi n) = \cos(-1.1\pi n) = \cos(1.1\pi n)\,.
When two discrete-time signals with different frequencies generate identical
samples, we say that the frequencies alias.
Consider the following:
x_1[n] = \cos(0.9\pi n + 0.1\pi)
x_2[n] = \cos(-2.9\pi n + 0.1\pi)
x_3[n] = \cos(5.1\pi n - 0.1\pi)
x_4[n] = \cos(6.9\pi n - 0.1\pi)
x_5[n] = \cos(-7.1\pi n - 0.1\pi)
Which of these signals (if any) are equal to x_1 at every sample?
Which of these signals (if any) are equal to x_2 at every sample?
Which of these signals (if any) are equal to x_3 at every sample?
Which of these signals (if any) are equal to x_4 at every sample?
Which of these signals (if any) are equal to x_5 at every sample?
