Final answer:
In digital signal processing, discrete-time cosine signals can appear identical due to aliasing. Signals with frequencies that differ by multiples of 2π will be identical. Hence, signals 1 and 2, as well as signals 3 and 4 from the question are identical due to these aliasing effects.
Step-by-step explanation:
The topic discussed in this question involves digital signal processing (DSP) and the concept of aliasing. Specifically, the question requires us to determine which discrete-time (DT) cosine signals are equal to each other. This involves assessing the frequency components of each cosine signal and relating them through the principles of DSP theory, specifically the sampling theorem and aliasing effects.
To compare the DT cosine signals given, we need to consider the periodic nature of the cosine function and recall that a cosine wave of frequency f and its alias at frequency -(f - 2π) will look the same in the discrete domain. Considering this, we can rewrite the provided signals to find that:
- xi[n] = cos(2πn/3) is identical to xi[n] = cos(4πn/3) because cos(4πn/3) = cos(2πn - 2πn/3) = cos(-2πn/3).
- xi[n] = cos(8πn/3) and xi[n] = cos(10πn/3) are also identical since cos(10πn/3) = cos(2πn*3 - 2πn/3) = cos(-2πn/3).
Therefore, cosine signals 1 and 2 are identical, as well as signals 3 and 4, due to the properties of cosine and aliasing.