Consider two continuous-time signals 𝑥(𝑡) and 𝑦(𝑡), both with energy 𝐸. If the cross-correlation function 𝑅𝑥𝑦(𝜏) reaches its maximum value of 𝐸 at 𝜏=0, what can be inferred about the relationship between 𝑥(𝑡) and 𝑦(𝑡)?Select one:a. They are perfectly correlated.b. They are orthogonal to each other.c. They are uncorrelated.d. They are anti-correlated
Question
Consider two continuous-time signals 𝑥(𝑡) and 𝑦(𝑡), both with energy 𝐸. If the cross-correlation function 𝑅𝑥𝑦(𝜏) reaches its maximum value of 𝐸 at 𝜏=0, what can be inferred about the relationship between 𝑥(𝑡) and 𝑦(𝑡)?Select one:a. They are perfectly correlated.b. They are orthogonal to each other.c. They are uncorrelated.d. They are anti-correlated
Solution
The correct answer is a. They are perfectly correlated.
Here's why:
The cross-correlation function Rxy(τ) measures the similarity between two signals as a function of the time lag applied to one of them. When τ=0, it means there is no time lag between the two signals.
If the cross-correlation function reaches its maximum value at τ=0, it means the two signals are most similar at this point. Since the maximum value of the cross-correlation function is equal to the energy E of the signals, this indicates that the signals are perfectly correlated.
Perfect correlation means that every time one signal increases or decreases, the other signal does the same. In other words, the signals move together.
Therefore, the relationship between x(t) and y(t) is that they are perfectly correlated.
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