Question

The condition for orthogonality of two functions \( x_1(t) \) and \( x_2(t) \) in terms of correlation is

• A. \( R_{12} = 0 \)
• B. \( R_{12} = 1 \)
• C. \( R_{12} = \infty \)
• D. \( R_{11} = 0 \) and \( R_{22} = 0 \)

Hint:

Orthogonality means the inner product of two functions is zero: \( \langle x_1(t), x_2(t) \rangle = \int x_1(t)x_2^*(t)dt = 0 \). For real functions, \(x_2^*(t) = x_2(t)\).
\(R_{12}(\tau)\) is the cross-correlation function. Often, \(R_{12}\) in a simplified context like this refers to the value related to the inner product for orthogonality check.

Answer 1

The Correct Option is A

Two real functions (or signals) \(x_1(t)\) and \(x_2(t)\) are orthogonal over a given interval \([T_1, T_2]\) if their inner product is zero:

\[ \int_{T_1}^{T_2} x_1(t) x_2(t) dt = 0 \] 

The term \(R_{12}\) in the options typically refers to a measure of cross-correlation. If \(R_{12}\) specifically denotes the value of this inner product (or cross-correlation at zero lag, \(R_{12}(0) = \int x_1(t)x_2(t)dt\) for energy signals over \((-\infty, \infty)\)), then the condition for orthogonality is \(R_{12}=0\).

\(R_{11}(0) = \int x_1^2(t)dt\) is the energy of signal \(x_1(t)\). \(R_{11}=0\) would mean \(x_1(t)\) is a zero signal.

Thus, \(R_{12}=0\) signifies that the functions are uncorrelated in the sense of their inner product being zero, which is the definition of orthogonality.

\[ \boxed{R_{12} = 0} \]