Why is optimal beam-forming vector w orthogonal to channel signature?

1. How a function can be approximated by a set of orthogonal functions?
2. What does it mean for a signal to be orthogonal?
3. What makes two signals orthogonal?
4. What is beamforming vector?

How a function can be approximated by a set of orthogonal functions?

As these functions are orthogonal to each other, any two signals x_j(t), x_k(t) have to satisfy the orthogonality condition. i.e. Let a function f(t), it can be approximated with this orthogonal signal space by adding the components along mutually orthogonal signals i.e. All terms that do not contain C_k is zero.

What does it mean for a signal to be orthogonal?

In general, a signal set is said to be an orthogonal set if (s_k,s_j) = 0 for all k ≠ j. A binary signal set is antipodal if s₀(t) = −s₁ (t) for all t in the interval [0,T]. Antipodal signals have equal energy E, and their inner product is (s₀,s₁) = −E.

What makes two signals orthogonal?

Two signals are orthogonal if 〈y(t),x(t)〉 = 0. (Pythagorean Theorem). If signals x(t) and y(t) are orthogonal and if z(t) = x(t) + y(t) then Ez = Ex + Ey.

What is beamforming vector?

Beamforming is a technique used to improve the signal-to-noise ratio of received signals, eliminate undesirable interference sources, and focus transmitted signals to specific locations. Beamforming is central to systems with sensor arrays, including MIMO wireless communications systems such as 5G, LTE, and WLAN.