Question

Functions ϕi (t),i=1,2,3 are given as,

t if t∈[0,1]
ϕ₁ (t)={ else
0
​

t if t∈[0,2]
ϕ₂ (t)={ else
0
​


t if t∈[1,2]
ϕ₃ (t)={ else
0
​
a) Plot these functions. Are these functions orthogonal to each other?

Answer 1

After analyzing the inner product of the given functions φ1(t), φ2(t), and φ3(t), it is determined that none of these functions are orthogonal to each other since their intervals overlap and their inner products are not zero.

Step-by-step explanation:

To address the question of whether the functions φ1(t), φ2(t), and φ3(t) are orthogonal, we need to understand the concept of orthogonality in the context of functions. Two functions are orthogonal over a certain interval if their inner product (integral of their product over the interval) is zero. The functions provided are piecewise functions defined over differing intervals, and a simple plot of each function can yield:

• φ1(t) = t for t ∈ [0,1] and is 0 elsewhere.
• φ2(t) = t for t ∈ [0,2] and is 0 elsewhere.
• φ3(t) = t for t ∈ [1,2] and is 0 elsewhere.

By evaluating their inner products, we can determine orthogonality:

1. For φ1(t) and φ2(t), the inner product is non-zero since they both share the interval [0,1] where the functions are not zero.
2. For φ1(t) and φ3(t), they only overlap for t = 1, and the inner product over an interval of just a point is zero, which is not sufficient to claim orthogonality.
3. For φ2(t) and φ3(t), they share the interval [1,2], and thus, the inner product is non-zero.

Therefore, none of the functions φ1(t), φ2(t), and φ3(t) are orthogonal to each other.