Question

# O Determine if the signals \phi_{1}(t)=t^{2} \phi_{2}(t)=6 t^{3}+4 defined over the interval -1 < t<1, form an orthogonal set. Justify your answer. .(7 points) An orthonormal signal set {o1(t), 2(t), \$3(t)} is defined over the inter val [-7, 7]and exactly represents a certain function f(t) over this interval as f(t)=\sqrt{2} \phi_{1}(t)-2 \phi_{2}(t)+\sqrt{3} \phi_{3}(t) Using Parseval's theorem, determine the energy of the signal f(t) over the interval [-7, 7]. ) Approximate a signal f(t)as a weighted sum of functions x1(t) and f(t) \approx c_{1} x_{1}(t)+c_{2} x_{2}(t) where the approximation error signal is e(t)=f(t)-c_{1} x_{1}(t)-c_{2} x_{2}(t) The signals f(t), e(t), a (t),Table 1 lists the relevant inner products. For example, (f,x1)the weighting coefficients c and c2 that minimizes the energyand r2(t) are real-valued and defined on the interval t1,t2).= 9. Determine the value of E_{e}=\int_{t_{1}}^{t_{2}} e^{2}(t) d t of the approximation error signal.  Fig: 1  Fig: 2  Fig: 3  Fig: 4  Fig: 5  Fig: 6  Fig: 7  Fig: 8  Fig: 9  Fig: 10  Fig: 11  Fig: 12  Fig: 13  Fig: 14  Fig: 15