Communication with modulation

Use the orthonormal waveforms in the above figure to approximate the function \[ x(t) = \sin{ \pi t/4 } \] over the interval \( 0 \le t \le 4 \) by the linear combination \[ \hat { x } (t)=\sum _{ n=1 }^{ 3 }{ { c }_{ n }{ \psi }_{ n }(t) }. \] Determine the expansion coefficients \(\{ c_{n}\} \) that minimize the mean-square approximation error \[ E=\int _{ 0 }^{ 4 }{ [x(t)-\hat { x } (t)]^{2} dt } .\] Then find \( \pi ( c_{1} + c_{2} + c_{3} ). \)

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