# 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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