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The purpose of this problem is to show that the representation of an arbitrary periodic signal by a Fourier series or, more generally, as a linear combination of any set of orthogonal functions is computationally efficient and in fact very useful for obtaining goad approximations of signals.$^{12}$Specifically, let $\left\{\phi_{1}(\mathrm{r})\right\}, i=0,\pm 1,\pm 2, \ldots$ be a set of orthonormal functions on the interval $a \leq t \leq b,$ and let $x(t)$ be a given signal. Consider the following approximation of $x(t)$ over the interval $a \leq t \leq b$: $\hat{x}_{n}(t)=\sum_{i=-N}^{+N} \boldsymbol{a}_{1} \boldsymbol{\phi}_{i}(t)$ Here, the $a_{t}$ are (in general, complex) constants. To measure the deviation between $x(t)$ and the series approximation $\hat{x}_{N}(t),$ we consider the error $e_{N}(t)$ defined as $e_{N}(t)=x(t)-\hat{x}_{N}(t)$.A reasonable and widely used criterion for measuring the quality of the approximation is the energy in the error signal over the interval of interest-that is, the integral of the square of the magnitude of the error over the inlerval $a \leq t \leq b$ : $E=\int_{a}^{b}\left|e_{N}(t)\right|^{2} d t$.(a) Show that $E$ is minimized by choosing $a_{i}=\int_{a}^{b} x(t) \phi_{i}^{*}(t) d t$.[Hint: Use eqs. $(\mathrm{P} 3.66-1)-(\mathrm{P} 3.66-3)$ to express $E$ in terms of $a_{i}, \phi_{1}(t),$ and $x(t)$. Then express $a,$ in rectangular coordinates as $a_{i}=b_{i}+j c_{i},$ and show that the equations $\frac{\partial E}{\partial b_{i}}=0 \quad$ and $\quad \frac{\partial E}{\partial c_{i}}=0, i=0,\pm 1,\pm 2, \ldots, N$ are satisfied by the $\left.a_{i} \text { as given by eq. }(P 3.66-4) .\right]$(b) How does the result of part \{a) change if $\boldsymbol{A}_{t}=\int_{a}^{b}\left|\phi_{i}(t)\right|^{2} d t$ and the $\left\{\phi_{1}(t)\right\}$ are orthogonal but not orthonornal?(c) Let $\phi_{n}(t)=e^{j n \alpha_{0} t},$ and choose any interval of length $T_{0}=2 \pi / \omega_{0} .$ Show that the $a_{1}$ that minimize $E$ are as given in eq. (3.50).(d) The set of Walsh functions is an often-used set of orthonormal functions. (See Problem 266.3 The set of five Walsh functions, $\phi_{0}(t), \phi_{1}(t), \ldots, \phi_{4}(t),$ is illustrated in Figure $P 3.66,$ where we have scaled time so that the $\phi_{i}(t)$ are nonzero and orthonormal over the interval $0 \leq t \leq 1 .$ Let $x(t)=\sin \pi t .$ Find the approximation of $x(t)$ of the form $\hat{x}(t)=\sum_{i=0}^{4} a_{i} \phi_{i}(t)$ such that $\int_{0}^{1}|x(t)-\hat{x}(t)|^{2} d t$ is minimized.(e) Show that $\hat{x}_{N}(t)$ in eq. $(P 3.66-1)$ and $e_{N}(t)$ in eq. $(P 3.66-2)$ are orthogonal if the $a_{i}$ are chosen as in $\operatorname{eqg}(\mathrm{P} 3.66-4)$.The results of parts (a) and (b) are extremely important in that they show that each coefficient $a$, is independent of all the other $a_{j}$ 's, $i \neq j$ Thus, if we add more terms to the approximation [e.g., if we compare the approximation $\left.\hat{x}_{N+1}(t)\right],$ the coefficients of $\phi_{1}(t), i=1, \ldots, N,$ that were previously determined will not change. In contrast to this, consider another type of series expansion, the polynomial Taylor series. The infinite Taylor series for $e^{\prime}$ is $e^{t}=1+t+t^{2} / 2 !+\ldots,$ but as we shall show, when we consider a finite polynomial series and the error criterion of eq. $(\mathbf{P} 3.66-3),$ we get a very different result.Specifically, let $\phi_{0}(t)=1, \phi_{1}(t)=t, \phi_{2}(t)=t^{2},$ and so on.(f) Are the $\phi_{1}(t)$ orthogonal over the merval $0 \leq t \leq 1 ?$(g) Consider an approximation of $x(t)=e^{t}$ over the therval $0 \leq t \leq 1$ of the form $\hat{x}_{n}(t)=a_{0} \phi_{0}(t)$.Find the value of $a_{0}$ that minimizes the energy in the emor signal over the interval.(h) We now wish to approximate $e^{t}$ by a Taylor series using two terms- i.e., $\hat{x}_{1}(t)=a_{0}+a_{1} t .$ Find the optimum values for $a_{0}$ and $a_{1}$. [Hint: Compute $E$ in terms of $a_{0}$ and $a_{1},$ and then solve the simultaneous equations $\frac{\partial \boldsymbol{E}}{\dot{\boldsymbol{\partial}} \boldsymbol{a}_{0}}=0 \quad$ and $\quad \frac{\partial \boldsymbol{E}}{\partial \boldsymbol{a}_{1}}=0$.Note that your answer for $a_{0}$ has changed from its value in part (g), where there was only one term in the series. Further, as you increase the number of terms in the series, that coefficient and all others will continue to change. We can thus see the advantage to be gained in expanding a function using orthogonal terms.]
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