A Continuous time Periodic Signal X T is Real Valued and Has a Fundamental Period T 8
Problem 1
A continuous-time periodic signal $x(t)$ is real valued and has a fundamental period $r=8 .$ The nonzern Founer series coefficients for $x(t)$ are $a_{1}=a_{-1}=2, a_{3}=a_{-3}^{*}=4 j$.
Express $x(t)$ in the form $x(t)=\sum_{k=11}^{\infty} A_{k} \cos \left(\omega_{k} t+\phi_{k}\right)$.
Problem 2
A discrete-time periodic signal $x\lfloor\pi$ ) is real valued and has a fundamental period $N=5 .$ The nonzero Fourier series coefficients for $x[n]$ are $$a_{0}=1, a_{2}=a_{2}^{*}=e^{j \pi / 4}, a_{4}=a_{4}^{*}=2 e^{j \pi / 3}$$.
Express $x[n]$ in the form $$x[n]=A_{0}+\sum_{k=1}^{\infty} A_{i} \sin \left(\omega_{k} n+\phi_{k}\right)$$.
Problem 3
For the continuous-time periodic signal $x(t)=2+\cos \left(\frac{2 \pi}{3} t\right)+4 \sin \left(\frac{5 \pi}{3} t\right)$, determine the fundamental frequency $\omega_{0}$ and the Fourier series coefficients $a_{k}$ such that $x(t)=\sum_{k=-\infty}^{\infty} a_{k} e^{j k \omega_{0} t}$.
Problem 4
Use the Fouriet series analysis equation (3.39) to calculate the coefficients $a_{k}$ for the continuous-time periodic signal $x(t)=\left\{\begin{array}{ll}1.5, & 0 \leq t<1 \\ -1.5, & 1 \leq t<2\end{array}\right.$ with fundamental frequency $\omega_{0}=\pi$.
Problem 5
Let $x_{1}(t)$ be a continuous-time periodic signal with fundamental frequency $\omega_{1}$ and Fourier coefficients $a_{k}$. Given that $x_{2}(t)=x_{1}(1-t)+x_{1}(t-1)$, how is the fundamental frequency $\omega_{2}$ of $x_{2}(\mathrm{r})$ related to $\omega_{1} ?$ Atso, find a relationship between the Fourier series coefficients $b_{k}$ of $x_{2}(t)$ and the coefficients $a_{k}$. You may use the properties listed in Table 3.1.
Problem 6
Consider three continuous-time periodic signals whose Fourier series representations are as follows: $$x_{1}(t)=\sum_{k=0}^{100}\left(\frac{1}{2}\right)^{k} e^{j k \frac{2 \pi}{50}}$$, $$x_{2}(t)=\sum_{k=-100}^{100} \cos (k \pi) e^{j k \frac{2 \pi}{50} t}$$, $$x_{3}(t)=\sum_{k=-100}^{100} j \sin \left(\frac{k \pi}{2}\right) e^{j k \frac{2 \pi}{50} t}$$.
Use Forurier series properties to help answer the following questions:
(a) Which of the three signals is/are real valued?
(b) Which of the three signals is/are even?
Problem 7
Suppese the periodic signal $x(t)$ has fundamental period $T$ and Fourier coefficients $a_{k}$. In a variety of situations, it is easier to calculate the Fourier series coefficients $b_{x}$ for $g(t)=d x(t) f d t,$ as opposed to calculating $a_{k}$ directly. Given that $\int_{T}^{2 T} x(t) d t=2$, find an expression for $a_{k}$ in terms of $b_{k}$ and $T .$ You may use any of the properties listed in Table 3.1 to help find the expression.
Problem 8
Suppose we are given the following information about a signal $x(t)$:
1. $x(t)$ is real and odd.
2. $x(t)$ is periodic with period $T=2$ and has Fourier coefficients $a_{k}$.
3. $a_{k}=0$ for $|k|>1$.
4. $\left.\frac{1}{2}\right|_{0} ^{2}|x(t)|^{2} d t=1$.
Specify two different signals that satisfy these conditions.
Problem 9
Use the analysis equation (3.95) to evaluate the numerical values of one period of the Fourier series coefficients of the periodic signal $$x[n]=\sum_{m=-\infty}^{\infty}\{4 \delta[n-4 m]+8 \delta[n-1-4 m]\}$$.
Problem 10
Let $x[n]$ be a real and odd periodic signal with period $N=7$ and Fourier coefficients $a_{k},$ Given that $a_{15}=j, a_{10}=2 j, a_{17}=3 j$, determine the values of $a_{0}, a_{-1}, a_{-2},$ and $a_{-3}$.
Problem 11
Suppose we are given the following information about a signal $x[n]$:
1. $x[n]$ is a real and even signal.
2. $x[n]$ has period $N=10$ and Fourier coefficients $a_{k}$.
3. $a_{11}=5$.
4. $\frac{1}{10} \sum_{n=0}^{9}|x[n]|^{2}=50$.
Show that $x[n]=A \cos (B n+C),$ and specify numerical vatues for the constants $A$, $B,$ and $C$.
Problem 12
Each of the two sequences $x_{1}[n]$ and $x_{2}[n]$ has a period $N=4,$ and the corresponding Fourier series coefficients are specified as $\left.x_{1} | n\right] \longleftrightarrow a_{k}, \quad x_{2}[n] \leftrightarrow \rightarrow b_{k}$, where $a_{0}=a_{3} \div \frac{1}{2} a_{1}=\frac{1}{2} a_{2}=1 \quad$ and $\quad b_{1}=b_{1}=b_{2}=b_{3}=1$.
Using the multiplication property in Table 3.1 , determine the Founer series coefficients $c_{k}$ for the signal $g[n]=x_{1}[n] x_{2}[n]$.
Problem 13
Consider a continuous-time LTI system whose frequency response is $$H(j \omega)=\int_{-\infty}^{x} h(t) e^{-j \omega t} d t=\frac{\sin (4 \omega)}{\omega}$$.
If the input to this system is a periodic signal $$x(t)=\left\{\begin{array}{ll}1, & 0 \leq t<4 \\-1, & 4 \leq t<8\end{array}\right.$$ with period $T=8,$ determine the corresponding system output $y(t)$.
Problem 14
When the impulse train $x[n]=\sum_{k=-\infty}^{\infty} \delta[n-4 k]$ is the input to a particular LTI system with frequency response $H\left(e^{j \omega}\right),$ the output of the system is found to be $y[n]=\cos \left(\frac{5 \pi}{2} n+\frac{\pi}{4}\right)$. Determine the values of $H\left(e^{j k \pi / 2}\right)$ for $k=0,1,2,$ and $3$.
Problem 15
Consider a continuous-time ideal lowpass filter $S$ whose frequency response is $$H(j \omega)=\left\{\begin{array}{ll}1, & |\omega| \leq 100 \\0, & |\omega|>100\end{array}\right.$$.
When the input to this filter is a signal $x(t)$ with fundamental period $T=\pi / 6$ and Fourier series coefficients $a_{k},$ it is found that $x(t) \stackrel{s}{\longrightarrow} y(t)=x(t)$.
For what values of $k$ is it guaranteed that $a_{k}=0 ?$
Problem 16
Determine the output of the filter shown in Figure $\mathrm{P} 3.16$ for the following periodic inputs:
(a) $x_{1}[n]=(-1)^{n}$
(b) $x_{2}[n]=1+\sin \left(\frac{3 \pi}{8} n+\frac{\pi}{4}\right)$
(c) $x_{3}[n]=\sum_{k=-\infty}^{\infty}\left(\frac{1}{2}\right)^{n-4 k} u[n-4 k]$
Problem 17
Consider three continuous-time systems $S_{1}, S_{2},$ and $S_{3}$ whose responses to a cornplex exponential input $e^{j \text { 5t }}$ are specified as $$\begin{array}{l}S_{1}: e^{j 5 t} \longrightarrow t e^{j 5 t,} \\S_{2}: e^{j 5 t} \longrightarrow e^{j 5(t-1),} \\S_{3}: e^{j 5 t} \longrightarrow \cos (5 t).\end{array}$$
For each system, determine whether the given information is sufficient to conclude that the system is definitely not LTI.
Problem 18
Consider three discrete-time systems $S_{1}, S_{2},$ and $S_{3}$ whose respective responses 6 a complex exponential input $e^{j \pi / 2 n}$ are specified as $S_{1}: e^{j \pi n / 2} \rightarrow e^{j \pi n / 2} u[n]$, $S_{2}: e^{j \pi n / 2} \longrightarrow e^{j 3 \pi r 2}$, $S_{3}: e^{j \pi n / 2} \rightarrow 2 e^{j 5 \pi n / 2}$.
For each system, determine whether the given information is sufficient to conclude that the system is definitely not LTT.
Problem 19
Consider a causal LTI system implemented as the $R L$ circuit shown in Figure P3.19. A current source produces an input current $x(t),$ and the system output is considered to be the current $y(t)$ flowing through the inductor.
(a) Find the differential equation relating $x(t)$ and $y(t)$.
(b) Determine the frequency response of this system by considering the outpul of the system to inputs of the form $x(t)=e^{j \alpha i}$.
(c) Determine the output $y(t)$ if $x(t)=\cos (t)$.
Problem 20
Consider a causal LTI system implemented as the $R L C$ circuit shown in Figure P3.20. In this circuit, $x(t)$ is the input voltage. The voltage $y(f)$ across the capacitor is considered the system output.
(a) Find the differential equation relating $x(f)$ and $y(t)$.
(b) Determine the frequency response of this system by considering the output of the system to inputs of the form $\boldsymbol{x}(t)=e^{j \omega t}$.
(c) Deternine the output $y(t)$ if $x(t)=\sin (t)$.
Problem 21
A continuous-time periodic signal $x(t)$ is real valued and has a fundamental period $T=8 .$ The nonzero Fourier series coefficients for $x(t)$ are specified as $a_{1}=a_{-1}^{*}=j, a_{5}=a_{-5}=2$.
Express $x(t)$ in the form $x(t)=\sum_{k=0}^{\infty} A_{k} \cos \left(w_{k} t+\phi_{k}\right)$.
Problem 22
Determine the Fourier series representations for the following signals:
(a) Each $x(t)$ illustrated in Figure $P 3.22(a)-(f)$.
(b) $x(t)$ periodic with period 2 and $x(t)=e^{-t} \quad$ for $\quad-1<t<1$
(c) $x(t)$ periodic with period 4 and $x(t)=\left\{\begin{array}{ll}\sin \pi t, & 0 \leq t \leq 2 \\ 0, & 2<t \leq 4\end{array}\right.$
Problem 23
In each of the following, we specify the Fourier series coefficients of a continuous time signal that is periodic with period $4 .$ Determine the signal $x ( t$ ) in each case.
(a) $a_{k}=\left\{\begin{array}{ll}0, & k=0 \\ (j)^{k} \frac{\sin k \pi^{14}}{k \pi}, & \text { otherwise }\end{array}\right.$
(b) $a_{k}=(-1)^{k} \frac{\sin k \pi / 8}{2 k \pi}$
(c) $a_{k}=\left\{\begin{array}{ll}j k, & |k|<3 \\ 0, & \text { otherwise }\end{array}\right.$
(d) $a_{k}=\left\{\begin{array}{ll}1, & k \text { even } \\ 2, & k \text { odd }\end{array}\right.$
Problem 24
Let $x(t)=\left\{\begin{array}{ll}t, & 0 \leq t \leq 1 \\ 2-t, & 1 \leq t \leq 2\end{array}\right.$ be a periodic signal with fundamental period $T=2$ and Fourier coefficients $a_{k}$.
(a) Determine the value of $a_{0}$.
(b) Determine the Fourier series representation of $d x(t) ldt$.
(c) Use she result of part (b) and the differentiation property of the continuous-time Fourier series to help determine the Fourier series coefficients of $x(t)$.
Problem 25
Consider the following three continuous-time signals with a fundamental period of $T=1 / 2$: $\mathbf{x}(t)=\cos (4 \pi t)$, $y(t)=\sin (4 \pi t)$, $z(t)=x(t) y(t)$.
(a) Determine the Fourier series coefficients of $x(t)$.
(b) Determine the Fourier series coefficients of $y(t)$.
(c) Use the results of parts (a) and (b), along with the multiplication property of the continuous-time Fourier series, to determine the Fourier series coefficients of $z(t)=x(t) y(t)$.
(d) Determine the Fourier series coefficients of $z(t)$ through direct expansion of $z(t)$ in trigonometric form, and compare your result with that of part $(c)$.
Problem 26
Let $x(t)$ be a periodic signal whose Fourier series coefficients are $a_{k}=\left\{\begin{array}{ll}2, & k=0 \\ j\left(\frac{1}{2}\right)^{|k|}, & \text { othervise }\end{array}\right.$.
Use Fourier series properties to answer the following questions:
(a) Is $x(t)$ real?
(b) Is $x(t)$ even?
(c) Is $d x(t) / d t$ even?
Problem 27
A discrete-time periodic signal $x[n]$ is real valued and has a fundamental period $N=5 .$ The nonzero Fourier series coefficients for $x[n]$ are $a_{0}=2, a_{2}=a_{-2}^{*}=2 e^{j \pi / 6}, \quad a_{4}=a_{-4}^{*}=e^{j \frac{\pi}{3}}$
Express $x[n]$ in the form $x[n]=A_{0}+\sum_{k=1}^{\infty} A_{k} \sin \left(\omega_{k} n+\phi_{k}\right)$.
Problem 28
Determine the Fourier series coefficients for each of the following discrete-time periodic signals. Plot the magnitude and phase of each set of coefficients $a_{k}$.
(a) Each $x[n]$ depicted in Figure $P 3.28\{a,-(c)$
(b) $x[n]=\sin (2 \pi n / 3) \cos (\pi n / 2)$
(c) $x[n]$ periodic with period 4 and $x[n]=1-\sin \frac{\pi \pi}{4} \quad$ for $0 \leq n \leq 3$
(d) $x[n]$ periodic with period 12 and $x[n]=1-\sin \frac{\pi n}{4} \quad$ for $0 \leq n \leq 11$
Problem 29
In each of the following, we specify the Fourier series coefficients of 1 signal that is periodic with period 8 Determine the signal $x[n]$ in each case.
(a) $a_{k}=\cos \left(\frac{k \pi}{4}\right)+\sin (\frac{3 k \pi}{4})$
(b) $a_{k}=\left\{\begin{array}{ll}\sin \left(\frac{k \pi}{3}\right), & 0 \leq k \leq 6 \\ 0, & k=7\end{array}\right.$
(c) $a_{k}$ as in Figure $P 3.29(a)$
(d) $a_{k}$ as in Figure $P 3.29(b)$
Problem 30
Consider the following three discrete-time signals with a fundamental period of 6:
$x | n]=l-\cos \left(\frac{2 \pi}{6} n\right)$, $y[n]=\sin \left(\frac{2 \pi}{6} n+\frac{\pi}{4}\right)$, $z[n]=x[n] y[n]$.
(a) Determine the Fourier series coefficients of $x[n]$.
(b) Determine the Fourier series coefficients of $y[n]$.
(c) Use the results of parts $(a)$ and $(b),$ along with the multiplication property of the discrete-time Fourier series, to determine the Fourier series coefficients of $z[n]=x[n] y[n]$.
(d) Determine the Fourier series coefficients of $z[n]$ through direct evaluation, and compare your result with that of part (c).
Problem 31
Let $x[n]=\left\{\begin{array}{ll}1, & 0 \leq n \leq 7 \\ 0, & 8 \leq n \leq 9\end{array}\right.$ be a pcriodic signal with fundamental period $N=10$ and Fourier series coefficients $a_{k} .$ Also, let $g[n]=x[n]-x[n-1]$.
(a) Show that $g[n]$ has a fundamental period of 10.
(b) Determine the Fourier series coefficients of $g[n]$.
(c) Using the Fourier series coefficients of $g \text { in }]$ and the First-Difference property in Table $3.2,$ determine $a_{k}$ for $k \neq 0$.
Problem 32
Consider the signal $x[n]$ depicted in Figure $P 3.32$ This signal is periodic with period $N=4 .$ The signal ean be cxpressed in terms of a discrete-time Founier series as
Problem 33
Consider a causal continuous-time LTI system whose input $x(t)$ and output $y(t)$ are related by the following differential equation: $\frac{d}{d t} y(t)+4 y(t)=x(t)$
Find the Fourier series representation of the output $y(t)$ for each of the following inputs
(a) $x(t)=\cos 2 \pi t$
(b) $x(t)=\sin 4 \pi t+\cos (6 \pi t+\pi / 4)$
Problem 34
Consider a continuous-time LTI system with impulse response
Problem 35
Consider a continuous-time LTI system $S$ whose frequency response is $H(j \omega)=\left\{\begin{array}{ll}1, & \{\omega | \geq 250 \\ 0 . & \text { otherwise }\end{array}\right.$. When the input to this system is a signal $x$ ( $t$ ) with fundamental period $T=\pi / 7$ and Fourier series coefficients $a_{k},$ it is found that the output $y(t)$ is identical to $x(t)$. For what values of $k$ is it guaranteed that $a_{k}=0 ?$
Problem 36
Consider a causal discrete-time LTI system whose input $x[n]$ and output $y[n]$ are related by the following difference equation: $y[n]-\frac{1}{4} y[n-1]=x[n]$
Find the Fourier series representation of the output $y[n]$ for each of the following inputs:
(a) $x[n]=\sin \left(\frac{3 \pi}{4} n\right)$
(b) $x[n]=\cos \left(\frac{\pi}{4} n\right)+2 \cos \left(\frac{\pi}{2} n\right)$
Problem 37
Consider a discrete-time LTI system with impulse response $h[n]=\left(\frac{1}{2}\right)^{|n|}$.
Find the Fourier series representation of the output $y[n]$ for each of the following inputs:
(a) $x[n]=\sum_{k--\infty}^{\infty} \delta[n-4 k]$
(b) $x[n]$ is periodic with period 6 and $x[n]=\left\{\begin{array}{ll}1, & n=0,\pm 1 \\ 0, & n=\pm 2,\pm 3\end{array}\right.$
Problem 38
Consider a discrete-time LTI system with impulse response $h[n]=\left\{\begin{array}{ll}1, & 0 \leq n \leq 2 \\ -1, & -2 \leq n \leq-1 \\ 0, & \text { otherwise }\end{array}\right.$.
Given that the input to this system is $x[n]=\sum_{k=-\infty}^{+\infty} \delta[n-4 k]$ determine the Fourier series coefficients of the output $y[n]$.
Problem 39
Consider a discrete-time LTI system $S$ whose frequency response is $H\left(e^{j \omega}\right)=\left\{\begin{array}{ll}1, & |\omega| \leq \frac{\pi}{8} \\ 0, & \frac{\pi}{8}<| \omega_{i}<\pi\end{array}\right.$ Show that if the input $x[n]$ to this system has a period $N=3,$ the output $y[n]$ has only one nonzero Fourier series coefficient per period.
Problem 40
Let $x(t)$ be a periodic signal with fundamental period $T$ and Fourier series cocfficients $a_{k} .$ Derive the Fourier series coefficients of each of the following signais in terms of $a_{k}$.
Problem 41
Suppose we are given the following information about a continuous-time periodic signal with period 3 and Fourier coefficients $a_{k}$.
1. $a_{k}=a_{k+2}$.
2. $a_{k}=a_{-k}$.
3. $\int_{-05}^{0.5} x(t) d t=1$.
4. $\int_{0.5}^{15} x(t) d t=2$.
Determine $x(t)$.
Problem 42
Let $x(t)$ be a real-valued signal with fundamental period $T$ and Fourier series coefficients $a_{k}$.
(a) Show that $a_{k}=a_{-k}^{*}$ and $a_{0}$ must be real.
(b) Show that if $x(t)$ is even, then its Fourier series coefficients must be real and even.
(c) Show that if $x(f)$ is odd, then its Fourier series coefficients are imaginary and odd and $a_{0}=0$.
Problem 43
(a) A continuous-time periodic signal $x(t)$ with period $T$ is said to be odd harmonic if, in its Fourier series representation
Problem 44
Suppose we are given the following information about a signal $x(t)$.
1. $x(t)$ is a real signal.
2. $x(t)$ is periodic with period $T=6$ and has Fourier coefficients $a_{k}$.
3. $a_{k}=0$ for $k=0$ and $k>2$.
4. $x(t)=-x(t-3)$.
5. $\frac{1}{6}\left[\begin{array}{l}3 \\ -3\end{array}|x(r)|^{2} d t=\frac{1}{2}\right.$
6. $a_{1}$ is a positive real number.
Show that $x(t)=A \cos (B t+C),$ and determine the values of the constants $A, B$, and $C$.
Problem 45
Let $x(t)$ be a real periodic signal with Fourier series representation given in the sine-cosine form of eq. (3.32)$;$ i.e.
$x(t)=a_{0}+2 \sum_{k=1}^{\infty}\left[B_{k} \cos k \omega_{0} t-C_{k} \sin k \omega_{0} t\right]$
(a) Find the exponential Fourier series representation of the even and odd parts of $x(t) ;$ that is, find the coefficients $\alpha_{k}$ and $\beta_{k}$ in terms of the coefficients in eq. $(P 3.45-1)$ so that $8 w\{x(t)\}=\sum_{k=\cdots \infty}^{+\infty} \alpha_{k} e^{j k \omega_{0} f}$, $O d\{x(t)\}=\sum_{k=-\infty}^{\infty} \beta_{k} e^{j k_{w_{0}}}$
(b) What is the relationship between $\alpha_{k}$ and $\alpha_{-k}$ in part $(a) ?$ What is the relationship between $\beta_{k}$ and $\beta_{-k} ?$
(c) Suppose that the signals $x(t)$ and $z(t)$ shown in Figure $P 3.45$ have the sine-cosine series representations
$x(t)=a_{0}+2 \sum_{k=1}^{\infty}\left[B_{k} \cos \left(\frac{2 \pi k t}{3}\right)-C_{k} \sin \left(\frac{2 \pi k t}{3}\right)\right]$, $z(t)=d_{0}+2 \sum_{k=1}^{\infty}\left[E_{k} \cos \left(\frac{2 \pi k r}{3}\right)-F_{k} \sin \left(\frac{2 \pi k t}{3}\right)\right]$.
Sketch the signal $y(t)=4\left(a_{0}+d_{0}\right)+2 \sum_{k=1}^{\pi}\left\{\left[B_{k}+\frac{1}{2} E_{k}\right] \cos \left(\frac{2 \pi k t}{3}\right)+F_{k} \sin \left(\frac{2 \pi k t}{3}\right)\right\}$.
Problem 46
In this problem, we derive two important properties of the continuous-time Fourier series: the multiplication property and Parseval's relation. Let $x(t)$ and $y(t)$ both be continuous-time periodic signals having period $T_{0}$ and with Fourier series representations given by
(a) Show that the Fourier series coefficients of the signal $z(t)=x(t) y(t)=\sum_{k=-\infty}^{+\infty} c_{k} e^{j k \omega_{0}}$ are given by the discrete convolution $c_{k}=\sum_{n=-\infty}^{+\infty} a_{n} b_{k-n}$.
(b) Use the result of part (a) to compute the Fourier series coefficients of the signals $x_{1}(t), x_{2}(t),$ and $x_{3}(t)$ depicted in Figure $P 3.46$.
(c) Suppose that $y(t)$ in eq. $(P 3.46-1)$ equals $x^{*}(t) .$ Express the $b_{k}$ in the equation in terms of $a_{k}$, and use the result of part (a) to prove Parseval's relation for periodic signals- that is, $\frac{1}{T_{0}} \int_{0}^{T_{0}}|x(t)|^{2} d t=\sum_{k=-\infty}^{+\infty}\left|a_{k}\right|^{2}$.
Problem 47
Consider the signal $x(t)=\cos 2 \pi r$.
since $x(t)$ is periodic with a fundamental period of $1,$ it is also periodic with a period of $N,$ where $N$ is any positive integer. What are the Fourier series coefficients of $x(t)$ if we regard it as a periodic signal with period $3 ?$
Problem 48
Let $x[n]$ be a periodic sequence with period $N$ and Fourier series representation $x[n]=\sum_{k=\langle N\rangle} a_{k} e^{j k(2 \pi i N) n}$.
The Fourier series coefficients for each of the following signals can be expressed in terms of $a_{k}$ in eq. $(P 3.48-1) .$ Derive the expressions.
(a) $x\left[n-n_{0}\right]$
(b) $x[n]-x[n-1]$
(c) $\left.x[n]-x\left[n-\frac{N}{2}\right] \text { (assure that } N \text { is even }\right)$
(d) $x[n]+x\left[n+\frac{N}{2}\right]$ (assume that $N$ is even; note that this signal is periodic with period $N / 2\}$
(e) $x^{*}[-n]$
(f) $(-1)^{n} x[n]$ (assume that $N$ is even)
(g)$(-1)^{n} x[n]$ (assume that $\dot{N}$ is odd: note that this signal is periodic with period $2 N$
(b) $y[n]=\left\{\begin{array}{ll}x[n], & n \text { even } \\ 0, & n \text { odd }\end{array}\right.$
Problem 49
Let $x[n]$ be a periodic sequence with period $N$ and Fourier series representation $x[n]=\sum_{k=<N>} a_{k} e^{j k(2 \pi i N) \eta}$.
(a) Suppose that $N$ is even and that $x[n]$ in eq. $(P 3.49-1)$ satisfies $x[n]=-x\left[n+\frac{N}{2}\right]$ for all $n$. Show that $a_{k}=0$ for all even integers $k$.
(b) Suppose that $N$ is divisible by $4 .$ Show that if $x[n]=-x\left[n+\frac{N}{4}\right]$ for all $n$, ther $a_{k}=0$ for every value of $k$ that is a multiple of 4.
(c) More generally, suppose that $\mathrm{N}$ is divisible by an integer $\mathrm{M}$. Show that if $\sum_{r=0}^{(N(M)-1} x\left[n+r \frac{N}{M}\right]=0$ for all $n$, then $a_{k}=0$ for every value of $k$ that is a multiple of $M$.
Problem 50
Suppose we are given the following information about a periodic signal $x[n]$ with period 8 and Fourier coefficients $a_{k}$.
1. $a_{k}=-a_{k-4}$.
2. $x | 2 n+1]=(-1)^{n}$.
Sketch one period of $x[n]$.
Problem 51
Let $x[n]$ be a periodic signal with period $N=8$ and Fourier series coefficients $a_{r}=-a_{k-4} .$ A signal $y|n|=\left(\frac{1+(-1)^{n}}{2}\right) x[n-1]$ with period $N=8$ is generaled. Denoting the Fourier series coefficients of $y[n]$ by $b_{k},$ find a function $f[k]$ such that $b_{k}=f[k] a_{k}$.
Problem 52
$x[n]$ is a real periodic signal with period $N$ and complex Fourier series coefficients $a_{k},$ Let the Cartesian form for $a_{k}$ be denoted by $a_{k}=b_{k}+j c_{k}$. where $b_{k}$ and $c_{k}$ are both real .
(a) Show that $a_{-k}=a_{k}^{*} .$ What is the relation between $b_{k}$ and $b_{\ldots k} ?$ What is the relation between $c_{k}$ and $c_{-k} ?$
(b) Suppose that $N$ is even. Show that $a_{N / 2}$ is real.
(c) Show that $x[n]$ cart also be expressed as a trigonometric Fourier series of the form $x[n]=a_{0}+2 \sum_{k=1}^{(N-1)^{2}} b_{k} \cos \left(\frac{2 \pi k n}{N}\right)-c_{k} \sin \left(\frac{2 \pi k n}{N}\right)$ if $N$ is odd or as $x | n]=\left(a_{0}+a_{N / 2}(-1)^{n}\right)+2 \sum_{k=1}^{(N-2)(2)} b_{k} \cos \left(\frac{2 \pi k n}{N}\right)-c_{k} \sin \left(\frac{2 \pi k n}{N}\right)$ if $N$ is even.
Problem 53
Let $x[n]$ b a real periodic signal with period $N$ and Fourier coefficients $a_{s}$.
(a) Show that if $N$ is even, at least two of the Fourier coefficients within one period of $a_{k}$ are real.
(b) Show that if $N$ is odd, at least one of the Fourier coefficients within one period of $a_{k}$ is real.
Problem 54
Consider the function $a[k]=\sum_{n=0}^{N-1} e^{j(2 \pi t N) k n}$.
(a) Show that $a[k]=N$ for $k=0, \pm N, \pm 2 N, \pm 3 N, \ldots$
(b) Show that $a[k]=0$ whenever $k$ is not an integer multiple of $N$. (Hint: Use the finite sum formula.
(c) Repeat parts (a) and (b) if $a[k]=\sum_{n=<N>} e^{j(2 \pi i N) k n}$.
Problem 55
Let $x[n]$ be a periodic signal with fundamental period $N$ and Fourier series coefficients $a_{k} .$ In this problem, we derive the time-scaling property $x_{(m)}[n]=\left\{\begin{array}{ll}x\left[\frac{n}{m}\right], & n=0, \pm m, \pm 2 m, \cdots \\ 0, & \text { elsewhere }\end{array}\right.$ listed in Table 3.2.
(a) Show that $x_{(m)}[n]$ has period of $m N$.
(b) Show that if $x[n]=v[n]+w[n]$, then $x_{(m)}[n]=v_{(m)}[n]+w_{(m)}\{n]$.
(c) Assuming that $x[n]=e^{j 2 \pi k_{0}, t, N}$ for some integer $k_{0},$ verify that $x_{(m)}[n]=\frac{1}{m} \sum_{i=0}^{m-1} e^{j 2 \pi\left(k_{0}+1 N\right) n t(m N)}$.
(d) Using the results of parts (a), (b), and (c), show that if $x[n]$ has the Fourier coefficients $a_{k},$ then $x_{(m ;}[n]$ must have the Fourier coefficients $\frac{1}{m} a_{k}$.
Problem 56
Let $x[n]$ be a periodic signal with period $N$ and Fourier coefficients $a_{k}$.
(a) Express the Fourier coefficients $b_{k}$ of $|x[n]|^{2}$ in terms of $a_{k}$.
(b) If the coefficients $a_{k}$ are real, is it guaranteed that the coefficients $b_{k}$ are also real?
Problem 57
(a) Let $x[n]=\sum_{x=0}^{N} a_{k} e^{j k(2 \pi i N) n}$ and $y[n]=\sum_{k=0}^{N-1} b_{k} e^{j k(2 \pi f N) n}$ be periodic signals. Show that $x[n] y[n]=\sum_{k=0}^{N \cdot 1} c_{k} e^{j k(2 \pi t N) n}$, where $c_{k}=\sum_{i=0}^{N} a_{i} b_{k-t}=\sum_{l=0}^{N-1} a_{k-l} b_{t}$.
(b) Generalize the result of part (a) by showing that $c_{k}=\sum_{l=<N>} a_{l} b_{k-l}=\sum_{l=<N} a_{k-l} b_{l}$.
(c) Use the result of part (b) to find the Fourier series representation of the following signals, where $x[n]$ is given by eq. $(P 3.57-1)$
(i) $\quad x[n] \cos \left(\frac{6 \pi n}{N}\right)$
(ii) $\quad x[n] \sum_{r=-\infty}^{+\infty} \delta[n-r N]$
(iii) $x[n]\left(\sum_{r=-\infty}^{+\infty} \delta\left[n-\frac{r N}{3}\right]\right)$ (assume that $N$ is divisible by 3 )
(d) Find the Fourier series representation for the signal $x[n] y[n],$ where $x[n]=\cos (\pi n / 3)$ and $y[n]=\left\{\begin{array}{ll}1, & |n| \leq 3 \\ 0, & 4 \leq|n| \leq 6\end{array}\right.$ is periodic with period 12.
(e) Use the result of part (b) to show that $\sum_{n=<N>} x[n] y(n]=N \sum_{l=<N>} a_{l} b_{-l}$ , and from this expression, derive Parseval's relation for discrete-time periodic signals.
Problem 58
Let $x[n]$ and $y[n]$ be periodic signals with common period $N,$ and let $z[n]=\sum_{r=<N>} x[r] y[n-r]$ be their periodic convolution
(a) Show that $z[n]$ is also periodic with period $N$.
(b) Verify that if $a_{k}, b_{k},$ and $c_{k}$ are the Fourier coefficients of $x[n], y[n],$ and $z[n]$, respectively, then $c_{k}=N a_{k} b_{k}$.
(c) Let $x[n]=\sin \left(\frac{3 \pi n}{4}\right)$ and $y[n]=\left\{\begin{array}{ll}1, & 0 \leq n \leq 3 \\ 0, & 4 \leq n \leq 7\end{array}\right.$ be two signals that are periodic with period $8 .$ Find the Fourier series representation for the periodic convolution of these signals.
(d) Repeat part (c) for the following two periodic signals that also have period 8 : $x[n]=\left\{\begin{array}{ll}\sin \left(\frac{3 \pi n}{4}\right) . & 0 \leq n \leq 3 \\ 0, & 4 \leq n \leq 7\end{array}\right.$, $y[n]=\left(\frac{1}{2}\right)^{n}, 0 \leq n \leq 7$.
Problem 59
(a) Suppose $x[n]$ is a periodic signal with period $N$, Show that the Fourier series coefficients of the periodic signal $g(t)=\sum_{k=-\infty}^{\infty} x[k] \delta(t-k T)$ are periodic with period $N$.
(b) Suppose that $x(t)$ is a periodic signal with period $T$ and Fourier series coefticients $a_{k}$ with period $N .$ Show that there must exist a periodic sequence $g[n]$ such that $x(t)=\sum_{k=\infty}^{\infty} g[k] \delta(t-k T / N)$.
(c) Can a continuous periodic signal have periodic Fourier coefficients?
Problem 60
Consider the following pairs of signals $x[n]$ and $y[n] .$ For each pair, determine whether there is a discrete-time $\mathrm{LTI}$ system for which $y[n]$ is the output when the corresponding $x[n]$ is the input. If such a system exists, determine whether the system is unique (i.e., whether there is more than one LTI system with the given input output pair). Also, determine the frequency response of an LTI system with the desired behavior. If no such LTI system exists for a given $x[n], y[n]$ pair, explain why.
(a) $x[n]=\left(\frac{1}{2}^{n}\right), y[n]=\left(\frac{1}{4}^{n}\right)$
(b) $x[n]=\left(\frac{1}{2}^{n}\right) u[n], y(n]=\left(\frac{1}{4}\right)^{n} u[n]$
(c) $x[n]=\left(\frac{1}{2}^{n}\right) u[n], y[n]=4^{n} u[-n]$
(d) $x[n]=e^{j n k}, y[n]=2 e^{j n / 8}$
(e) $x[n]=e^{\sin x} u[n] . y[n]=2 e^{\sin 8 x} u[n]$
(f) $x[n]=j^{n}, y[n]=2 j^{n}(1-j)$
(g) $x[n]=\cos (\pi n / 3) . y[n]=\cos (\pi n \beta)+\sqrt{3} \sin (\pi n / 3)$
(h) $x[n]$ and $y_{1}[n]$ as in Figure $P 3.60$
(i) $x[n]$ and $y_{2}[n]$ as in Figure $P 3.60$
Problem 61
As we have seen, the techniques of Fourier analysis are of value in examining continuous-time LTI systems because periodic complex exponentials are eigenfunctions for LTT systems. In this problem, we wish to substantiate the following statement: Although some LTI systems may have additional eigenfunctions, the complex exponentials are the only signals that are eigenfunctions of every LTI system.
(a) What are the eigenfunctions of the LTI system with unit impulse response $h(t)=\delta(t) ?$ What are the associated eigenvalues?
(b) Consider the LTI system with unit impulse response $h(t)=\delta(t-T)$ Find a signat that is not of the form $e^{3 t}$, but that is an eigenfunction of the system with eigenvalue 1. Similarly, find the eigenfunctions with eigenvalues $1 / 2$ and 2 that are not complex exponentials. (Hint: You can find impulse trains that meet these requirements.)
(c) Consider a stable LTI system with impulse response $h(t)$ that is real and even. Show that cos we and sin $wt$ are eigenfunctions of this system.
(d) Consider the LTI system with impulse response $h(t)=u(t)$. Suppose that $\phi(t)$ is an eigenfunction of this system with eigenvalue $\lambda$. Find the differential equation that $\phi(t)$ must satisfy, and soive the equation. This result, together with those of parts (a) through (c), should prove the validity of the statement made at the beginning of the problem.
Problem 62
One technique for building a dc power supply is to take an ac signal and full-wave rectify it. That is, we put the ac signal $x(t)$ through a system that produces $y(t)=$ $|x(t)|$ as its output.
(a) Sketch the input and output wave forms if $x(t)=\cos t .$ What are the fundamental periods of the input and output?
(b) If $x(t)=\cos t,$ determine the coefficients of the Fourier series for the output $y(t)$.
(c) What is the amplitude of the de component of the input signal? What is the amplitude of the de component of the output signal?
Problem 63
Suppose that a continuous-time periodic signal is the input to an LTI system. The signal has Fourier series representation $x(t)=\sum_{k=-\infty}^{\infty} \alpha^{|k|} e^{j k(\pi / 4)}$, where $\alpha$ is a real number between 0 and $1,$ and the frequency response of the system is $H(j \omega)=\left\{\begin{array}{ll}1, & \{\omega \leq W \\ 0, & |\omega|>W\end{array}\right.$. How large must $W$ be in order for the output of the system to have at least $90 \%$ of the average energy per period of $x(f) ?$
Problem 64
As we have seen in this chapter, the concept of an eigenfunction is an extremely important tool in the study of LIT systerns. The same can be said for linear. but time-varying, systems. Specifically, consider such a system with input $x(t)$ and output $y(t) .$ We say that a signal $\phi(t)$ is an eigenfunction of the system if $\phi(t) \rightarrow \lambda \phi(t)$. That is, if $x(t)=\phi(t),$ then $y(t)=\lambda \phi(t),$ where the complex constant $\lambda$ is called the eigenvalue associated with $\phi(t)$. (a) Suppose that we can represent the input $x(t)$ to our system as a linear combination of eigenfunctions $\phi_{f}(r),$ each of which has a corresponding eigenvalue $\lambda_{t}$ that is, $x(t)=\sum_{l=-2}^{\infty} c_{k} \phi_{k}(t)$.
Express the output $y(t)$ of the systern in terms of $\left\{c_{k}\right\} .\left\{\phi_{k}(t)\right\},$ and $\left\{\lambda_{k}\right\}$.
(b) Consider the system characterized by the differential equation $y(t)=t^{2} \frac{d^{2} x(t)}{d t^{2}}+t \frac{d x(t)}{d t}$Is this system linear? Is it time invariant?
(c) Show that the functions $\phi_{\mathrm{t}}(t)=t^{k}$ are eigenfunctions of the system in part (b). For each $\phi_{k}(t),$ determine the corresponding eigenvatue $\lambda_{k}$.
(d) Determine the output of the system if $x(t)=10 t^{-10}+3 t+\frac{1}{2} t^{4}+\pi$.
Problem 65
Two functions $u$ ( $f$ ) and $v(t)$ are said to be orthogonal over the interval $(a, b)$ if $\int_{a}^{b} v(t) w^{*}(t) d t=0$.
If, in adctiton, $\int_{a}^{b}|u(t)|^{2} d t=1=\int_{a}^{b}|u(t)|^{2} d t$,the functions are said to be rormalized and hence are called orthonormal. A set of functions $\left\{\phi_{k}(t)\right\}$ is called an orthogonal (orthonormal) set if each pair of functions in the set is orthogonal (orthonormal).
(a) Consider the pairs of signals $u(t)$ and $v(t)$ depicted in Figure $P 3.65 .$ Determine whether each pair is orthogonal over the interval (0,4).
(b) Are the functions $\sin m \omega_{0} \ell$ and $\sin n \omega_{0} f$ orthogonal over the interval $(0, T)$, where $T=2 \pi / \omega_{0}$ ? Are they also orthonormal?
(c) Repeat part (b) for the functions $\phi_{r m}(r)$ and $\phi_{n}(t),$ where $\phi_{i}(t)=\frac{1}{\sqrt{T}}\left[\cos k \omega_{0} t+\sin k \omega_{0} t\right]$.
(d) Show that the functions $\phi_{k}(t)=e^{j k \omega_{0}^{\prime}}$ are orthogonal over any interval of length $T=2 \pi / \omega_{0}$. Are they orthonormal?
(e) Let $x(t)$ be an arbitrary signal, and let $x_{0}(t)$ and $x_{e}(t)$ be, respectively, the odd and even parts of $x(t) .$ Show that $x_{o}(t)$ and $x_{e}(t)$ are orthogonal over the interval $(-T, T)$ for any $T$.
(f) Show that if $\left\{\phi_{k}(t)\right\}$ is a set of orthogonal signals over the interval $(a, b),$ then the set $\left\{(1 / \sqrt{A_{i}}) \phi_{k}(t)\right\},$ where $A_{k}=\int_{a}^{b}\left|\phi_{k}(t)\right|^{2} d t$, is orthonormal.
(g) Let $\left\{\phi_{i}(t)\right\}$ be a set of orthonormal signals on the interval $(a, b),$ and consider a signal of the form signal of the form $x(t)=\sum_{i} a_{i} \phi_{i}(t)$, where the $a_{i}$ are complex constants. Show that $\int_{a}^{b}|x(t)|^{2} d t=\sum_{i}\left|a_{i}\right|^{2}$.
(h) Suppose that $\phi_{1}(t), \ldots, \phi_{N}(t)$ are nonzero only in the time interval $0 \leq t$ $\leq T$ and that they are orthonornat over this time interval. Let $L$, denote the LTI system with impulse response $h_{i}(t)=\phi_{i}(r-t)$.
Show that if $\phi_{l}(t)$ is applied to this system, then the output at time $T$ is 1 if $i=j$ and 0 if $i \neq j .$ The system with impulse response given by eq. $\{P 3.65-2)$ was referred to in Problems 2.66 and 2.67 as she matched filter for the signal $\phi_{1}(t)$.
Problem 66
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.]
Problem 67
As we discussed in the text, the origins of Fourier analysis can be found in problems of mathematical physics. In particular, the work of Fourier was motivated by his investigation of heat diffusion. In this problem, we illustrate how the Fourier series enter into the investigation. $^{13}$
Consider the problem of determining the temperature at a given depth beneath the surface of the earth as a function of time, where we assume that the temperature at the surface is a given function of time $T(f)$ that is periodic with period 1 . (The unit of time is one year.) Let $T(x, r)$ denote the temperature at a depth $x$ below the surface at time $t .$ This function obeys the heat diffusion equation $\frac{\partial T(x, t)}{\partial t}=\frac{1}{2} k^{2}-\frac{\partial^{2} T(x, t)}{\partial x^{2}}$ with auxiliary condition $T(0, t)=T(t)$, Here, $k$ is the heat diffusion constant for the earth $(k>0) .$ Suppose that we expand $T(t)$ in a Fourier series:$T(t)=\sum_{n=-\infty}^{+\infty} a_{n} e^{\prime n 2 m t}$.
Similarly, let us expand $T(x, t)$ at any given depth $x$ in a Fourier series in $t$. We obtain $T(x, t)=\sum_{n=-\infty} b_{n}(x) e^{j n 2 \pi t}$, where the Fourier coefficients $b_{n}(x)$ depend upon the depth $x$.
(a) Use eqs. $(P 3.67-1)-(P 3.67-4)$ to show that $b_{n}(x)$ satisfies the differential equation $\frac{d^{2} b_{n}(x)}{d x^{2}}=\frac{4 \pi j n}{k^{2}} b_{n}(x)$ with auxiliary condition $b_{n}(0)=a_{a}$.
since eq. $(\mathbf{P} 3.67-5 \mathrm{a})$ is a second-order equation, we reed a second auxiliary condition. We argue on physical grounds that, far below the earth's surface, the variations in temperature due to surface fluctuations should disappear. That is, $\lim _{x \rightarrow \infty} T(x, t)=$ a constant.
(b) Show that the solution of eqs. $(P 3.67-5)$ is $b_{n}(x)=\left\{\begin{array}{ll}\left.a_{n} \exp |-\sqrt{2 \pi|n|}(1+j) x / k\right\}, & n \geq 0 \\ a_{n} \exp \{-\sqrt{2 \pi|n|}(1-j) x / k\} & n \leq 0\end{array}\right.$.
(c) Thus, the temperature oscillations at depth $x$ are damped and phase-shifted versións of the ternperature oscillations at the surface. To see this more clearly, let $T(t)=a_{0}+a_{1} \sin 2 \pi t$ (so that $a_{0}$ represents the mean yearly temperature). Sketch $T(t)$ and $T(x, t)$ over a one-year period for $x=k \sqrt{\frac{\pi}{2}}$. $a_{0}=2,$ and $a_{1}=1 .$ Note that at this depth not only'are the temperature oscillations significantly damped, but the phase shift is such that it is warmest in winter and coldest in summer. This is exactly the reason why vegetable cellars are constructed!
Problem 68
Consider the closed contour shown in Figure $P 3.68$. As illustrated. we can view this curve as being traced out by the tip of a rotating vector of varying length. Let $r(\theta)$ denote the length of the vector as a function of the angle $\theta .$ Then $r(\theta)$ is periodic in $\theta$ with period $2 \pi$ and thus has a Fourier series representation. Let $\left\{a_{k}\right\}$ denote the Fourier coefficients of $\boldsymbol{N}(\boldsymbol{\theta})$.
(a) Consider now the projection $x(\theta)$ of the vector $r(\theta)$ onto the $x$ -axis, as indicated in the figure. Determine the Fourier coefficients for $x(\theta)$ in cerms of the $a_{k}$ 's.
(b) Consider the sequence of coefficients $b_{k}=a_{k} e^{j k \pi / 4}$.
Sketch the figure in the plane that corresponds to this set of coefficients.
(c) Repear part (b) with $b_{k}=a_{k} \delta[k]$.
(d) Sketch figures in the plane such that $r(\theta)$ is not constant. but does have each of the following properties:
(i) $r(\theta)$ is even.
(ii) The fundamental period of $r(\theta)$ is $\pi$.
(iii) The fundamental period of $r(\theta)$ is $\pi / 2$.
Problem 69
In this problem, we consider the discrete-time counterpart of the concepts introduced in Problems 3.65 and $3.66 .$ In analogy with the continuous-time case, two discrete time signals $\phi_{k}[n]$ and $\phi_{m}[n]$ are said to be orthogonal over the interval $\left(N_{1}, N_{2}\right\}$ if $\sum_{n=N_{1}}^{x_{2}} \phi_{k}[n] \phi_{m}^{*}[n]=\left\{\begin{array}{ll}A_{k} & k=m \\ 0, & k \neq m\end{array}\right.$.
If the value of the constants $A_{k}$ and $A_{m}$ are both $1,$ then the signals are said to be orthonarmal.
(a) Consider the signals $\phi_{k}[n]=\delta[n-k], k=0,\pm 1,\pm 2, \ldots, \pm N$. Show that these signats are orthonormal over the interval $(-N, N)$.
(b) Show that the signals $\phi_{k}[n]=e^{j k(2 \pi t) / n}, k=0,1, \ldots, N-1$, are orthogonal over any interval of length $N$.
(c) Show that if $x[n]=\sum_{i=1}^{M} a_{i} \phi_{i}[n]$, where the $\phi_{i}[n]$ are orthogonal over the interval $\left(N_{1}, N_{2}\right),$ then $\sum_{n=N_{1}}^{N_{2}}\left\{\left.x[n\}\right|^{2}=\sum_{i=1}^{M}\left|a_{i}\right|^{2} A_{t}\right.$.
(d) Let $\phi_{i}[n], i=0,1, \ldots, M,$ be a set of orthogonal functions over the interval $\left(N_{1}, N_{2}\right),$ and let $x[n]$ be a given signal. Suppose that we wish to approximate $x[n]$ as a linear combination of the $\phi_{i}[n] ;$ that is, $\hat{x}[n]=\sum_{i=0}^{M} a_{i} \phi_{1}[n]$, where the $a_{i}$ are constant coefficients, Let $e[n]=x | n]-\{[n]$, and show that if we wish to minimize $E=\sum_{n=N_{1}}^{N_{2}}|e[n]|^{2}$, then the $a_{i}$ are given by $a_{i}=\frac{1}{A_{1}} \sum_{n=N_{1}}^{N_{2}} x[n] \phi_{i}^{i}[n]$.
Hint: As in Problem 3.66 , express $E$ in terms of $a_{i}, \phi_{i}[n], A_{i}$, and $x[n]$, write $a_{i}=b_{i}+j c_{i},$ and show that the equations $\frac{\partial E}{\partial b_{i}}=0 \quad$ and $\quad \frac{\partial E}{\partial c_{i}}=0$ are satisfied by the $a$, given by eq. $(\mathrm{P} 3,69-2)$. Note that applying this result when the $\left.\phi_{1}\{n] \text { are as in part (b) yields eq. }(3.95) \text { for } a_{k} .\right]$
(e) Apply the result of part (d) when the $\phi_{1}[n]$ are as in part (a) to determine the coefficients $a_{i}$ in terms of $x[n]$.
Problem 70
(a) In this problem, we consider the definition of the two-dimensional Fourier series for periodic signals with two independent variables. Specifically, consider a signal $x\left(t_{1}, t_{2}\right)$ that satisfies the equation
Problem 71
Consider the mechanical system shown in Figure P3.71. The differential equation relating velocity $v(t)$ and the input force $f(t)$ is given by $B v(t)+K \int v(t) d t=f(r)$.
(a) Assuming that the output is $f_{x}(t),$ the compressive force acting on the spring. write the differential equation relating $f_{s}(r)$ and $f(t) .$ Obtain the frequency response of the system, and argue that it approximates that of a lowpass filter.
(b) Assuming that the output is $f_{d}(f)$, the compressive force acting on the dashpot, write the differential equation relating $f_{d}(t)$ and $f(t) .$ Obtain the frequency response of the system, and argue that it approximates that of a hyphpass filter.
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Source: https://www.numerade.com/books/chapter/fourier-series-representation-of-periodic-signals/
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