ECEN 314 - Exam I
Exam 1
Exam 1
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| Cartes-fiches | 16 |
|---|---|
| Langue | English |
| Catégorie | Electrotechnique |
| Niveau | École primaire |
| Crée / Actualisé | 09.02.2020 / 23.02.2020 |
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Determine the value of \(P_\infty\) and \(E_\infty\) for a signal \(x(t)\).
\(E_\infty=\int_{-\infty}^\infty|x(t)|^2dt\)
\(P_\infty=\displaystyle{\lim_{T \to \infty}}\frac{1}{2T}\int_{-T}^T|x(t)|^2dt\)
What is the unit step function, \(u(t)\).
\(u(t)=\begin{cases} 0, t<0\\1,t\ge0\end{cases}\)
Determine the value of \(P_\infty\) and \(E_\infty\) for a discrete signal \(x[n]\).
\(E_\infty=\displaystyle{\sum_{n=-\infty}^{\infty}|x[n]|^2}\)
\(P_\infty=\displaystyle{\lim_{N \to \infty}}\frac{1}{2N+1}\sum_{n=-N}^{N}|x[n]|^2\)
Determine the value of \(P\) and \(E\) for a discrete signal \(x[n] \) over the timer priod \(t_1< t< t_2\).
\(N_0=|n_1|+|n_2|\)
\(E=\displaystyle{\sum_{n=n_1}^{n_2}|x[n]|^2}\)
\(P=\displaystyle{\frac{1}{N_0+1} \sum_{n=n_1}^{n_2}|x[n]|^2}\)
Determine the value of \(P\) and \(E\) for a continuos time signal \(x(t) \) for the time period \(t_1 < t < t_2\).
\(T_0=|t_1|+|t_2|\) get the time between \(t_1\) and \(t_2\).
\(E=\int_{-\frac{T_0}{2}}^{\frac{T_0}{2}}|x(t)|^2dt\)
\(P=\displaystyle{\frac{1}{T_0}\int_{-\frac{T_0}{2}}^{\frac{T_0}{2}}|x(t)|^2dt}\)
What is the magnitude of \(\alpha e^{j\omega+\theta}\)?
\(|\alpha e^{j\omega+\theta}|\rightarrow\\\text{[Re]}=\alpha cos({j\omega+\theta}),\\\text{[Im]}=\alpha sin({j\omega+\theta}),\\\sqrt{[Re]^2+[Im]^2}\rightarrow\\\\\\\\sqrt{\alpha cos^2({j\omega+\theta})+\alpha sin^2({j\omega+\theta})}=\alpha \)
What are the following summation formulas?
\(\displaystyle{\sum_{n=1}^A}\ 1\\\displaystyle{\sum_{n=1}^A}\ n\)
\(\displaystyle{\sum_{n=1}^A}\ 1=A\\\displaystyle{\sum_{n=1}^A}\ n=\frac{n(n+1)}{2}\)
Let x(t) be a signal with x(t) = 0 for t > 5.
For what range will x(t) be equal to 0.
x(-t)
x(t + 1)
x(t + 2) + x(t - 2)
x(-t + 2)x(t + 1)
x(-t):
\(-t>5\Rightarrow t>-5\)
x(t + 1):
\(t+1>5\Rightarrow t>4\)
x(t + 2) + x(t - 2):
\(t+2>5\Rightarrow t>3\\ t-2>5\Rightarrow t>7\)
x(-t + 2)x(t + 1):
\(-t+2>5\Rightarrow t>-3\\ t+1>5\Rightarrow t>4\)
Determine the fundamental period of the signal:
\( x(t) = 2cos(10t + 1)-sin(4t -1).\)
\(\frac{2\pi}{10}=\frac{\pi}{5}\\\frac{2\pi}{4}=\frac{\pi}{2}\)
\(\frac{\text{LCM of Numerators}}{\text{HCF of Denominators}}\)
\(\frac{\text{LCM}(\pi,\pi)=\pi}{\text{HCF}(2,5)=1}=\pi\)
What is the even signal of \(x(t)\).
\(Ɛv\{x(t)\}=x_e(t)=\frac{1}{2}[x(t)+x(-t)]\)
Using this linear system S:
\(x(t)=e^{2jt}\rightarrow y(t)=e^{j3t}\\x(t)=e^{-2jt}\rightarrow y(t)=e^{-j3t}\)
What is \(y(t)\) for \(x(t) = \cos(2(t+1))\)
Simply the equation to \(\alpha x(t)\):
\(\cos(2(t+1))=\frac{1}{2}[e^{j2(t+1)}+e^{-j2(t+1)}]\\\frac{1}{2}[e^{j2t}e^{j2}+e^{-j2t}e^{-j2}]\\\alpha=\frac{1}{2}e^{j2},\ x(t)=e^{2jt}\\x(t)\rightarrow y(t)\Rightarrow\frac{1}{2}[e^{j3t}e^{j2}+e^{-j3t}e^{-j2}]\\\frac{1}{2}[e^{j3t+j2}+e^{-j3t-j2}]\\\frac{1}{2}[e^{j(3t+2)}+e^{j(-3t-2)}]\\\ \\\cos(3t+2)\)
What is the difference between a memory and memoryless system?
And determine which of the following are memoryless.
\(1.\ y(t)=x(t-2)+x(t+2)\\ 2.\ y(t)=[\cos(3t)]x(t)\\ 3.\ y(t)=\begin{cases} 0,\ t\lt0 \\x(t)+x(t-2),\ t\ge0 \end{cases}\\ 4.\ y(t)=\frac{dx(t)}{dt}\\ 5.\ y(t)=\begin{cases} 0,\ x(t)\lt0 \\x(t)+x(t-2),\ x(t)\ge0 \end{cases}\\\)
Memoryless: The output only depends on the current input.
Memory: The output depedns on either future or past values of the input.
1. Not Memoryless, Since y(t) depends on the past and future.
2. Memoryless
3. Not memoryless, Since y(t) depends on the past.
4. Memoryless
5. Not memoryless, Since y(t) depends on the past.
How to check for time invariance in a system \(y(n)=nx(n)\)?
And determine which of the following are time invariant.
\(1.\ y(t)=x(t-2)+x(t+2)\\2.\ y(t)=[\cos(3t)]x(t)\\3.\ y(t)=\begin{cases}0,\ t\lt0\\x(t)+x(t-2),\ t\ge0\end{cases}\\4.\ y(t)=\frac{dx(t)}{dt}\\5.\ y(t)=\begin{cases}0,\ x(t)\lt0\\x(t)+x(t-2),\ x(t)\ge0\end{cases}\\\)
\(y(n,k)=y(n-k)\\y(n,k)=T[x(n-k)]=nx(n-k)\\y(n-k)=(n-k)x(n-k)\\nx(n-k)\ne(n-k)x(n-k)\)
Not time invariant.
1. Time variant
2. Time variant
3. Time variant
4. Time variant
5. Time variant
How do you check for linearity for a system \(x(t)\)?
\(T[a_1x_1(t)+a_2x_2(t)]=a_1T[x_1(t)]+a_2T[x_2(t)]\)
How do you check if the following systems are casual or not causl?
\(y(t)=x(t)+x(t-2)\\y(t)=x(t-2)\\y(t)=x(t)\\y(t)=x(t)+x(t+2)\\y(t)=x(t+2)+x(t-2)\)
Casual:
Present and Past
Present only
Not Casual:
Present, Past, and Future
Present and Future
1. Causal, Present and Pase
2. Casual, Present and Past
3. Casual, Present Only
4. Not Casual, Present and Future
5. Not Casual, Present, Past, and Future
What does stability of a system mean?
For finite i/p, there must be finit o/p.
