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Fenster schliessen

Determine the value of \(P_\infty\) and \(E_\infty\) for a signal \(x(t)\).

\(E_\infty=\int_{-\infty}^\infty|x(t)|^2dt\)

\(P_\infty=\displaystyle{\lim_{T \to \infty}}\frac{1}{2T}\int_{-T}^T|x(t)|^2dt\)

Fenster schliessen

What is the unit step function, \(u(t)\).

\(u(t)=\begin{cases} 0, t<0\\1,t\ge0\end{cases}\)

Fenster schliessen

Determine the value of \(P_\infty\) and \(E_\infty\) for a discrete signal \(x[n]\).

\(E_\infty=\displaystyle{\sum_{n=-\infty}^{\infty}|x[n]|^2}\)

\(P_\infty=\displaystyle{\lim_{N \to \infty}}\frac{1}{2N+1}\sum_{n=-N}^{N}|x[n]|^2\)

Fenster schliessen

Determine the value of \(P\) and \(E\) for a discrete signal \(x[n] \) over the timer priod \(t_1< t< t_2\).

\(N_0=|n_1|+|n_2|\)

\(E=\displaystyle{\sum_{n=n_1}^{n_2}|x[n]|^2}\)

\(P=\displaystyle{\frac{1}{N_0+1} \sum_{n=n_1}^{n_2}|x[n]|^2}\)

Fenster schliessen

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 \(t_2\).

\(E=\int_{-\frac{T_0}{2}}^{\frac{T_0}{2}}|x(t)|^2dt\)

\(P=\displaystyle{\frac{1}{T_0}\int_{-\frac{T_0}{2}}^{\frac{T_0}{2}}|x(t)|^2dt}\)

Fenster schliessen

What is the magnitude of \(\alpha e^{j\omega+\theta}\)?

\(|\alpha e^{j\omega+\theta}|\rightarrow\\\text{[Re]}=\alpha cos({j\omega+\theta}),\\\text{[Im]}=\alpha sin({j\omega+\theta}),\\\sqrt{[Re]^2+[Im]^2}\rightarrow\\\\\\\\sqrt{\alpha cos^2({j\omega+\theta})+\alpha sin^2({j\omega+\theta})}=\alpha \)

Fenster schliessen

What are the following summation formulas?

\(\displaystyle{\sum_{n=1}^A}\ 1\\\displaystyle{\sum_{n=1}^A}\ n\)

\(\displaystyle{\sum_{n=1}^A}\ 1=A\\\displaystyle{\sum_{n=1}^A}\ n=\frac{n(n+1)}{2}\)

Fenster schliessen

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):

\(-t+2>5\Rightarrow t>-3\\ t+1>5\Rightarrow t>4\)

Fenster schliessen

Determine the fundamental period of the signal:

\( x(t) = 2cos(10t + 1)-sin(4t -1).\)

\(\frac{2\pi}{10}=\frac{\pi}{5}\\\frac{2\pi}{4}=\frac{\pi}{2}\)

\(\frac{\text{LCM of Numerators}}{\text{HCF of Denominators}}\)

\(\frac{\text{LCM}(\pi,\pi)=\pi}{\text{HCF}(2,5)=1}=\pi\)

Fenster schliessen

What is the even signal of \(x(t)\).

\(Ɛv\{x(t)\}=x_e(t)=\frac{1}{2}[x(t)+x(-t)]\)

Fenster schliessen

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 \(\alpha x(t)\):

\(\cos(2(t+1))=\frac{1}{2}[e^{j2(t+1)}+e^{-j2(t+1)}]\\\frac{1}{2}[e^{j2t}e^{j2}+e^{-j2t}e^{-j2}]\\\alpha=\frac{1}{2}e^{j2},\ x(t)=e^{2jt}\\x(t)\rightarrow y(t)\Rightarrow\frac{1}{2}[e^{j3t}e^{j2}+e^{-j3t}e^{-j2}]\\\frac{1}{2}[e^{j3t+j2}+e^{-j3t-j2}]\\\frac{1}{2}[e^{j(3t+2)}+e^{j(-3t-2)}]\\\ \\\cos(3t+2)\)

Fenster schliessen

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.

Fenster schliessen

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

Fenster schliessen

How do you check for linearity for a system \(x(t)\)?

\(T[a_1x_1(t)+a_2x_2(t)]=a_1T[x_1(t)]+a_2T[x_2(t)]\)

 

Fenster schliessen

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

Fenster schliessen

What does stability of a system mean?

For finite i/p, there must be finit o/p.