# Lernkarten

Karten 16 Karten 1 Lernende English Grundschule 09.02.2020 / 23.02.2020 Keine Angabe
0 Exakte Antworten 16 Text Antworten 0 Multiple Choice Antworten

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.