Time left 1:29:18 Question 1 Not yet answered Marked out of 8.00 P Flag question Solve for the following signals The FT of $$ x(t)=2 \operatorname{sinc}(4 t) $$ is: Choose... The FT of $$ x(t)=2 \pi(2 t-2) $$ is: Choose... The FT of $$ x(t)=2 \wedge\left(\frac{t-3}{2}\right) $$, where $$ \wedge $$ is the triangular function Choose... The FT Of $$ x(t)=\pi\left(\frac{t}{2}\right) e^{-j 10 \pi t} $$ Choose... Question 2 Not yet answered Marked out of 10.00 P Flag question Consider the following signal: $$ x(t)=4 \cos ^{2}(20 \pi t)+2 \sqrt{2} \sin \left(30 \pi t+\frac{\pi}{4}\right) $$ The value of $$ a_{1} $$ is Choose... The dc-value of the signal $$ x(t) $$ is Choose... The average power is $$ \square $$ The value of $$ X_{-4} $$ is Choose... The fundamental Frequency $$ f_{O} $$ is Choose...

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### Question 1

1. **The FT of $$ x(t)=2 \operatorname{sinc}(4 t) $$**:
   The Fourier Transform (FT) of the sinc function is a rectangular function. The FT of $$ \operatorname{sinc}(at) $$ is $$ \frac{1}{|a|} \operatorname{rect}\left(\frac{f}{2a}\right) $$. Here, $$ a = 4 $$, so:
   $$
   \text{FT}\{2 \operatorname{sinc}(4t)\} = 2 \cdot \frac{1}{4} \operatorname{rect}\left(\frac{f}{8}\right) = \frac{1}{2} \operatorname{rect}\left(\frac{f}{8}\right)
   $$

2. **The FT of $$ x(t)=2 \pi(2 t-2) $$**:
   This is a linear function. The FT of $$ at + b $$ is given by $$ 2\pi j \delta'(f) + 2\pi b \delta(f) $$. Here, $$ a = 2\pi $$ and $$ b = -2\pi $$:
   $$
   \text{FT}\{2\pi(2t-2)\} = 2\pi j \delta'(f) - 4\pi^2 \delta(f)
   $$

3. **The FT of $$ x(t)=2 \wedge\left(\frac{t-3}{2}\right) $$**:
   The triangular function has a known FT. The FT of $$ \wedge(at) $$ is $$ \frac{1}{a^2} \operatorname{tri}\left(\frac{f}{a}\right) $$. Here, $$ a = \frac{1}{2} $$:
   $$
   \text{FT}\{2 \wedge\left(\frac{t-3}{2}\right)\} = 2 \cdot 2 \operatorname{tri}(2f) e^{-j 3f} = 4 \operatorname{tri}(2f) e^{-j 3f}
   $$

4. **The FT of $$ x(t)=\pi\left(\frac{t}{2}\right) e^{-j 10 \pi t} $$**:
   The FT of $$ x(t) e^{-j \omega_0 t} $$ is $$ X(f - \frac{\omega_0}{2\pi}) $$. The FT of $$ \pi\left(\frac{t}{2}\right) $$ is $$ 2 \operatorname{rect}(f) $$:
   $$
   \text{FT}\{\pi\left(\frac{t}{2}\right) e^{-j 10 \pi t}\} = 2 \operatorname{rect}(f + 5)
   $$

### Question 2

1. **The value of $$ a_{1} $$**:
   The Fourier series coefficient $$ a_n $$ for $$ x(t) = A \cos^2(\omega_0 t) $$ can be calculated using the formula:
   $$
   a_n = \frac{1}{T_0} \int_{0}^{T_0} x(t) e^{-j n \omega_0 t} dt
   $$
   Here, $$ A = 4 $$ and $$ \omega_0 = 20\pi $$. The average value of $$ \cos^2 $$ is $$ \frac{1}{2} $$, so:
   $$
   a_0 = 2 \quad \text{and} \quad a_1 = 0
   $$

2. **The dc-value of the signal $$ x(t) $$**:
   The DC value is the average value over one period:
   $$
   \text{DC value} = \frac{1}{T_0} \int_{0}^{T_0} x(t) dt
   $$
   The average value of $$ 4 \cos^2(20 \pi t) $$ is $$ 2 $$ and the average value of $$ 2\sqrt{2} \sin(30 \pi t + \frac{\pi}{4}) $$ is $$ 0 $$:
   $$
   \text{DC value} = 2
   $$

3. **The average power**:
   The average power $$ P $$ can be calculated as:
   $$
   P = \frac{1}{T_0} \int_{0}^{T_0} |x(t)|^2 dt
   $$
   The average power of $$ 4 \cos^2(20 \pi t) $$ is $$ 8 $$ and for $$ 2\sqrt{2} \sin(30 \pi t + \frac{\pi}{4}) $$ is $$ 2 $$:
   $$
   P = 8 + 2 = 10
   $$

4. **The value of $$ X_{-4} $$**:
   The Fourier series coefficient $$ X_{-4} $$ can be calculated similarly. Since $$ \cos^2 $$ and $$ \sin $$ functions do not contribute to negative frequencies, $$ X_{-4} = 0 $$.

5. **The fundamental frequency $$ f_{O} $$**:
   The fundamental frequency is given by:
   $$
   f_{O} = \frac{1}{T_0}
   $$
   The period $$ T_0 $$ is the least common multiple of the periods of the components:
   $$
   T_0 = \frac{1}{20} \quad \text{and} \quad T_0 = \frac{1}{30} \quad \Rightarrow \quad f_{O} = \text{lcm}(20, 30) = 60 \text{ Hz}
   $$

### Summary of Answers
1. FT of $$ x(t)=2 \operatorname{sinc}(4 t) $$: $$ \frac{1}{2} \operatorname{rect}\left(\frac{f}{8}\right) $$
2. FT of $$ x(t)=2 \pi(2 t-2) $$: $$ 2\pi j \delta'(f) - 4\pi^2 \delta(f) $$
3. FT of $$ x(t)=2 \wedge\left(\frac{t-3}{2}\right) $$: $$ 4 \operatorname{tri}(2f) e^{-j 3f} $$
4. FT of $$ x(t)=\pi\left(\frac{t}{2}\right) e^{-j 10 \pi t} $$: $$ 2 \operatorname{rect}(f + 5) $$

For Question 2:
1. $$ a_{1} = 0 $$
2. DC value = 2
**Question 1:** 1. The Fourier Transform of $$ x(t) = 2 \operatorname{sinc}(4t) $$ is $$ \frac{1}{2} \operatorname{rect}\left(\frac{f}{8}\right) $$. 2. The Fourier Transform of $$ x(t) = 2\pi(2t - 2) $$ is $$ 2\pi j \delta'(f) - 4\pi^2 \delta(f) $$. 3. The Fourier Transform of $$ x(t) = 2 \wedge\left(\frac{t-3}{2}\right) $$ is $$ 4 \operatorname{tri}(2f) e^{-j 3f} $$. 4. The Fourier Transform of $$ x(t) = \pi\left(\frac{t}{2}\right) e^{-j 10\pi t} $$ is $$ 2 \operatorname{rect}(f + 5) $$. **Question 2:** 1. $$ a_{1} = 0 $$ 2. DC value = 2 3. Average power = 10 4. $$ X_{-4} = 0 $$ 5. Fundamental frequency $$ f_{O} = 60 $$ Hz

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