3.3. 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 wo and the Fourier series coefficients a such that x(t)=\sum_{k=-\infty}^{\infty} a_{k} e^{j k \omega_{0} t}

3.11. 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 ak. 3. a11 = 5. \text { 4. } \frac{1}{10} \sum_{n=0}^{9}|x[n]|^{2}=50 Show that x[n] = A cos(Bn + C), and specify numerical values for the constants A,B, and C.

Consider a pulse s(t) = sinc(at)sinc (bt). where a z b. (a) Sketch the frequency-domain response S(f) of the pulse. (b) Suppose that the pulse is to be used over an ideal real-base band channel with one-sided bandwidth 400 Hz. Choose a and b so that the pulse is Nyquist for 4PAM signaling at 1200 bits/s and exactly fills the channel bandwidth. (c) Now, suppose that the pulse is to be used over a passband channel spanning the frequency range 2.4-2.42 GHz. Assuming that we use64QAM signaling at 60 Mbits/s, choose a and b so that the pulse is Nyquist and exactly fills the channel bandwidth. (d) Sketch an argument showing that the magnitude of the transmitted waveform in the preceding settings is always finite.

Problem 2. Consider the system described by the following difference equation, y[n]=y[n-1]-0.25 y[n-2]+x[n], \quad n \geq 0 with initial conditions y[-1]=1 and y[-2]=3 and input x[n] = 2(0.9)^n u[n]. For this system: a. Draw a block diagram using multipliers, adders, and shifters that represents the system. b. Determine the initial-condition response, yi[n]. c. Is this system stable? Explain. d. Determine the analytical form of the impulse response function, h[n]. e. Calculate the forced response for the system, yf [n]. f. Finally, determine the analytical form for the total response of the system, y[n].

3) Given that x(t) has the Fourier transform X(jw), express the Fourier transforms of the signals listed below in terms of X(jw). You may find useful the Fourier transform properties listed in the attached Table X2(t) = x(3t- 6)

Convert each signal to the finite sequence form {a, b, c, d, e}. (a) u[n] – u[n – 4] (b) u[n] – 2 u[n – 2] + u[n - 4] (c) n u[n] – 2(n –2)u[n – 2] + (n – 4)u[n – 4] (d) n u[n] – 2(n – 1) u[n – 1] + 2(n – 3) u[n – 3] – (n – 4) u[n – 4]

The energy band diagram of germanium shows the location of a particular energy levelE, at T =600 K. [10 Points] What is the type of semiconductor represented in the energy diagram?Justify your answer. Calculate the number of free electrons at energy level E,. | Calculate the intrinsic carrier concentration. Calculate the total number of free electrons. | If Na is considered as zero, calculate the number of Donor atoms (ND).

Consider the periodic signal x(t) shown in the figure below. a) Using the table of Fourier Series, find the Fourier coefficients for the exponential form for the signal x(t). Evaluate all coefficients. (2 marks) b) Sketch the frequency spectrum (magnitude and phase) for the signal x(t), showing the decomponent and the first four harmonics (for k between -4 and +4). (2 marks)

(A) Figure P14.15a represents a simple voltage divider circuit. A voltmeter with an internal resistance of 1 M disconnected to measure VA, the voltage across R2. Calculate the voltage measured for Va and the error from the actual value. If the same voltage divider is constructed using 200kN resistors, as shown in Figure P14.15(b), calculate the measured VA and the error from the actual value.

(a) (21 points) Find the Fourier transform of each of the signals given below:Hint: you may use Fourier Transforms derived in class. \text { i. (optional) } x_{1}(t)=2 \text { rect }\left(\frac{-t-3}{2}\right) \cos (10 \pi t) \text { ii. } x_{2}(t)=e^{(2+3 j) t} u(-t+1) \text { iii. } x_{3}(t)=\left\{\begin{array}{ll} 1+\cos (\pi t), & |t|<1 \\ 0, & \text { otherwise } \end{array}\right. (b) (6 points) Find the inverse Fourier transform of the signal shown below: (c) (8 points) Two signals f1(t) and f2(t) are defined as f_{1}(t)=\operatorname{sinc}(2 t) f_{2}(t)=\operatorname{sinc}(t) \cos (3 \pi t) Let the convolution of the two signals be f(t) = (f1 * f2)(t) i. Find F(jw), the Fourier transform of f(t). ii. Find f(t).