3. (21 pts) Determine whether or not each of the following systems is invertible. If the system is invertible, find the inverse system. If it is not, find two input signals to the system that produce the same output.

Determine the Fourier-series expansion of the following signals: \text { 1. } x(t)=\cos (2 \pi t)+\cos (4 \pi t) \text { 2. } x(t)=\cos (2 \pi t)-\cos (4 \pi t+\pi / 3) \text { 3. } x(t)=2 \cos (2 \pi t)-\sin (4 \pi t) \text { 4. } x(t)=\sum_{n=-\infty}^{\infty} \Lambda(t-2 n)

4. Sampling, Aliasing, and Reconstruction: In this problem we want to consider the signal x(t) = cos(275000 t) and make plots at different sampling rates. For each part below you will create a plot. All of your plots should be appear on the same figure as subplots, one below the other. Use the subplot (4,1,p) command which will create 4 rows and 1 column of plots. The value of p gives the specific plot number from top to bottom. a) For the first subplot make a plot of x(1) sampled at 100 kHz from 0 to 1 msec (0.001 seconds). Plot both the sample values as stems and the linear interpolation between the samples (Use stem(t, x); hold on; plot (t,x); hold off;) b) Repeat part (a) in the second subplot using a sampling frequency of 25 kHz. c) Repeat part (a) in the third subplot using a sampling frequency of 10 kHz. d) Repeat part (a) in the fourth subplot using a sampling frequency of 8 kHz. e) The plot in part (d) is under-sampled and shows aliasing. Find the frequency of the aliased signal and add it to this plot. For plotting the aliased signal use a 100 kHz sampling frequency so it appears continuous.

Convert each signal to the finite sequence form {a, b, c, d, e}. \text { (b) } n \cdot u[n]-n \cdot u[n-5] \text { (c) } u[n-1] \cdot u[4-n] \text { (d) } 2 \delta[n-1]-4 \delta[n-3] \text { (а) } \quad u[n]-\delta[n-3]-u[n-4]

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.

(a) Given the second-order frequency response function: H(j \omega)=\frac{100}{(j \omega)^{2}+51 j \omega+50} (i) Analytically determine the straight line approximations (asymptotes) of the Bode Log-Magnitude plot of the frequency response function. (ii) Sketch the Bode magnitude (dB) of the frequency response function, clearly labeled, and indicate the frequency () where the OdB-line is crossed on your sketch. You are not required to sketch the phase. An LTI system subjected to an input x() has a response y(t), and its frequency response function is: H(j \omega)=\frac{2}{\frac{1}{3}(j \omega)^{2}+\frac{5}{3} R j \omega+2} \text { where } R \text { is a resistance }(\Omega) \text {. } For what range of values of the resistance R is the system under damped? \text { (ii) If } R=1 \Omega \text {, what is the impulse response, } h(t) \text { of the system? } \text { (ii) If } R=1 \Omega \text {, what is the impulse response, } h(t) \text { of the system? } \text { (iii) If } R=1 \Omega \text {, what is the step response, } s(t) \text { of the system? }

B. (1 points) Would you get the same set of output values if you used cross correlation instead of convolution, using the same kernel and image as in part A(YES or NO)

h) the best range of wavelengths for distinguishing dry land (soil+vegetation) from water areas is (choose one): visible light (400-700nm) or infrared (700-1200nm).Enter either visible or infrared.

Given point (0,0,1) in the right image, what is the equation of the correspondingepipolar line in the left image? Enter A,B,C or D. A) -10 y + 20 = 0 B)-100 x + 20 = 0 C) 10 x + 20 y + 100 = 0 D) -10 x + 20 y + 100 = 0

1. (28 pts) We have been introduced to several general properties of systems, namely: Causality • Linearity • Memory • Time Invariance • Stablility