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Q1: 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)See Answer
Q2: Consider the following analogue signal s(t): s(t)=20 \cos (10 \pi t)+20 \cos ^{2}(10 \pi t) Where time t is expressed in seconds. Analyse this signal by answering the following questions: a) Use Fourier transform to find the frequency spectrum S(f) of s(t). You may directly use the common Fourier Transforms listed at the end of Chapter 3.(1,5 Marks) b) Using 5Hz/cm as horizontal scale, represent S(f) graphically on cartesian axes.See Answer
Q3: Assignment You are going to estimate the bandwidth of FM/PM signals by finding the frequency spectrum of the given FM/PM signals. Compare your estimates with the Carson's rule estimates. Use a mini lab report format to write the report. \text { Assume a signal } m(t)=2 \cos (2 \pi t)+3 \sin (4 \pi t) 1. FM b) Plot the magnitude spectrum of EM(t). Estimate the bandwidth required. c) Compute the bandwidth of pEM(t) by using the Carson's rule. ) Plot the magnitude spectrum of pPM(t). Estimate the bandwidth required. c) Compute the bandwidth of PM(t) by using the Carson's rule.See Answer
Q4: 1. Mystery System. Copyright University of Colorado Denver. This is an exam question. The use of any external resources such as Chegg is not allowed. Download the file black box.p from CANVA Sand save it in the folder where you have your MATLAB code. This file defines a system y=black box (x); Do not try to load this file into MATLAB just have it in the folder where you are running other code from so MATLAB can find the file. a) Use MATLAB to evaluate y=blackbox (x); where x =- [1 2 3 4] c) Use MATLAB to evaluate y blackbox(x); where x =[0 1 2 3 4 d) Is the blackbox system linear? Justify your answer using your answers from (a) - (c) e) Is the blackbox system invariant? Justify your answer using your answers from (a)-(c) Write equation for the blackbox system that gives output y[n] in terms of input x[n]. If you cannot write an equation, describe clearly in words what the black box system is doing. (Hint if the answer is not obvious input other signals into the system and see what comes out). b) Use MATLAB to evaluate y-blackbox (x); where x =2*[1 2 3 4]See Answer
Q5: 3. Filtering. Copyright University of Colorado Denver. This is an exam question. The use of anyexternal resources such as Chegg is not allowed. Download and load into MATLAB the fileRemoteExam2Problem3.mat the file contains a filter impulse response h. a) Use the freqz function to make a plot of the frequency response of the filter. b) Make a plot of signal x(1) = 100 cos(275t +%)sampled at 1000 Hz from t=0 to1=1 second c) Apply the filter to x(t) and plot the output of the filter from t=0 to 1=1 second d) Consider the AM modulator and filter cascaded system below. What should be the value of Wm make the output z[n] close to zero (small amplitude)? See Answer
Q6: 4. Frequency Identification. Copyright University of Colorado Denver. This is an exam question. The use of any external resources such as Chegg is not allowed. Download the fileRemoteExam2Problem4.mat from the CANVAS Exams folder and load it into MATLAB. This file has the variable x. Make a stem plot of x using stem. b) Write an expression for X(e") by taking the DTFT of x[n]. c) Plot the magnitude of X(e) in MATLAB for 0<ôST (Hint: use abs () to magnitude and make sure your plot has at least 1000 points. One way to do this is W =linspace (0, pi, 1000); d) x[n] is the signal measured from a motion sensor on a patient's chest,vertical position of the sensor. The sampling rate is two samples per second (fs = 2). One breath is one up and down motion. Use your plot from part(c) to fill in the table below. Based on your answers diagnose the patient by picking the most likely condition that best matches the sensor data See Answer
Q7: 4. Filter Design. Recall that an ideal low pass filter (here with cutoff w pi/12 ) has the following impulse and frequency response We want to modify this filter to be an ideal band pass filter with passband centered at w = pi/4 and bandwidth 2pi/12 a) Make a sketch of the ideal band pass filter frequency response magnitude (Har (e)) with properties as described above b) Write an expression for H (e) in terms of H (e)(Hint: use the frequency shift property of the DTFT c) Write an expression for bandpass filter impulse response h {n} use MATLAB to evaluate your expressions for h [N] for n = 1 to 100 use freqz to plot the frequency response of this 100 point bandpass filter ( do not use firl to create your own bandpass filter)See Answer
Q8: 2True and False: A Gambler's Ruin (10 pts) [True | False] (1 point) The DTFT is sampling the DFT in the frequency domain. [True | False] (1 point) You cannot guarantee that an FIR filter can achieve a linearphase response. [True | False] (1 point) Zero padding adds new information into the DFT. In fact, zero padding will have the same result as sampling the signal more (i.e., a signal with 10 samples and 90 zeros is the same as a signal with 100 samples). [True | False] (1 point) The frequency mapping H(s) to H(z) is nonlinear, where H(2)is obtained with the bilinear transformation. [True | False] (1 point) IIR filters can be designed stable; however, the actual implementation in hardware can be unstable. [Short Answer] (2 point) What is DFT leakage? [Short Answer] (3 points) In a sentence or two, describe the difference between an analog-to-digital converter and a discrete-to-continuous converter.See Answer
Q9: 1) Consider the rectangular pulse signal x(t)=\left\{\begin{array}{ll} 1, & |t|<T_{1} \\ 0, & |t|>T_{1} \end{array}\right. Find the Fourier transform of this signal and sketch it. Make sure to find the amplitude at the origin and the zeros of X(jw) See Answer
Q10: 2) Consider the following a continuous-time signals: h(t) = 8 (t-to), find y(t)a. dx(t)b. y(t) =find H(jw)dtSee Answer
Q11: Convert each signal to the finite sequence form {a, b, c, d, e}. \text { (a) } u[n]-u[n-4] \text { (b) } u[n]-2 u[n-2]+u[n-4] \text { (c) } n u[n]-2(n-2) u[n-2]+(n-4) u[n-4] \text { (d) } n u[n]-2(n-1) u[n-1]+2(n-3) u[n-3]-(n-4) u[n-4]See Answer
Q12: Consider the system shown in the figure with the input signal x (t) = 2 cos(100t) and the filterH (ela) with a cutoff frequency . [20 points] Design the system by selecting values for the sampling frequency ws, the cutofffrequency Nc, and the final filter cutoff frequency W1. Explain and justify in details allsteps and assumptions considered for your selection. ] Find and plot the output signalY, (jw)d, See Answer
Q13: Given the signal x[n]=\left(\frac{1}{2}\right)^{n} u[n-2] \text { Plot the magnitude of the discreet-time Fourier transform }\left|X\left(e^{j \omega}\right)\right| \text { . } Plot the signal x3[n], and its DTFT.See Answer
Q14: Consider the following signal Write x(t) in terms of the pulse signal pr (t). ]Determine and express Fourier transformX (w).See Answer
Q15: The figure below shows the Fourier transform of a real bandpass signal, i.e., a signal whose frequencies are not centered around the origin. We want to sample this signal. Let F, in Hz (a) (4 pts) One option is to sample this signal at the Nyquist rate. Then to recover the signal, we pass its sampled version through a low pass filter. What is the Nyquist rate of this signal? (b) (9 pts) Since the signal might have high frequency components, Nyquist rate for this signal can be high. In other words, we need to have a lot of samples of the signal,which means that the sampling scheme is costy. It turns out that for this type of signal,we can sample it at a sampling frequency lower that the Nyquist rate and we can still recover the signal, however in this case, we will use a band pass filter. To see this, we have the following two options for the sampling frequency: • Fs= 0.5 Hz; • Fs = 1 Hz; For each case, draw the spectrum of the signal after sampling it. To recover the signal,which F, can we use? How we should choose the frequencies of the band pass filter?What is the minimum F, we can use and still recover the signal?See Answer
Q16: 4–4 A DSB-SC signal has a carrier frequency of 900 kHz and A. = 10. If this signal is modulated by a waveform that has a spectrum given by Fig. P3–3. Find the magnitude spectrum for this DSB-SC signal.See Answer
Q17: 2. In this problem, we will encode bits into linearly modulated signals using 8-PAM at symbol rate 10kHz.Use the abstract alphabet A = {0,1,..., 7}, the signal constellation C = {-7,–5, –3, –1, 1,3, 5, 7},and the mapping f(i) = 2i–7. Let p1(t) = I_T/2,T/2(t) and let p2(t) =It/2,T/2(t)(1+cos(2nt/T))/2. (a) (20 pts) Let an E A for n E Z and suppose that a has the empirical frequency properties described in problem 1. Let xn = f(an) and let u;(t) be the linearly modulated signal encoding x with pulse P:(t). Find the power spectral densities of u1 and u2. Create a single log-log plot with both PSDS.The signals are real, so the PSD is even and it is only necessary to plot positive frequencies. (b) (20 pts) Consider the finite bit sequence b = (1,0,0,1,1, 1, 1,0,0, 1, 1,0, 1,1, 1), so b e {0,1}15. (I generated this bit sequence randomly.) Translate this into a symbol sequence in (ao, a1, a2, a3, a4) EAš and a sequence of amplitudes (xo, x1, X2, X3, X4) C5 and let vi(t) be this linear modulated signal with pulse p;(t) that encodes this data. Find the energy spectral densities of vi and v2.Create a single log-log plot with both ESDS. (c) (20 pts) Repeat the previous part for b = (1,1,1, 0,0,0,1, 1, 1,0,0,0, 1, 1, 1). Which of the bit sequences produces an ESD plot more similar to the PSD plot from part (a) and why?See Answer
Q18: (a) (10 pts) Let xn = 1 for all n E Z be a sample sequence. Let v(t) be the interpolation of these samples using sinc. Compute v(1/2) from the series representation of v(t). Hint: use the Taylor series of arc tangent. (b) (10 pts) Now let yn = (-1)" for n < 0, yn = 0 for n > 1, and let w(t) interpolate these samples using sinc. What is w(1/2)?See Answer
Q19: f) At a sampling rate f=15HZ the sampler outputs a signal v2(t). From what you found earlier,and using 5Hz/em as horizontal scale, represent V2(f) graphically on cartesian axes between(2 Marks)-45 and 45HZ.See Answer
Q20: E Sampling rate fs = 15HZ ) At a sampling rate f,-15HZ the sampler outputs a signal v2(t). From what you found earlier,and using 5Hz/cm as horizontal scale, represent V2(f) graphically on cartesian axes between-45 and 45Hz.(2 Marks)See Answer

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