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## Problem #I Linear Amplifiers (40 pts) (SHOW ALL WORK) In the Problem, all resistor values are in ohms, voltages are volts and currents are amps. Amp “A” is voltage-to-current, Amps “B” and “C” are current-to-voltage. Use /1 = 0.01(V1), v2 = 100(/2) and V3 = 50(/3). Use Vin shown in the table. Find all the values listed in the table. Hint: Observe that R3, R4 and R5 are m parallel.

## 1. The time domain signal x(t) = 2 Cos 2π 20 t + 2 Cos 2π 40 t is subject to ideal sampling. (a) Sketch the magnitude spectrum X(f) (show magnitude and frequencies) (1) (b) Identify the minimum sampling frequency fsmin (in Hz) (2) (c) By using fsmin of part (b), Sketch the Sampled spectrum Xs(f) up to the center frequenc of ±1 fsmin (4) (Do show precise magnitudes and all frequencies up to around ±1 fsmin ) (d) In case when you use sampling frequency of 90 Hz, decide the width of the guard band in [Hz], if you have. (3)

## 7. Filtering of COVID Data: In this problem we will filter and make plots of data from the 2020-2021 pandemic. Your final answer will be one plot, each part below describes an addition to make to the plot. Go to https://covidtracking.com/data/download and click on the Colorado link to open a spreadsheet with data from Colorado. Copy and paste the data from the column “positiveIncrease” which represents the new cases each day. Paste this column vector into an m-file and give the vector a name like x. a) Make stem plot of the data with time (days) on the x-axis. Use flipud to make the vector samples start from the oldest date and the last sample is the newest date. ,,,,,,,,,,,,,,

## 6. Difference Equation: Please do Problem P-5.4 on page 188

## 4. Sampling, Aliasing, and Reconstruction: In this problem we want to consider the signal 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(t) 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.

## 3. Audio Sampling and Recording: For audio recording, different sampling rates and number of bits per sample are used depending on the fidelity requirements. Stereo sound means there are two simultaneous audio signals that may be slightly different. Some standards are described in the table below:

## 2. Discrete to Continuous Conversion: Discrete time MATLAB signals can be converted to continuous time signals with the function soundsc which put out an analog signal for listening. In the following MATLAB code a discrete time sinusoid is generated and then played out with soundsc: nn=0:219009; xx=(7/pi)*cos(0.4*pi*nn+2.03); soundsc(xx,16000) a) Determine the analog frequency (in Hz) that will be heard. b) What is the duration of the signal in seconds? c) Now the code is changed to nn=0:219009; xx=(7/pi)*cos(0.4*pi*nn+2.03); soundsc(xx,32000) What will be the new analog frequency and tone duration that is heard? d) Now the code is changed to nn=0:219009; xx=(7/pi)*cos(0.4*pi*nn+2.03); soundsc (xx,8000) What will be the new analog frequency and tone duration that is heard?

## 1. Sampling: Consider the signal x t t ( ) 10 10cos(2 400 ) that is to be sampled at a rate of 1000 samples/second. a) Find the normalized frequency ˆ b) Write an expression for the sampled signal x n[ ]. c) Make a sketch of the spectrum of the continuous signal x(t). d) Make a sketch of the spectrum of the discrete signal x n[ ] vs ˆ , make sure your plot includes the shifted copies due to sampling. e) Use MATLAB to make a plot of x n[ ] showing the first 50 samples. Use the stem command instead of plot to illustrate the samples as discrete values.

## 5. For the system shown below: a) Find the time domain response, y(t), to a step input, r(t)=u(t) b) Determine the rise time, Tr, of the system

## 4. For the following state equation: a).Find,,, b)Find the natural response showing,,,,,,,, c) find the forced response showing,,,,,