Homework is assigned each Wednesday and due the followingWednesday. It will be graded roughly, perhaps on a scale of 1-4.Late homework will not be accepted. You are allowed to workon the homework in small groups, but you must write up your ownhomework to hand in. The homework will be discussed, but notsolved, in the weekly section meeting.
signals and systems homework solutions
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Grading policy: ONE late homework submission is allowed without penalty. It should be submitted by the next class after the due date.
Late submissions beyond the first one will NOT be accepted.
ONE lowest score for the homework assignments will be automatically dropped.
Honor code: You are encouraged to discuss homework assignments with each other, but write up solutions on your own! Sharing or copying a solution will result in a zero score for the relevant assignment for both parties.
If a significant part of your solution is due to someone else or from other sources (books, forums, etc), you should acknowledge the source! Failure to do so will result in a zero score for the relevant assignment.
This lab is meant to go with lectures that provide a basic introduction to signals such as continuous and discrete-time signals, operations on signals such as scaling of the time axis, shift in time. In this Lab, you will learn to use wavrecord to record an audio clip or look for an audio clip online in wav form and try to plot that signal as a function of time. You will simulate echo in an auditorium and listen to the sound.
Exercise 3 - Even though the audio signal is originally a continuous-time signal, we cannot represent continuous-time signals on a digitial computer directly. In this example, you have sampled the continuous-time signal and you are representing it as a discrete-time signal. Create the signal $x[2n]$ and play the signal by using the sound command and use the same sampling frequency as before. See that it sound shriller. Think about why this should be the case.
Exercise 4 - Sum of sinusiods of the same frequency. One of the useful results about sinusoids and cosinusoids is that sums of sinusoids and cosinusoids of the same frequency but with arbitrary amplitude and phase result in another sinsuoid or cosinusoid of the same frequency. In this exercise you will observe this using MATLAB. Let $x_1(t) = A_1 \sin(\omega t + \theta_1)$ and let $x_2(t) = A_2 \sin(\omega t + \theta_2)$. For $A_1 = 1, A_2 = \sqrt3, \omega = 4 \pi, \theta_1 = 0, \theta_2 = \pi/4$, plot $x_1(t)$ and $x_2(t)$ as a function of $t$ for $-10 \leq t \leq 10$ . Use 200 samples along the time axis, i.e., time should be incremented in steps of 0.01. Observe that each of the signals is periodic with the time period that you expect. Now plot the sum of the signals and observe that it is also a sinusoid with the exact same frequency (time period). Now, analytically find an expression for $x_3(t)$ in the form $B \sin(\omega t + \phi)$, i.e., find $B$ and $\phi$. Plot this signal for the same range of values of $t$ and observe that the signals coincide. Use the hold on command in MATLAB and plot the two signals in different colors. We can easily extend this result to see that a linear combination of any number of sinusoids of the same frequency results in another sinusoid of the same frequency.
Exercise 5 - Sum of sinusoids of different frequencies. Let us now consider the same example as above, except that we consider the case when the frequencies of sinusoids are different. Let $x_1(t) = A_1 \sin(\omega_1 t + \theta_1)$, $x_2(t) = A_2 \sin(\omega_2 t + \theta_2)$ and $x_3(t) = A_3 \sin(\omega_2 t + \theta_3)$. For $A_1 = 1, A_2 = 1, A_3 =1$, $\omega_1 = 4 \pi, \omega_2 = 2 \sqrt5 \pi, \omega_3 = 2 \sqrt3 \pi$, $\theta_1 = 0, \theta_2 = \pi/4, \theta_3 = \pi/6$, plot $x_1(t),x_2(t)$ and $x_3(t)$ as a function of $t$ for $-10 \leq t \leq 10$. Let $x_4(t)$ be the sum of the three signals. Is $x_4(t)$ periodic? Why or why not? Adding sinusoids of different frequencies arises in modeling wireless communication channels when the receiver or transmitter is moving. In wireless channels, typically the transmitting signals propagate via several reflections (similar to echo) and the net result is that if a sinusoid is transmitted, the received signal will be the sum of sinusoids with different amplitudes and phases. When there is relative mobility between the transmitter and the receiver, these reflected paths also undergo different Doppler shifts and, hence, their frequencies will be slightly different.
Exercise 7 - Consider the signal $x[n] = e^-0.1n \ U[n+20]$. Let $y[n] = \sum_-\infty^n x[m]$. Write a MATLAB program to compute $y[n]$ and plot it. Use the stem function to plot discrete-time signals. Notice that $y[n] = y[n-1]+x[n]$. Using this can you simplify your program? Read the help on the filter command and implement this using the filter command.
Fourier's results says that a large class of periodic signals can be represented as linear combinations of sinusoids and cosinusoids or, equivalently, complex exponentials. More precisely, let $x(t)$ be a continuous-time periodic signal which satisfies some conditions known as the Dirichlet conditions (let us worry about these conditions now). Let the time period of the signal be $T$. We will refer to $\omega_0 = \frac2\piT$ as the fundamental frequency of this signal. It turns out that the signal $x(t)$ can be expressed as
Think about what you have accomplished and how this means that different signals can be written as linear combinations of complex exponentials. What questions come to your mind? Note these questions down so that we can talk about them on Tuesday.
Write a program in MATLAB that will compute the discrete-time Fourier series (DTFS) coefficients given a discrete-time signal $x[n]$ and its time period $N$. Using this program compute the DTFS coefficients of the signals in Problem 3.28 a (parts a and c only). Compare with the analytical expressions you got when working out the homework for parts a and c. Plot the magnitude and phase of $X[k]$ as a function of $k$ for $0 \leq k \leq N-1$. Use the stem function in MATLAB for the plot.
The assignment will be due on Tuesday along with the rest of the homework.What to turn in ? - MATLAB code and a total of 4 plots (magnitude and phase of the Fourier series coefficients for part a and part c).
You may have heard the exciting news that scientists were able to record gravity waves from the collision of two black holes. What a historic moment for science! In my opinion, it is as much a victory for engineering as it is for science, since the engineering that has gone behind the development of detectors to detect extremely weak signals and the algorithms that are needed to detect the weak signal in the presence of noise are remarkable! One way for people to get a feel for the gravity wave is to convert the gravity wave into a sound signal and such as signal is shown below.
The type of signal recorded is called a chirp signal. A chirp signal is a sinusoidal signal whose frequency linearly increases with time. Assume that a chirp signal is of length $T$ seconds and the frequency of the sinusoid increases from $\omega_1$ at the $t=0$ to $\omega_2$ at $t=T$. Express such a signal mathematically. Let $x_L(t)$ and $x_W(t)$ be the two signals recorded at the LIGO observatories in Livingston, Louisiana and Hangford, Washington. Mathematically, what is the relationship between the two signals?
I would like to emphasize that the signal recorded in the LIGO detectors is *not* the audio signal created by the collision of black holes. The signal actually corresponds to the expansion and shrinking of distances between objects, i.e., warping of the space. We can nevertheless pretend that these signals are audio signals and ``listen" to them through speakers. This should also emphasize the utility of the tools we are learning in this course to model a variety of phenomena from gravity waves to audio signals!
Other HW. Download the files: Problem description , Matlab file , Tada sound .Study the problem described in the linked notes. Use the Matlabprogram to experiment with different parameter values (frequency, filters, etc) ----------------------------------------------------------- EEE 203 FINAL EXAM Material: System properties (L,TI,C,M,S),e.g., given a system determine if it is TI. Output of a system to"composite" inputs from its output to elementary inputs.Linear systems: General description; systemproperties in terms of the impulse response; convolution; e.g., given a linear system determine if it iscausal. Also given h and x find the convolution h*x. Fourier Series expansions; expansion of periodicsignals (impulse-trains, pulse-trains); properties of FS (simplify the computation of FSusing linearity, time shifts etc.); filtering (computation of FS for the output);Parseval's relation (finite FS approximation properties, computefilter/other parameters to retain/reject a certain percentage of the inputsignal energy etc). Fourier Transforms, Basic Properties. (Compute theFourier Transform of "composite" signals using shifting/convolution/... properties. Computefrequency-domain properties without explicitly computing the transform itself). Fourier Transforms, Applications: Filtering (Use of the convolution property, idealfilters, basic notions of non-ideal filters. Steady-state response to sinusoids. Transfer functions of systems described by ODEs with constant coefficients.). Sampling and Reconstruction(Frequency domain analysis of sampling systems). Discrete time signals andsystems. Fundamental properties,difference equations, transfer functions. Steady-state response toasymptotically (steady-state) periodic inputs.
Beginning in fall, 2020, this course will live primarily in Canvas. However, this site contains many reference materials that you may find useful including MATLAB examples and previous homework, quizzes, and exams. 2ff7e9595c
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