Engineering Homework Help

Please use matlab for this homework and attach a .m file for the codes.

Seismic Deconvolution (15 pts)The application for this problem is trying to identify the seismic properties of different layers of sedimentand earth in the ground. This can be done by digging a hole, and placing a transmitter at the red circlelabelled ”TX”. The transmitted signalx[n] travels through the earth before being detected at the red circlelabelled ”RX”, and gives the detected signaly[n]. The goal is to calculate the impulse response of the earthh[n] given knowledge of the transmitted signalx[n] and the detected signaly[n]. From this impulse responsedifferent properties of the earth can be directly derived. Suppose the transmit and receive hardware havethe following specifications:•Sampling periodTs= 1 ms•Total record timeT= 400 ms1. One of the most common transmit waveforms for seismic problems is the Ricker wavelet, given by theexpressionx(t) = (1−2π2f20t2)e−π2f20t2for a certain center frequencyf0. With the provided specifications, a center frequency of 30 Hz, and atime delayof 50 ms:(a) Plot the transmit pulsex(t)(b) Plot the Magnitude response|X(ω)|between 0 and 500 Hz (Notice where the maximum is in thefrequency domain).2. The propagation through the earth is governed by the green’s function, equivalently the impulse responsehi[n]. This impulse response is given in the provided MATLAB functionseismicgreensfunction.m.The greens function is a simplification of a realistic greens function, so thathi[n] =L∑k=0Ake−σknu[dk−dk−1]or a decaying exponential at each layer, represented by the tophat functionsu[dk−dk−1] for the distancefunctiondat each layerk= 0,…,L. The outputy[n] is then calculated by convolution asy[n] =hi[n]? x[n] =F−1{Hi(ω)X(ω)}Page 2

Consider you know there are three layers of earth with the properties:Aσd170.51020.2170.3(a) Plot the outputy[n](b) Plot the impulse responsehi[n], as calculated usinghi[n] =F−1{Y(ω)X(ω)}(deconvolution). You mayfind the MATLAB functionsfftandifftuseful.(c) Due to the ill-posed nature of deconvolution, in a noisy environment, the deconvolution oftenundergoesTikhonov regularization. This can be accomplished by adding a factorγto the de-nominator. Perform Tikhonov regularization (without adding noise) and plot ̃hi[n] =F−1{Y(ω)X(ω)+γ}withγ= 0.1.3. The provided fileseismicdata.matcontains the outputy[n] of an unknown seismic readinghi[n] withthe same known Ricker wavelet inputx[n], and is corrupted by noise. It is known that there are threedifferent layers, but the specific size and material composition are unknown. The possibilities are:•Limestone and Wet Sediment:Aσd1010.31210.3110.3•Homogeneous Sandstone:Aσd110.5110.2110.3•Sediment and Coal:Aσd1070.51220.2110.3(a) Which material did this reading come from? (Compare the reconstructed impulse response to theresponse calcualted from theseismicgreensfunction.mfunction)(b) Plot the impulse response calculated from the deconvolution (using the Tikhonov regularizationwithγ= 0.1)(c) Plot the impulse response calculated from the deconvolutionWITHOUTusing any regularizationPage 3

Ultrasound Processing (10 pts)The application for this problem is ultrasound imaging, such as what you might have seen in a hospital.An ultrasound pulse is transmitted, reflects off of different tissue, and is detected by the ultrasoundprobe. One efficient way of reconstructing an ultrasound image is to plot the envelope of the signal,which is what we will do in this problem. The data for this problem is found inrfdataphantom.mat.The ultrasound signal is found in RFdata (time×angle), the speed of sound in meters per second isgiven by csound, and the sampling rate by fs.4. First we will examine the features of the detected ultrasound signal.(a) Plot the ultrasound signal at the first angle(b) Plot the magnitude response|X(ω)|at this first angle between−7.5 and 7.5 MHz.5. We can calculate the analytic signal using the MATLAB functionhilbert(a) Plot the magnitude of the analytic signal (also known as the envelope) at the first angle(b) Plot the magnitude response of the analytic signal (at this first angle)|A(ω)|between−7.5 and 7.5MHz.We can form the log compressed signal by using the expressionlogEnvelope = log10(1 +CF|a(t)|)/log10(1 +CF)using a compression factor ofCF= 6. Lets assume we haveMangles. The lateral position can then becalculated asxi= ((M−1)/2)θ0+iθ0whereθ0is theanglevariable corrresponding to the step size. The axial position can be calculatedusing a step-size of ∆ =c2fs, wherecis the speed of sound andfsis the sampling rate. If we assumethere areNtime samples, then the expression isyi=y0+ ∆N−i∆wherey0is thestartpositionaxialvariable.6. Plot the log-compressed ultrasound image (logEnvelope) using the lateral (xi) and axial (yi) scalings.Use a gray colormap to make it look like you expect.Page 4

Near Vertical Incidence Skywave Detection (15 pts)The application for this problem is Near Vertical Incidence Skywave (NVIS) propagation, which isused for non line-of-sight communications for either military or ham radio applications. The goal is tocommunicate with someone beyond the horizon. In order to accurately transmit to the receiver (and notbe intercepted by adversaries), the signal is fired at a significant vertical angle, so that it bounces offthe ionosphere before travelling towards the intended recipient. Before accurate communication can beachieved, an estimate of the height of the ionosphere must be known, so that fairly simple trigonometrycan tell you which angle to transmit at to illuminate a certain area. This estimate is done by sending atransmission vertically, and detecting how long it takes to be received. This process can be implementedefficiently in high noise environments usingmatched filtering. It is this technique which we will nowexplore.7. The matched filter is a fairly simple filter, which makes its effectiveness that much more surprising. If thetransmit pulse was a signalx(t), then the matched filter responsem(t) =x(T−t) (i.e. a time-reversedversion of the transmitted signal). The output of the matched filter is thend(t) =r(t)? m(t) =d(t)? x(T−t)for a received signalr(t) and matched filter outputd(t). The location of the peak in thed(t) signal isthen the location of the transmit pulsex(t) in the received signalr(t). To begin, we will use a simplesquare wave:(a) Assume you have a received signalr(t) that is 20 ms long with a sampling period of 14.286μs.Center your signal so that the time vector goes from−10 to 10 milliseconds. Plot the rectangularpulsex(t) =u(t+ 500e-6)−u(t−500e-6).(b) If we assume the speed of light is 3e8 [m/s], what is thetime delayt0of the signal to propagateto the ionosphereand backif the ionosphere is 150 km up?(c) Calculate the response asr(t) =x(t−t0) for the time delay corresponding to 150 km. Plot thematched filter responsed(t) =r(t)? x(T−t) for−10e-3< t <10e-3 (You may find the MATLABfunctionconvwith the option ’same’ helpful)(d) Now we will add noise to the received signal. Add normally distributed random noisen(t) (youmay find the MATLAB functionrandnhelpful) to the received signal with amplitude 3. Plot thenoisy received signalˆr(t) =x(t−t0) +n(t)(e) Plot the matched filter filter response to the noisy received signal (with the correct x-axis scaling)ˆd(t) = ˆr(t)? x(T−t)Page 5

(f) Find the index of the peak of the matched filterˆd(t) (the MATLABmaxmay be helpful). Whatheight does this correspond to? (You can run it several times to see how much randomness there isin the estimate, or you can userng(453)at the top to seed the random number generator)8. For the final step, consider the signals in the providedNVIS.mat. There is are three different transmitsignals, TXA, TXB, and TXC. There are also three different received signals, RXD, RXE, and RXF.Each transmit signal is unique to one of the received signals, and is only used once. The first transmitsignal TXA, is what is known as a chirp signal. This signal has a frequency that varies in time, andthis particular chirp has a quadratic dependence on time, sof(t)∼t2. The ionosphere height was 162km for this transmit. The second transmit signal TXB is a Ricker wavelet, similar to what was usedfor the seismic processing problem. However, this time we have used the analytic version of the signal.The ionosphere height was 140 km for this transmit. The final signal TXC is a square wave (which wehave also converted to an analytic signal), similar to that used in the previous problem. The ionosphereheight was 150 km for this transmit. The sample rate and time vector are identical with the previousproblem (7). Using matched filtering (convolve only the real parts):(a) Which transmit signal was used for the RXD signal? What estimated height was the ionosphere forthis measurement? Plot the real part of the RXD and the matched filter response for the correcttransmit signal.(b) Which transmit signal was used for the RXE signal? What estimated height was the ionosphere forthis measurement? Plot the real part of the RXE and the matched filter response for the correcttransmit signal.(c) Which transmit signal was used for the RXF signal? What estimated height was the ionosphere forthis measurement? Plot the real part of the RXF and the matched filter response for the correcttransmit signal.(d) Which signal (chirp, Ricker, square) performed the best for matched filtering applications?