(a) Consider a pulse, x(t), with a Gaussian shape given by x(r) = exp(-(t/b)2) Thus the full-width of this pulse measuredfrm teeoamplitude points is given by 2b. It can be sbown that the Fourier transform, X(jo), of the pulse x(r) is given by bto Recall from homework 5 that the frequency response function of a transmission line of length L with resistance, capacitance, inductance and conductance per unit length of R, C, L and G, respectively, is given by V(Lo) V,(0) r(ju) = Tbe electrical parameters associated with the 1865 trans-Atlantic submarine cable have been estimated to be: R = 2.2 x 10-3A/m, C = 8.0 x 10-11F/m. L = 4.1 x 10-7H/m. G = 1.0 x 10-1"Q-1/m and 4-310, m. Using Matlab and these transmission line parameters, plot lH jo) /H(N) and X(Jo)/XU vs (radi's) together on the same graph assuming a pulse width 2b 0.1 seconds. Use the axis command axis(0 1000 1] for your plot. The transmission line acts as a filter. Based on this plot, do you expect the pulse that emerges from the end of the transmission line to look like the input pulse? Explain your answa. Also plot Hj) in rads vs co. Use the axis command axis( 0 100 0-4] for your plot. Use the Matlab unwrap' command on your phase data, ZH(jco), so that the pbase is a continuous functioo of a. Use the Matlab comm andhelp unwrap' to find out about the 'unwrap. command. (b) The pulse, yfr), emerging from the end of the transmission line will be given by For a fixed value of t the above integral can be approximaled by the following sum provided is suficiently small and N is sufficiently large. Write a Matlab program to compute the output pulse, y(r), over the range -0.15 s15a 200 equally space times using the above sum. Choose = 0.05 rad/soc and N = 5000. On the same plot, plot the input pulse-r(1) and the normalized output pulse, r)/max r) vs time t. Use the axis command axis(I-0.2 1.6 0 1.1 foe your plot. Commeat on the implication of your result. (c) Compute Hjo) when R G L C and show that when this condition is met that Recall ve,t) is the voltage measure along the transmission line at timer and a distance z from the beginning of the line. This results indicates that when the condition is satisfied, the launched signal is attenuated as it propagates along the cable, but that it's shape does not change (and thus the width of a launched pulse will rem ain constant as it propagates along the cable)-E = is known as the Heaviside condition. In general, > For example, the electrical parameters associated with the 1865 trans-Atlantic submarine cable have been estimated to be: R-2.2 x 10-3/m, C = 8.0 x 10-"F/m. L-4. 1 x 10-7H/m, and G-1.0 x 10-10-1/m. By periodically splicing special sections of cable, with high inductive values, into the normal cable (a process known as inductive loading), the Heaviside condition can be met. The American Telegraph and Telephone Company (AT&T) commercially deployed Heaviside's inductive loading technique in 1900, and the technique became widely used for both telegraph and telephone lines. This invention (a form of egualization) made enormous sums of money for AT&T and revolutionized telecommunications at the beginning part of the 20th century. (d) Use the transmission line parameters given in part (c) above, but change L to 1.76 10-3 H/m so that the Heaviside condition is satisfied. With this new value for L repeat parts (a) and (b). Use the Matlab axis command axis(IO 10-12 0) for the phase plot. Comment on your results (e) Compute the speed at which an electrical pulse will propagate down a cable when the Heavisde condition is satisfied. Express your answer in terms of L and C, and find a numerical value for the speed corresponding to the values of L and C given in this problem. Compare this speed to the speed of Light in a vacuum. (a) Consider a pulse, x(t), with a Gaussian shape given by x(r) = exp(-(t/b)2) Thus the full-width of this pulse measuredfrm teeoamplitude points is given by 2b. It can be sbown that the Fourier transform, X(jo), of the pulse x(r) is given by bto Recall from homework 5 that the frequency response function of a transmission line of length L with resistance, capacitance, inductance and conductance per unit length of R, C, L and G, respectively, is given by V(Lo) V,(0) r(ju) = Tbe electrical parameters associated with the 1865 trans-Atlantic submarine cable have been estimated to be: R = 2.2 x 10-3A/m, C = 8.0 x 10-11F/m. L = 4.1 x 10-7H/m. G = 1.0 x 10-1"Q-1/m and 4-310, m. Using Matlab and these transmission line parameters, plot lH jo) /H(N) and X(Jo)/XU vs (radi's) together on the same graph assuming a pulse width 2b 0.1 seconds. Use the axis command axis(0 1000 1] for your plot. The transmission line acts as a filter. Based on this plot, do you expect the pulse that emerges from the end of the transmission line to look like the input pulse? Explain your answa. Also plot Hj) in rads vs co. Use the axis command axis( 0 100 0-4] for your plot. Use the Matlab unwrap' command on your phase data, ZH(jco), so that the pbase is a continuous functioo of a. Use the Matlab comm andhelp unwrap' to find out about the 'unwrap. command. (b) The pulse, yfr), emerging from the end of the transmission line will be given by For a fixed value of t the above integral can be approximaled by the following sum provided is suficiently small and N is sufficiently large. Write a Matlab program to compute the output pulse, y(r), over the range -0.15 s15a 200 equally space times using the above sum. Choose = 0.05 rad/soc and N = 5000. On the same plot, plot the input pulse-r(1) and the normalized output pulse, r)/max r) vs time t. Use the axis command axis(I-0.2 1.6 0 1.1 foe your plot. Commeat on the implication of your result. (c) Compute Hjo) when R G L C and show that when this condition is met that Recall ve,t) is the voltage measure along the transmission line at timer and a distance z from the beginning of the line. This results indicates that when the condition is satisfied, the launched signal is attenuated as it propagates along the cable, but that it's shape does not change (and thus the width of a launched pulse will rem ain constant as it propagates along the cable)-E = is known as the Heaviside condition. In general, > For example, the electrical parameters associated with the 1865 trans-Atlantic submarine cable have been estimated to be: R-2.2 x 10-3/m, C = 8.0 x 10-"F/m. L-4. 1 x 10-7H/m, and G-1.0 x 10-10-1/m. By periodically splicing special sections of cable, with high inductive values, into the normal cable (a process known as inductive loading), the Heaviside condition can be met. The American Telegraph and Telephone Company (AT&T) commercially deployed Heaviside's inductive loading technique in 1900, and the technique became widely used for both telegraph and telephone lines. This invention (a form of egualization) made enormous sums of money for AT&T and revolutionized telecommunications at the beginning part of the 20th century. (d) Use the transmission line parameters given in part (c) above, but change L to 1.76 10-3 H/m so that the Heaviside condition is satisfied. With this new value for L repeat parts (a) and (b). Use the Matlab axis command axis(IO 10-12 0) for the phase plot. Comment on your results (e) Compute the speed at which an electrical pulse will propagate down a cable when the Heavisde condition is satisfied. Express your answer in terms of L and C, and find a numerical value for the speed corresponding to the values of L and C given in this problem. Compare this speed to the speed of Light in a vacuum