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# Fourier Analysis, Signals, Bandwidth, Formulas

The solution for every part it should include the original formulas & a steps of calculations.

Problem 1 (10 points): Consider the example given in textbook pages 91-92.

Figure 1 (on page 92) part (d) shows the low pass filtered signal when only the first four harmonic terms are passed through the filter. The low pass filter rejects the other higher frequency components.

1. a) (7 points) compute the percentage of power for the signal shown in part (d) relative to the original signal shown in part (a) of the figure.
2. b) (3 points) compute the bandwidth of the signal shown in part (d) of the figure. Assume A = 2 volts and T = 1 msec.

Hint: you may want to use the table of coefficients , , and given in the PowerPoint slides provided by instructor.

Problem 2 (15 points): Consider the power spectral density function shown in Figure below for the periodic signal g(t). The figure shows the specific frequency components and the corresponding power.

1. a) (10 points) specify one corresponding time-domain function g(t).
2. b) (5 points) sketch g(t) for values of t ranging from -1 to 1 seconds.

Problem 3 (15 points) A file of size = 100 KBytes is to be sent from Node 1 to Node 5 traversing four identical intermediate links as shown in Figure below. The data link protocol operating on node 1 requires that = 50 bits be added as overhead to any frame transmitted on the link. Assume the link transmission rate is = 80 kb/s while the link propagation time, prop is equal to 2 msec. Node processing delay is ignored. An intermediate node can start transmitting the received frame immediately when it is fully received.

1. a) (5 points) If the file is NOT segmented at node 1, but rather sent as a whole together with the overhead , compute the total transmission time, total1, till the file is completely received at Node 5.
2. b) (10 points) For this part assume the file at node 1 is divided into frames each of size Of course, as before V overhead bits are added to F data bits at node 1 before transmission on link 1. The figure below shows the case of dividing the frame into frames.

Note that is the time to transmit the frame on the link, while prop is the time it signal to propagate and reach the other end of the link.

If the frame size F is chosen to be very small, then m is going to be large and these links are going to be transmitting overhead bits most of the time. The other extreme is when the frame size is equal to the original file size as in part (a) of this problem, then m is equal to 1. In this case the file transmission time is still not optimal. It is required to determine the best frame size that results in minimum transmission time.

Hint: Let the frame size F vary from 1 bit all the way to original file size in bits, and for each value compute the corresponding T_total2. Refer to textbook problem 39 page 190 for the more general case.

Problem 4 (15 points):

A typical telephone channel has the bandwidth of 3100 Hz can support dial-up modems that can variable speeds depending on the quality of the links on both ends of the data call.

1. a) (5 points) Assuming the dial-up modem is working at rate of 33.6 kb/s, compute the minimum needed Eb/N0 in dBs.
2. b) (4 points) For part (a), what is the theoretical number of bits per symbol needed to achieve this 33.6 kb/s?
3. c) (6 points) What are (1) the baud rate and (2) number of bits per symbol used on a standard telephone channel that operates at 33.6 kb/s. Specify the standard model that

implements the above specifications for a dial-up modem.

Hint: Refer to textbook pages 145-147.

Problem 5 (10 points):

Consider a 256 4kHz channels ADSL system operating over twisted pair as shown in Figure. The system utilizes a variety of modulation and coding schemes on each of the channels depending on the reported SNR for the channel. Assume the available modulation schemes are BPSK, QPS, 16-QAM, and 64-QAM, while the utilized coding schemes provide coding rates equal to 1/4, 1/3, 1/2, 3/4, and 5/6.

Assume the POTS uses the lower 6 channels and 2 channels are reserved for control. Furthermore, ADSL uses a modulation that produces 4000 bauds per second on the channel.

If 3/4 of the available ADSL channels are utilized for the downstream, compute the minimum and maximum possible downlink throughput numbers for the ADSL system?

Hint: Please refer to textbook subsection on ADSL modem.

Last Updated on September 19, 2021

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