As indicated in the chapter text, any wave can be represented as a sum of sine waves of various frequencies, amplitudes, and phases. This problem explores the effects of channel bandwidth, shape of the spectrum, and phase distortion on the shape of a waveform. A square wave is made up of sine waves according to the following equation:

S = sin(2????f t) + 1∕3 sin(3 × 2????f t) + 1∕5 sin(5 × 2????f t)+…

It is difficult to plot sine waves with any accuracy. Instead, we will use a triangular waveform shape as an approximation for the sine wave.

a. On a sheet of graph paper, carefully construct a triangular wave that starts at 0, rises to a maximum value of 15, falls to a minimum value of −15, then returns to 0. Your waveform should extend over 15 units on the time scale. Now construct a second triangle wave with an amplitude of 5 and a time span of 5. Your second waveform should start at 0. Add the amplitudes of the new waveform and the previous one to produce a new waveform which is the sum of the two. What do you observe?

b. Now create a third waveform of amplitude 3 and time span 3, and add it to the previous result. What do you observe? If the bandwidth is limited so that only the first two waves can pass through the channel, what is the effect on the waveform?

c. Next, start with a fresh sheet of graph paper. Draw the fundamental triangle wave. Draw the second triangle wave, but this time shift the phase 90°, so that the positive peak of the second wave coincides with the initial zero position of the fundamental. Add the two wave-forms. What effect did the phase shift have on the summed waveform shape?

d. On another fresh sheet of graph paper, draw the fundamental waveform one more time. Draw the second waveform, this time with a height of 3 instead of 5. Also draw the third waveform with a height of 4 instead of 3. All wave-forms start at 0, that is, with no phase shift distortion. Add the three wave-forms. What effect did the altered spectrum shape have on the waveform?

e. Based on the original amplitudes of the waves and the modified amplitudes used in part (d), draw the spectrum of the channel.