Analyzing a Square wave in terms of sin components

A. A signal generator outputs a unipolar square wave with a period of 0.50 ms. The output of the generator is passed through an ideal low pass filter that has a cut off frequency of 4.5 kHz, to a spectrum analyzer. What frequencies would be seen on the spectrum analyzer screen?

b. The signal generator output in part (a) is changed to a rectangular pulse train in which the period is still 0.50 ms but the pulse length is 0.10 ms. What frequencies would be displayed on the spectrum analyzer screen?

c. The waveform output by the signal generator in part (b) is changed so that the rectangular pulses have the same width but no longer are periodic. They occur at random intervals of time. Explain in words how the display on the spectrum screen would change.

Solution Preview

A regular periodic square or rectangular pulse train (i.e. time domain signal) can be split into its individual "Fourier" frequency components using Fourier Analysis techniques. For an Odd function unipolar square or rectangular signal, the time domain signal {f(t)} can be written in terms of odd harmonic frequency sinusoidal components such that

Where, a0 is the DC level of the signal (which would exist for a unipolar signal but not a bipolar signal) and n is any odd integer with n tending to infinity for a real perfect square wave signal.

This follows the classic and well known Sin(x)/x function or Sinc(x) function as shown in the attachment.

Thus it shows that there is a general shaping envelope to the frequency ...

Solution Summary

Detials of an unipolar Square Wave in the time domain are used, such as its period to determine the output frequency components seen on a spectrum analyser. The pulse duration is then specified and the output components as seen on the spectrum analyser deduced

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Please provide necessary explanations leading to the answer.
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F(r)=[(r^2)-(r^-2)]^-2
Find the Limit
lim