Homework on Fourier Analysis (solutions)

Feel free to work on the homework in groups. The work you hand in, however, should reflect your understanding of the material. You should show all of your calculations (neatly) and justify all of your answers for full credit.

According to the Schaum's Outline Mathematical Handbook, the Fourier expression for a "sawtooth wave" is

1. Is this a Fourier Series or a Fourier Transform? Explain how you know this.

This is a series, because it is a sum of discrete frequencies, as opposed to an integral of continuous frequencies.

2. What is the frequency f₀ of the sawtooth wave described by the expression above? (Hint: Don't forget that the argument of a sinusoidal function will be wt, where w is the angular frequency. You are asked for the frequency f.)

The frequency of a wave described by a Fourier series is the same as that of the lowest-frequency term (fundamental) in the series. The first term has the lowest frequency, with w = p and a frequency of 0.5 Hz. (w = 2pf)

3. Sketch the Fourier spectrum for this wave, including frequencies up to 10f₀.

4. The Fourier spectrum for a triangular "pulse" is sketched below. Is this pulse represented by a Fourier Series or a Fourier Transform? Explain how you know this.

We should use a transform, because a single pulse is not a periodic signal and so it cannot be adequately represented by a sum of discrete frequencies; a continuous spectrum must be used instead.

5. Indicate on the spectrum the range of frequencies comprising the bandwidth of the pulse.

Make your own sketch of the Fourier spectrum for a square pulse. What should the approximate spread in frequencies be if the pulse width is 1 ns? (A nanosecond, 1 ns, is 10^-9 s.) Justify your answer.

A pulse width of 1 ns corresponds to an information transfer rate of 1 GHz. Since bandwidth is roughly equal to bitrate, the frequency spread should be approximately 1 GHz.

1.	Is this a Fourier Series or a Fourier Transform? Explain how you know this.
	This is a series, because it is a sum of discrete frequencies, as opposed to an integral of continuous frequencies.
2.	What is the frequency f₀ of the sawtooth wave described by the expression above? (Hint: Don't forget that the argument of a sinusoidal function will be wt, where w is the angular frequency. You are asked for the frequency f.)
	The frequency of a wave described by a Fourier series is the same as that of the lowest-frequency term (fundamental) in the series. The first term has the lowest frequency, with w = p and a frequency of 0.5 Hz. (w = 2pf)
3.	Sketch the Fourier spectrum for this wave, including frequencies up to 10f₀.

4.	The Fourier spectrum for a triangular "pulse" is sketched below. Is this pulse represented by a Fourier Series or a Fourier Transform? Explain how you know this.
	We should use a transform,* because a single pulse is not a periodic signal and so it cannot be adequately represented by a sum of discrete frequencies; a continuous spectrum must be used instead.*
5.	Indicate on the spectrum the range of frequencies comprising the bandwidth of the pulse.

	Make your own sketch of the Fourier spectrum for a square pulse. What should the approximate spread in frequencies be if the pulse width is 1 ns? (A nanosecond, 1 ns, is 10^-9 s.) Justify your answer.
	A pulse width of 1 ns corresponds to an information transfer rate of 1 GHz. Since bandwidth is roughly equal to bitrate, the frequency spread should be approximately 1 GHz.