Homework on Signal Analysis, 2000 version (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.

1.-5. Match the signals on the left with the Fourier spectra on the right. Assume that any repeating signal continues indefinitely, that the frequency scales on spectra A and D are the same, and that the frequency scales on spectra B and E are the same.

Periodic signals can be represented by a discrete series of frequency components. These components will have frequencies which are multiples of the primary frequency of the signal. Therefore, a higher frequency signal will have its components more spread out along the frequency axis relative to a lower frequency signal of the same wave shape.
Non-periodic signals are represented by a continuos spectrum of frequency components. The transform of a square pulse resembles a diffraction pattern. As the pulse becomes more narrow in the time domain, the transform will spread in the frequency domain.
By contrast, a gaussian pulse transforms to a gaussian. However, it still has an inverse relationship between the width of the pulse in the time domain and in the frequency domain.From these guidelines the various signals can be matched with their respective frequency representations
1. B
2. D
3. A
4. E
5. C

6. An antenna (open-ended) produces radiation in the form of a standing wave. If the antenna is 10 meters long, what are the four longest wavelengths of radiation that it can emit? Would this be a good antenna to produce radio waves? If so, would it be for FM or AM stations?
The boundary conditions of an open-ended antenna require that standing waves have a minima (node) at one end and a maxima (antinode) at the other. Wavelengths which meet this requirement are given by the expression L = n l / 4 for n = 1, 3, 5, … Therefore the four longest wavelength which can be emitted are 4 L = 40 m,
4/3 L = 13.3 m,
4/5 L = 8 m, and
4/7 L = 5.7 m The corresponding frequencies are given by f = c / l. This gives a range of frequencies between 7.5 MHz and 52.5 MHz which falls between the AM and FM bands. Therefore, AM wavelengths are too long to be radiated by this antenna though FM could be radiated. FM is not likely, since it is a waste of antenna length.

7. What are the advantages and disadvantages of using the NRZ data format vs. the RZ or Manchester formats?
Because the NRZ format uses only one signal amplitude per bit, it is able to send twice as many bits than RZ or Manchester with a given bandwidth. However, the simplistic one value per bit scheme makes no distinction between bits. Therefore, if the bit rate of the data stream is not known, NRZ format data can not be interpreted.

8. To send data at a rate of a TeraHertz, what is the longest rise time you can have for the signal? 1 Terahertz = 10¹² bits / seconds For data to be read the rise time must be less than .7 of the signal period rise time < .7 (1 / f ) = 7 x 10^-13 sec