EE342 Problem Set 9

DUE W 04/05/2000

1. Consider the continuous-time signal f(t)=(1/4)^t u(t).
   • Sample f(t) every T = 1/2 sec and write f[k] in closed-form.
   • Determine the DTFT of f[k] in closed form. Note if any aliasing occurred during sampling.
   • Sample f(t) every T = 1/2 sec from t = 0 sec to t = To = 2 sec. Plot the resulting truncated discrete-time signal fw[k].
   • Compute the DFT of fw[k] and plot its spectra (vs rOmega_o) on the same plots as the DTFT spectra of f[k].
   • Comment on similarities and differences between the two spectra and why they occur.
2. DTMF (Dual Tone Multi-Frequency) encoding is used on touch tone phones to represent each number on the telephone keypad as a combination of two tones. When a number key is pressed the tone of the row and the tone of the column are generated together. The keypad tone matrix is shown in the diagram below. As an example, pressing the '2' button generates two tones concurrently at 697Hz and 1336Hz. The frequencies were chosen carefully such that no frequency is a multiple of another, the difference between any two frequencies does not equal any of the frequencies, and the sum of any two frequencies does not equal any of the frequencies.

   1	2	3	697Hz
   4	5	6	770Hz
   7	8	9	852Hz
   *	0	#	941Hz
   1209Hz	1336Hz	1477Hz

   Consider the data stored in the data file tones.dat. The data was obtained by recording the tones generated when four numbers were pressed on the telephone keypad. After downloading the file from my web page into a working matlab directory, it can be loaded into a matlab vector named tones by typing load tones.dat. Verify the vector is there and is the correct size by typing whos. You can then listen to the tones if your computer has a sound card and if your version of matlab supports the sound() function by typing sound(tones,Fs) where tones is the vector of recorded data and Fs is the sampling frequency. The signal was sampled at 11,025Hz resulting in 10,320 samples. To illustrate the use of the DFT and the effect of aliasing perform the following operations. Let matlab connect the data points in all plots with the plot() function rather than stem() to keep the plots from looking too messy.
   • Listen to the signal if possible and plot it versus time in seconds. Clearly label on the plot where each number was pressed.
   • Seperate the signal data into four sets of data with each set of data corresponding to each of the individual four numbers. Plot each number's signal versus time.
   • Using the DFT, plot the magnitude and phase spectra of each number pressed versus frequency in Hertz. Zoom in to show the prevelent frequencies and indicate what they are. Be careful of images and indicate only the frequencies that are likely to exist in the original signal before sampling.
   • Using the tone matrix shown above and your DFT results for each number, determine the series of four numbers dialed.
   • Reduce the sampling rate to 1102.5Hz by keeping only every tenth sample in the original (total) signal containing all four numbers pressed. Plot the undersampled signal versus time.
   • Using the DFT, plot the magnitude and phase spectra of the undersampled signal versus frequency in Hertz. Note that here we'll have one magnitude and one phase spectrum plot for the combined four numbers dialed.
   • Comment on the effects of undersampling in terms of what you see in the DFT spectra. If possible, listen to the original signal as well as the undersampled signal to hear the effects and comment on what you hear as well.