Using the pure musical tone A₃, we examine the consequences of choosing fewer samples to estimate its power spectral density. In the following, you can hear the original note x(t), view the (sampled) signal and its windowed version y(t), and estimate the power spectral densities S_xx(ω=2πf) and S_yy(ω=2πf). Use the zoom controls to examine both time domain and frequency domain characteristics.

1. Listen to the musical tone so that you will know what signal you are dealing with and what you might expect in a spectral analysis.


2. What “theoretical” spectrum should you expect for S_xx(ω=2πf)? Why do you think that this is the proper choice?


3. What power spectral density do you observe when you use 2¹⁵ samples of the tone? Does this match your expectation for the “theoretical” spectrum?


4. What power spectral density do you observe when you use 2¹⁰ samples of the tone?


5. Describe in words how the spectral estimate of the power spectral density changes as one goes from 2¹⁵ to 2¹⁰ samples.


6. How many cycles of the sinusoid are in the time-domain window with 2¹⁰ samples?


7. As you continue to reduce the number of samples down to 2⁸, describe the potential problems that you might expect in estimating the frequency of a musical note that is “near” A₃.