Laboratory Exercise 12.3
Can the use of windows other than the rectangular (Block) window help us to estimate power spectral densities? We start with the pure musical tone A3 to examine the interplay between the number of samples and the form (shape) of the window w[n] that is used to estimate the power spectral density.

In the following, you can hear the original note, view the (sampled) signal x[n] and its windowed version y[n] = w[n]x[n], and estimate the power spectral densities Sxx(ω=2πf) and Syy(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 Sxx(ω=2πf)? Why do you think that this is the proper choice?

  3. Using 215 samples, how does the estimate of the power spectral density change when a window other than the Block window is used to select and weight the data that are to be analyzed?

  4. Which of the eight windows leads to the narrowest main peak of the spectral estimate? Which is the next best? Which is the widest main peak?

  5. Repeat the above two questions but now with 210 samples.

  6. What is the reason to use the width of the main peak in comparing various windows?
Until now, we have looked at the main lobe associated with the window. Now let us look at the sidelobes.
  1. Press the   Linear / Logarithmic   button. The window shows the same display as before except that the log10() of the power spectral density is displayed. This can help to pick out spectral details.

  2. Examine the side lobes of the spectrum of Sxx(ω=2πf) with 215 samples and a Block window.
In the following it may help to use a reasonable amount of zoom on the frequency axis.
  1. Reduce the number of samples to 211. How do the sidelobes change with fewer samples?

  2. How do the sidelobes of the Bartlett window compare to the Block window? Does this match the prediction from theory? Hint: A Bartlett window can be considered as the convolution of two Block windows.

  3. Considering the eight windows shown, which window has the smallest sidelobes?

  4. What is the relevance of sidelobes in evaluating different window choices?

Choose a window w[n]:


Samples: Synthetic note:      

x[n] & Sxx(ω): Nx = 0 Block window
y[n] & Syy(ω): Ny = 0 Block window

Zoom (t): Zoom (f):