We return to the situation studied in Laboratory Exercise 12.3 and ask if the use of windows other than the Block window can help us estimate the spectral components in a signal. In the following, you can hear the synthesized signal, view the (sampled) signal x[n] and its windowed version y[n] = w[n]x[n], 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. The   Linear / Logarithmic   button (below) switches the power spectral density amplitude display from linear to log₁₀(). This can help to pick out spectral details.

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 power spectral density do you observe when you use 2¹⁵ samples of the signal? How many spectral components do you think this signal has?


3. What power spectral density do you observe when you use 2¹² samples of the signal?


4. The HM (Half Maximum) criterion requires that the “dip” between the two peaks is at least—that is below—50% of the values at the peaks. With 2¹² samples, which windows do not meet the HM criterion?


5. How does this list change if there are 2¹¹ samples?

Until now, the tones we have heard are “pure”, uncorrupted by noise. In the following we will add independent, zero-mean, Gaussian noise with a certain value of σ to achieve a signal-to-noise ratio, SNR as defined in Equation 8.3. To answer the following questions, use a reasonable amount of zoom on the frequency axis.

6. Listen to the musical tone at each noise level so that you will know what signal you are dealing with and what you might expect in a spectral analysis.


7. What spectrum do you observe based upon 2¹⁵ samples with additive, independent Gaussian noise? Use a Block window and examine the spectral estimate for different values of the SNR


8. Can you distinguish between the two peaks (using the HM criterion) when SNR = 0.1?


9. If you now reduce the number of samples to 2¹¹ can you distinguish between the two peaks?


10. If you use 2¹² samples with which windows can you distinguish two peaks? Can you “reliably” estimate their positions? Assume that reliable means within 5%. Note that you are not being asked to estimate the amplitude of the spectral components but rather their positions.


11. How many samples (approximately) are required to distinguish two peaks and reliably estimate their positions with all of the windows?


12. With the number found above and SNR = 0.1, how would you describe the differences among the spectral estimates as you change the window shape?

Now choose SNR = 10 and perform the following experiments.

13. Can you distinguish between the two peaks with 2¹¹ samples?


14. If you use 2¹² samples with which windows can you distinguish two peaks? Can you “reliably” estimate their positions? Assume that reliable again means within 5%. Again, you are not being asked to estimate the amplitude of the spectral components but rather their positions.


15. How many samples (approximately) are required to distinguish two peaks and reliably estimate their positions with all of the windows?


16. What is your conclusion about the usefulness of the HM criterion?


17. Why are telescopes with large lenses necessary to resolve two proximate stars?