Time-domain vs. frequency-domain
The top three traces represent information in the time domain. The bottom two traces represent information in the frequency domain.
(Think of the frequency domain as the information that is visible on many audio systems, consisting of parallel vertical bars with lights thatdance up and down. These lights are often associated with a device referred to as a frequency equalizer. When the music contains a lot of drums, orother sounds at the bass end, the lights at the low (usually left) end of the frequency spectrum are very active. When the music contains a lot ofsymbols, or sounds at the treble end, the lights at the high (right) end of the frequency spectrum are very active. That is a form of real-time spectrumanalysis.)
Frequency spectrum analysis
The two bottom traces in Figure 3 result from performing frequency spectrum analysis on the top trace and the middle trace respectively.
The white noise spectrum
The trace in the gray area immediately below the center is an estimate of the spectral distribution of the white noise in the top trace. The spectrum analysiswas performed across the frequency range from zero frequency to the sampling frequency.
While not perfectly flat, as would be the case for perfectly white noise, you can see that the energy appears to be distributed across that entire range.
(If we wanted to improve our estimate, we could capture and analyze a much longer sample of the white noise.)
If you examine this trace carefully, you might notice that there is a point of near symmetry in the middle. The values that you see above that point (the folding frequency) are a mirror-image of the values that you see below that point. (I will have more to say about this later.)
The filtered noise spectrum
The bottom trace shows an estimate of the spectral distribution of the filtered noise in the center trace. Again, the spectrum analysis was performedacross the frequency range from zero frequency to the sampling frequency. Again also, there is a symmetry point in the middle with everything to the right ofthat point being a mirror image of everything to the left of that point.
Two spectral peaks are visible
Unlike the spectral analysis of the white noise, this spectral analysis shows two obvious peaks. One peak appears at one-fourth the sampling frequency, and the other peak appears at three-fourths the sampling frequency.
In other words, as we concluded from examining the center trace, the filtering process removed much of the energy above and below the designfrequency of the convolution filter.
(By changing the design frequency of the convolution filter, and repeating the process, we could move this peak up or down along thefrequency axis.)
What does the symmetry mean?
Without getting into a lot of detail at this point, the point of symmetry that I identified above is known as the Nyquist folding frequency. (See the earlier module titled Dsp00104-Sampled Time Series .)
In order to be able to identify the frequency of a sine wave, you must have at least two samples per cycle of the sine wave. The Nyquist folding frequencyis the frequency at which you have exactly two samples per cycle.