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Hopefully, by this point, you understand how multiplying two time series produces a new time series composed of the sum of all the products of theindividual sinusoids in the two original time series.
When each pair of sinusoids is multiplied together, they produce a new time series consisting of two other sinusoids whose frequencies are the sum anddifference of the original pair of frequencies.
When an average is computed for a fixed number of points on the new time series, the error in the average tends to be greater for cases where theoriginal frequency values were close together. This is because the period of one of the new sinusoids becomes longer as the original frequencies become closer.In general, the longer the period of the sinusoid, the more points are requiredto get a good estimate of its average value.
There are many operations in DSP where this matters a lot. As mentioned earlier, the computational requirements for DSP frequently boil down to nothingmore than multiplying a pair of time series and computing the average of the product. You will see many examples of this as you continue studying the modulesin this series of tutorials on DSP.
I am going to illustrate my point by showing you one such example in this module. This example will use a Fourier transform in an attempt to performspectral analysis and to separate two closely-spaced frequency components in a time series. As you will see, errors in the computed average can interfere withthis process in a significant way.
(This example will illustrate and explain the results using graphs. Future modules will provide more technical details on the DSP operationsinvolved.)
I will provide this illustration in several steps.
First, I will show you spectral data for several time series, each consisting of a single sinusoid. The time series will have different lengths but theindividual sinusoids will have the same frequency. This will serve as baseline data for the experiments that follow.
Then I will show you spectral data for several time series, each composed of the sum of two sinusoids. These time series will have different lengths. Thesinusoids in each time series will have the same frequencies. I will show you two cases that fall under this description. The frequency difference for the twosinusoids in each time series will be small in one case, and greater in another case.
Finally, I will show you spectral data for several time series, each composed of the sum of two sinusoids. These time series will be different lengths, andthe sinusoids in each time series will have different frequencies. In particular, the frequency difference between the two sinusoids in each timeseries will be equal to the theoretical frequency resolution for a time series of that particular length.
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