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Suppose, for example, that we have two time series, each of which is composed of two sinusoidal components as follows:
f(x) = cos(ax) + cos (bx)
g(x) = cos(cx) + cos(dx)
The product of the two time series is given by:
h(x) = f(x)*g(x)
= (cos(ax) + cos (bx)) * (cos(cx) + cos(dx))
where the asterisk (*) means multiplication.
Multiplying this out produces the following:
h(x) = cos(ax)*cos(cx)
+ cos(ax)*cos(dx)+ cos(bx)*cos(cx)
+ cos(bx)*cos(dx)
Thus, the time series produced by multiplying any two time series consists of the sum of a (potentially large) number of terms, each of which is the product of two sinusoids.
We probably need to learn a little about the product of two sinusoids. I will discuss this topic with a little more mathematical rigor in a future module. Inthis module, however, I will simply illustrate the topic using graphs.
Important: The product of two sinusoids is always a new time series, which is the sum of two new sinusoids.
The frequencies of the new sinusoids are different from the frequencies of the original sinusoids. Furthermore, the frequency of one of the new sinusoidsmay be zero.
As a practical matter, a sinusoid with zero frequency is simply a constant value. It plots as a horizontal straight lineversus time.
Think of it this way. As the frequency of the sinusoid approaches zero, the period, (which is the reciprocal of frequency), approaches infinity. Thus, the width of the first lobe of the sinusoid widens, causing every value in thatlobe to be the same as the first value.
This will become a very important concept as we pursue DSP operations.
More specifically, when you multiply two sinusoids, the frequency of one of the sinusoids in the new time series is the sum of the frequencies of the two sinusoids that were multiplied together. The frequency of the othersinusoid in the new time series is the difference between the frequencies of the two sinusoids that were multiplied together.
For the special case where the two original sinusoids have the same frequency, the difference frequency is zero and one of the sinusoids in the newtime series has a frequency of zero. It is this special case that makes digital filtering and digital spectrum analysis possible.
When we multiply two time series and compute the average of the resulting time series, we are in effect computing the average of the products of all theindividual sinusoidal components contained in the two time series. That is, the new time series contains the products of (potentially many) individual sinusoids contained in the two original time series. In the end, it all comesdown to computing the average value of products of sinusoids.
The product of any pair of sinusoids that have the same frequency will produce a time series containing the sum of two sinusoids. One of the sinusoidswill have a frequency of zero (hence it will have a constant value). The other sinusoid will have a frequency that is double the frequency of theoriginal sinusoids.
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