Java1478-fun with java, how and why spectral analysis works  (Page 4/9)

(The curve in Figure 4 can also be considered to consist of the sum of two parts. However, in Figure 4 , the bias value is zero.)

The net area

The net area contributed by the periodic part of the curve in Figure 5 continues to be zero. In effect, the non-zero net area under the curve in Figure 5 is the amount of the positive bias multiplied by the number of points. In other words, because I captured an even number of cycles of the periodic portionof the curve in my calculation of net area, only the bias contributes a non-zero value to the net area under the total curve.

Part of a cycle

Had I failed to capture an even number of cycles of the periodic portion of the curve, then the positive lobes might not completely cancel the negativelobes, and the net area under the curve would be influenced by the periodic portion of the curve in addition to the bias.

Why is this important?

These concepts are extremely important in this module as we learn how to do frequency spectral analysis.

As you will see in this module, in doing frequency spectral analysis, we will form a product between a target time series and a sinusoid. The purpose is tomeasure the power contained in the time series at the frequency of the sinusoid.

The product of the target time series and the sinusoid will produce the sum of a potentially infinite number of periodic functions, some of which may beriding on a positive or negative bias. We will measure the amount of bias by computing the average value of the product time series. The amount of bias willbe our estimate of the power contained in the target time series at the frequency of the sinusoid.

The Fourier transform

There is a mathematical process, known as the Fourier transform, which can be used to linearly transform information back and forth between two differentdomains. The information can be represented by sets of complex numbers in either or both domains.

The domains can represent a variety of different things. In DSP, the domains are often referred to as the time domain and the frequency domain , but that is not a requirement. For example, one of the domains could representthe samples that make up the pixels in a photograph and the other domain could represent something having to do with the manipulation of photographs.

Usually when dealing with the time domain and the frequency domain, the values that make up the samples in the time domain are purely real while thevalues that make up the samples in the frequency domain are complex containing both real and imaginary parts.

Time domain and frequency domain

For some purposes, it is preferable to have your information in the time domain. For other purposes, it is preferable to have your information in thefrequency domain. The Fourier transform allows you to transform your information back and forth between these two domains at will.

For example, it is possible to transform information from the time domain into the frequency domain, modify that information in the frequency domain, andthen transform the modified information back into the time domain. This is one way to accomplish frequency filtering.