2.2 Dsp00108-averaging time series  (Page 2/14)

Preview

This is a broad-ranging module. It begins with a discussion of averaging time series, ends with a discussion of spectral resolution, and covers severalrelated topics in between. Don't be alarmed, however, at the range of the module. The topics of time-series averaging and spectral resolution are verystrongly related.

I will discuss why we frequently need to average sampled time series, and explain some of the issues involved in that process.

I will also show you the impact of those averaging issues on DSP, using spectrum analysis as an example.

Discussion

It never ceases to amaze me how something as mathematically complex as DSP can be distilled down to the simplest of computational processes.

Which screw to turn ...

The mechanic and the screw

DSP reminds me of the old story about the customer who complained about the bill at the auto repair shop being too high. According to the customer, all themechanic did to fix the problem was turn one screw, and the bill was too high for the labor involved. The mechanic responded that he didn't charge for turningthe screw. Instead, he charged for knowing which screw to turn, and knowing which way and how far to turn it.

A very important module

This module, in conjunction with the earlier module titled Sampled Time Series may be the most important module in the entire collection because it provides a practical pseudo-mathematical framework for almost everythingthat follows.

Almost everything that you will do using DSP involves:

1. Multiplying the sample values in one set of samples by the corresponding sample values in a second set of samples.
2. Computing the average of the set of multiplication products.
3. Interpreting the value of the average relative to the task at hand.

Once you understand the ramifications of the "multiply and average" process, the solution to many DSP problems simply involves figuring out how to index yourway through the respective sample sets in order to apply the arithmetic appropriately. This is true for convolution, correlation, spectrum analysis,adaptive processing and many other forms of DSP as well.

Turning the screws in DSP

Knowing how to turn the screw is not the complicated part of DSP. Rather, the complicated part of DSP lies in knowing which screw to turn and which way toturn it. Once you know that, you will be surprised just how easy it is to actually turn the screw.

Computing the average value of a time series

As you will learn in this series of modules, a large majority of DSP operations consist simply of the following two steps:

1. Multiply one time series by another time series, to produce a third time series.
2. Compute the average value of the third time series.

In many cases, it is the average value of the third time series that provides the answer you are seeking.

The challenge is in knowing what the average value means, and how to interpret it.

Decomposition of time series

Almost everything that we will discuss in this series on DSP is based on the premise that every time series can be decomposed into a (potentially large) number of sinusoids, each having its own amplitude and frequency.