2.2 Dsp00108-averaging time series

Revised: Fri Oct 16 23:12:34 CDT 2015

This page is included in the following books:

• Digital Signal Processing - DSP

Table of contents

• Preface
  • Some of what you have previously learned
  • Viewing tip
    • Figures
• Preview
• Discussion
  • Which screw to turn...
    • The mechanic and the screw
    • A very important module
  • Turning the screws in DSP
    • Computing the average value of a time series
    • Decomposition of time series
    • A sum of products of sinusoids
    • Many sinusoidal products
    • Examples of products of sinusoids
    • More examples of the products of sinusoids
    • What is the average value of a sinusoid?
    • A short recap before continuing
  • Spectral analysis
    • Several steps are involved
    • The Fourier transform
    • Let's see some data
    • A spectral line
    • Two spectral lines
    • Zero-valued points in the spectra
    • The frequency resolution of the Fourier transform
• Summary
• Miscellaneous

Preface

This is one in a series of modules designed to teach you about Digital Signal Processing (DSP) using Java. The purpose of the miniseries is topresent the concepts of DSP in a way that can be understood by persons having no prior DSP experience. However, some experience in Java programmingwould be useful. Whenever it is necessary for me to write a program to illustrate a point, I will write it in Java.

Some of what you have previously learned

In a previous module, I explained the meaning of sampling, and discussed some of the problems that occur as a result of high-frequency componentsin the analog signal.

Measure and record the signal amplitude

I told you that to sample an analog signal means to measure and record its amplitude at a series of points in time. The values that you record constitute a sampled time series intended to represent the analog signal.

Avoiding frequency folding

I told you that to avoid problems, the sampling frequency must be a least twice as great as the highest frequency component contained in the analogsignal, and as a practical matter, should probably be somewhat higher.

Sinusoids, frequency, and period

I introduced you to sinusoids, taught you about sine and cosine functions, and introduced the concepts of period and frequency for sinusoids.

Decomposition of time series

I told you that almost everything we will discuss in this series on DSP is based on the premise that every time series can be decomposed into a largenumber of sinusoids, each having its own amplitude and frequency.

The notion of DSP

I told you that DSP is based on the notion that signals in nature can be sampled and converted into a series of numbers. The numbers can be fed into somesort of digital device, which can process the numbers to achieve some desired objective.

Viewing tip

I recommend that you open another copy of this module in a separate browser window and use the following links to easily find and view the Figureswhile you are reading about them.

Figures

• Figure 1 . Products of sinusoids.
• Figure 2 . Products of sinusoids.
• Figure 3 . More products of sinusoids.
• Figure 4 . Five Sampled Sinusoids.
• Figure 5 . Computed average value of a time series.
• Figure 6 . Expanded average value of a time series.
• Figure 7 . Computed average value of a time series.
• Figure 8 . Computed average value of a time series.
• Figure 9 . Computed average value of a time series.
• Figure 10 . Spectra of five different sinusoids of different lengths.
• Figure 11 . Spectra of five different sinusoids of different lengths.
• Figure 12 . Spectra of five different time series of different lengths.
• Figure 13 . Spectra of five different time series of different lengths
• Figure 14 . Average values of sinusoid products.
• Figure 15 . Illustration of frequency resolution.
• Figure 16 . Illustration of frequency resolution.