Java1478-fun with java, how and why spectral analysis works

Revised: Fri Oct 16 23:16:39 CDT 2015

This page is included in the following books:

• Digital Signal Processing - DSP

Table of contents

• Table of contents
• Preface
  • Viewing tip
    • Figures
• Multiply-add operations
  • Computational requirements for DSP
    • Typical notation
    • A sum-of-products operation
    • An alternative notation
• Preview
  • What is a time series?
    • A new time series is produced
    • Calculation of the mean or average
• The Fourier transform
  • Time domain and frequency domain
    • Example of time domain and frequency domain
    • Forward and inverse transforms
    • Sampled time series
    • Integration and summation
    • The FFT algorithms
    • DFT versus FFT
    • The DFT algorithm
    • Why does this work?
    • Products of sine and cosine functions
    • Product of two sine functions having the same frequency
    • Product of two cosine functions having the same frequency
    • Product of a sine function and a cosine function
    • Neither sine nor cosine
    • What about non-matching frequency components?
    • Sum and difference frequencies
    • A form of measurement error
    • Product of two sine functions at different frequencies
• Summary
• What's next?
• Miscellaneous

Preface

Programming in Java doesn't have to be dull and boring. In fact, it's possible to have a lot of fun while programming in Java. This module is onein a series that concentrates on having fun while programming in Java.

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. A typical sum-of-products operation.
• Figure 2. Alternative notation for a sum-of-products operation.
• Figure 3. Plot of values in a time series.
• Figure 4. Area under a periodic curve.
• Figure 5. Area under a periodic curve with an offset.
• Figure 6. Forward Fourier transform.
• Figure 7. Three trigonometric identities.
• Figure 8. Products of sine and cosine functions.
• Figure 9. Rewrite and simplify.
• Figure 10. Plot of sin(x) and sin(x)*sin(x).
• Figure 11. Plot of cos(x) and cos(x)*cos(x).
• Figure 12. Plot of sin(x), cos(x), and sin(x)*cos(x).
• Figure 13. Products of sine and cosine functions.
• Figure 14. Plot of sin(1.8x)*sin(2.2x).
• Figure 15. Plot of cos(1.8x)*cos(2.2x).
• Figure 16. Plot of sin(1.8x)*cos(2.2x).

Multiply-add operations

This module deals with a topic commonly know as Digital Signal Processing, (DSP for short) .

Computational requirements for DSP

The computational requirements for implementing DSP in a computer program are usually straightforward. Almost all DSP operations consist of multiplyingcorresponding values contained in two numeric series and then calculating the sum of the products. Sometimes, the final sum is divided by the total number ofvalues included in the sum to produce an average. This is often referred to as a sum-of-products or multiply-add operation .

(This is the digital equivalent of integrating the product of two continuous functions between specified limits.)

Typical notation

Such an operation can be indicated by the symbolic notation shown in Figure 1 (where the strange looking thing constructed of straight lines is the Greek letter sigma) .