2.1 Dsp00104-sampled time series

Revised: Fri Oct 16 23:11:24 CDT 2015

This page is included in the following books:

• Digital Signal Processing - DSP

Table of contents

• Preface
  • Viewing tip
    • Figures
• Preview
• Discussion
  • Sinusoids, time series, composition, and decomposition
  • The notion of sampling analog signals
    • What is meant by sampling?
    • The Nyquist folding frequency
  • A few comments about sampling
    • What do we really have?
    • Sampled sinusoids
    • Estimating the values in between the samples
    • A more common representation
    • The most common representation
  • What happens when the sampling frequency is reduced?
    • Some numbers
    • Comparison of frequencies with center frequency
    • Reduce the sampling frequency
    • A different approach
    • Introduce a sampling problem
    • Back to the case with no problems
    • Back to the case with the sampling problem
  • The bottom line
    • Using an analog low-pass pre-filter
    • Digital re-sampling
• 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 personshaving no prior DSP experience. However, some experience in Java programming would be useful. Whenever it is necessary for me to write a program toillustrate a point, I will write it 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 . Samples from five different sinusoids.
• Figure 2 . Rectangular representations of samples from five sinusoids.
• Figure 3 . Trapezoidal representations of samples from five sinusoids.
• Figure 4 . Most common representations of samples from five sinusoids.
• Figure 5 . Five sampled sinusoids
• Figure 6 . Result of re-sampling the five sinusoids.
• Figure 7 . Spectral analyses of five sinusoids with no sampling problems.
• Figure 8 . Spectral analyses of five sinusoids with sampling problem.
• Figure 9 . Spectral analyses of five sinusoids with no sampling problems.
• Figure 10 . Spectral analyses of five sinusoids with sampling problem

Preview

I will explain the meaning of sampling, and will explain some of the problems that arise when sampling and processing analog signals. Those problems generallyrelate to the relationship between the sampling frequency and the high-frequency components contained in the analog signal.

I will explain the concept of the Nyquist folding frequency , which is half the sampling frequency (more commonly called the sampling rate) .

I will illustrate the frequency folding phenomena by plotting sampled time series data as well as spectral data.

Discussion

Sinusoids, time series, composition, and decomposition

I introduced you to sinusoids and sampled time series in an earlier module . I taught you about sine and cosine functions, and the Java methods used to calculate their values. I also introduced the concepts of period and frequency for sinusoids.