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Introduction

Module by: Anders Gjendemsjø

Summary: Introduction to sampling

Note: Your browser may not currently support MathML. See our browser support page for additional details. You can always view the correct math in the PDF version.

Why sample?

This section introduces sampling. Sampling is the necessary fundament for all digital signal processing and communication. Sampling can be defined as the process of measuring an analog signal at distinct points.

Digital representation of analog signals offers advantages in terms of

  • robustness towards noise, meaning we can send more bits/s
  • use of flexible processing equipment, in particular the computer
  • more reliable processing equipment
  • easier to adapt complex algorithms

Claude E. Shannon

Figure 1: Claude Elwood Shannon (1916-2001)
Figure 1 (Shannon_3.jpeg)

Claude Shannon has been called the father of information theory, mainly due to his landmark papers on the "Mathematical theory of communication". Harry Nyquist was the first to state the sampling theorem in 1928, but it was not proven until Shannon proved it 21 years later in the paper "Communications in the presence of noise".

Notation

In this chapter we will be using the following notation

  • Original analog signal xtxt
  • Sampling frequency FsFs
  • Sampling interval TsTs (Note that: Fs=1Ts Fs 1 Ts )
  • Sampled signal xsn xs n . (Note that xsn=xnTs xs n x n Ts )
  • Real angular frequency ΩΩ
  • Digital angular frequency ωω. (Note that: ω=ΩTs ω Ω Ts )

The Sampling Theorem

The Sampling theorem:

When sampling an analog signal the sampling frequency must be greater than twice the highest frequency component of the analog signal to be able to reconstruct the original signal from the sampled version.

Finished? Have at look at: • Proof; • Illustrations; • Matlab Example; • Aliasing applet; • Hold operation; • System view; • Exercises

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