The Simplest Lowpass Filter

This chapter introduces analysis of digital filters applied to a very simple example filter. The initial treatment uses only high-school level math (trigonometry), followed by an easier but more advanced approach using complex variables. Several important topics in digital signal processing are introduced in an extremely simple setting, and motivation is given for the study of further topics such as complex variables and Fourier analysis [84].

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Subsections

• Introduction
  • What is a Filter?
  • Why learn about filters?
  
  
• The Simplest Lowpass Filter
  • Definition of the Simplest Low-Pass
  
  
• Finding the Frequency Response
  • Sine-Wave Analysis
  • Mathematical Sine-Wave Analysis
  • Amplitude Response
  • Phase Response
  
  
• An Easier Way
  • Complex Sinusoids
  • Complex Amplitude
  • Phasor Notation
  • Complex Sinusoids as Circular Motion
  • Rederiving the Frequency Response
  
  
• Summary
• Elementary Filter Theory Problems

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.