Analog Filters

For our purposes, an analog filter is any filter which operates on continuous-time signals. In other respects, they are just like digital filters. In particular, LTI analog filters can be characterized by their (continuous) impulse response , where is time in seconds. Instead of a difference equation, analog filters may be described by a differential equation. Instead of using the z transform to compute the transfer function, we use the Laplace transform (introduced in Appendix D). Every aspect of the theory of digital filters has its counterpart in that of analog filters. In fact, one can think of analog filters as simply the limiting case of digital filters as the sampling-rate is allowed to go to infinity.

In the real world, analog filters are usually electrical models, or ``analogues'', of mechanical systems working in continuous time. If the physical system is linear and time-invariant (LTI) (e.g., consisting of elastic springs and masses which are constant over time), an LTI analog filter can be used to model it. Before the widespread use of digital computers, physical systems were simulated on so-called ``analog computers.'' An analog computer was much like an analog synthesizer providing modular building-blocks (such as ``integrators'') that could be patched together to build models of dynamic systems.

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Subsections

• Example Analog Filter
• Capacitors
  • Mechanical Equivalent of a Capacitor is a Spring
  
  
• Inductors
  • Mechanical Equivalent of an Inductor is a Mass
  
  
• RC Filter Analysis
  • Driving Point Impedance
  • Transfer Function
  • Impulse Response
  • The Continuous-Time Impulse
  • Poles and Zeros
  
  
• RLC Filter Analysis
  • Driving Point Impedance
  • Transfer Function
  • Poles and Zeros
  • Impulse Response
  
  
• Relating Pole Radius to Bandwidth
• Quality Factor (Q)
  • Decay Time is Q Periods
  • Q as Energy Stored over Energy Dissipated
  
  
• Analog Allpass Filters
  • Lossless Analog Filters

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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.