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All-Pass Functions

All-Pass Functions

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The magnitude response of an all-pass (AP) filter is unity for all frequencies, thus all frequencies are passed without attenuation. The associated phase response, however, is useful for approximating a specified phase characteristic, and the AP group delay function is useful for approximating a specified delay characteristic. Moreover, if we indicate that a parameter α can be adjusted for a specific phase response, it is understood that α can likewise be adjusted for a specific group delay response. Approximating a linear phase by the AP phase corresponds to approximating a constant delay by the AP delay. Theoretically these approximations are not necessarily the same, but in practice the difference is often negligible. The phase (delay) properties of AP filters are so important for achieving a specified phase (delay) response of the electronic system that we devote an entire chapter to them, and we hope to consolidate much of the scattered information on this subject.

Chapter Contents:

  • 5.1 Applications of All-Pass Filters
  • 5.1.1 Delay Equalization
  • 5.1.2 Synthesis of Linear Delay for Pulse Compression
  • 5.1.3 Digital System Delay Equalization
  • 5.1.4 Phase-Splitting Networks
  • 5.1.5 Approximation Techniques
  • 5.2 All-Pass Transfer Function
  • 5.2.1 First-Order All-Pass Function
  • 5.2.2 Example of First-Order All-Pass Delay Calculation
  • 5.2.3 Butterworth Delay Equalization with a First-Order All-Pass Function
  • 5.2.4 Second-Order All-Pass Function
  • 5.2.5 Example of Second-Order All-Pass Delay Calculation
  • 5.2.6 Transient Responses of a Second-Order All-Pass System
  • 5.2.7 Butterworth Delay Equalization with a Second-Order All-Pass Function
  • 5.2.8 Effect of Delay Equalization on Transient Responses
  • 5.2.9 Transient Responses of Butterworth Filters with Linear Phase
  • 5.2.10 Narrow-Band Filter Equalization
  • 5.3 Least-Squares Approximation
  • 5.3.1 Numerical Integration
  • 5.3.2 Equalization of Filter Group Delay
  • 5.3.3 Starting Values for Computation
  • 5.3.4 Butterworth Delay Equalization with a Second-Order All-Pass Function
  • 5.4 Network Realizations
  • 5.4.1 First-Order Lattice Section
  • 5.4.2 Second-Order Lattice Section
  • 5.4.3 First-Order Bridge Networks
  • 5.4.4 Second-Order Bridge Networks
  • 5.4.5 Effect of Losses in a Second-Order Network
  • 5.4.6 Example of Bandpass Delay Equalization
  • References
  • Problems

Inspec keywords: all-pass filters; delay filters; approximation theory

Other keywords: constant delay approximation; all-pass filter; specified phase response; linear phase approximation; AP group delay function; magnitude response; specific group delay response; delay characteristic

Subjects: Interpolation and function approximation (numerical analysis); Filtering methods in signal processing; Interpolation and function approximation (numerical analysis); Signal processing theory

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