Frequency-Response Matching Using Digital Filter Design Methods

Given force inputs and velocity outputs, the frequency response of an ideal mass was given in Eq.(7.1.2) as

and the frequency response for a spring was given by Eq.(7.1.3) as

Thus, an ideal mass is an integrator and an ideal spring is a differentiator. The modeling problem for masses and springs can thus be posed as a problem in digital filter design given the above desired frequency responses. More generally, the admittance frequency response ``seen'' at the port of a general th-order LTI system is, from Eq.(8.3),

where we assume . Similarly, point-to-point ``trans-admittances'' can be defined as the velocity Laplace transform at one point on the physical object divided by the driving-force Laplace transform at some other point. There is also of course no requirement to always use driving force and observed velocity as the physical variables; velocity-to-force, force-to-force, velocity-to-velocity, force-to-acceleration, etc., can all be used to define transfer functions from one point to another in the system. For simplicity, however, we will prefer admittance transfer functions here.

---

Subsections

• Ideal Differentiator (Spring Admittance)
• Digital Filter Design Overview
• Digital Differentiator Design
• Fitting Filters to Measured Amplitude Responses
  • Measured Amplitude Response
  • Desired Impulse Response
  • Converting the Desired Amplitude Response to Minimum Phase
  
  
• Further Reading on Digital Filter Design

---

Previous: Delay Loop Expansion
Next: Ideal Differentiator (Spring Admittance)

Order a Hardcopy of Physical Audio Signal Processing

---

About the Author: 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.