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Embedded SystemsFPGAElectronics

Introduction to Digital Filters
    Elementary Audio Digital Filters
       Elementary Filter Sections
          The BiQuad Section

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The BiQuad Section

The term ``biquad'' is short for ``bi-quadratic'', and is a common name for a two-pole, two-zero digital filter. The transfer function of the biquad can be defined as

$\displaystyle H(z) = g \frac{1 + \beta_1 z^{-1}+ \beta_2 z^{-2}}{1 + a_1 z^{-1}+ a_2 z^{-2}} \protect$ (B.8)

where $ g$ can be called the overall gain of the biquad. Since both the numerator and denominator of this transfer function are quadratic polynomials in $ z^{-1}$ (or $ z$), the transfer function is said to be ``bi-quadratic'' in $ z^{-1}$ (or $ z$).

As derived in §B.1.3, for real second-order polynomials having complex roots, it is often convenient to express the polynomial coefficients in terms of the radius $ R$ and angle $ \theta$ of the positive-frequency pole. For example, denoting the denominator polynomial by $ A(z)=1 + a_1 z^{-1}+ a_2 z^{-2}$, we have

$\displaystyle A(z) = \left(1 - Re^{j\theta}z^{-1}\right)\left(1 - Re^{-j\theta}z^{-1}\right) = 1 - 2R\cos(\theta)z^{-1}+ R^2z^{-2}. $

This representation is most often used for the denominator of the biquad, and we think of $ \theta$ as the resonance frequency (in radians per sample-- $ \theta=2\pi f_c T$, where $ f_c$ is the resonance frequency in Hz), and $ R$ determines the ``Q'' of the resonance (see §B.1.3). The numerator is less often represented in this way, but when it is, we may think of the zero-angle as the antiresonance frequency, and the zero-radius affects the depth and width of the antiresonance (or notch).

As discussed on page [*], a common setting for the zeros when making a resonator is to place one at $ z = 1$ (dc) and the other at $ z = -1$ (half the sampling rate), i.e., $ \beta_1=0$ and $ \beta_2=-1$ in Eq.$ \,$(B.8) above $ \Rightarrow B(z) = 1-z^{-2}= (1-z^{-1})(1+z^{-1})$. This zero placement normalizes the peak gain of the resonator if it is swept using the $ a_1$ parameter.

Using the shift theorem for z transforms, the difference equation for the biquad can be written by inspection of the transfer function as

\begin{eqnarray*} v(n) &=& g\, x(n) \\ y(n) &=& v(n) + \beta_1 v(n-1) + \beta_2 v(n-2) \\ & & \qquad - a_1 y(n-1) - a_2 y(n-2) . \end{eqnarray*}

where $ x(n)$ denotes the input signal sample at time $ n$, and $ y(n)$ is the output signal. This is the form that is typically implemented in software. It is essentially the direct-form I implementation. (To obtain the official direct-form I structure, the overall gain $ g$ must be not be pulled out separately, resulting in feedforward coefficients $ [g,g\beta_1,g\beta_2]$ instead. See Chapter 9 for more about filter implementation forms.)

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Previous: Two-Pole Partial Fraction Expansion
Next: Biquad Software Implementations
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.


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