First Order Shelf Filter Design

The transfer function for the first-order shelf filter is given by

$\displaystyle \sigma(z;f_t,g_{\pi}) = \frac{b_0 + b_1 z^{-1}}{1 + a_1 z^{-1}}.$

The coefficients are given by

$\displaystyle b_0 = \frac{\beta_0 + \rho \beta_1}{1 + \rho \alpha_1}, \quad b_1... ..._0}{1 + \rho \alpha_1}, \quad a_1 = \frac{\rho + \alpha_1}{1 + \rho \alpha_1},$

where

$\displaystyle \beta_0$	$\displaystyle =$	$\displaystyle \frac{(1 + g_{\pi})+ (1 - g_{\pi})\alpha_1}{2}$
$\displaystyle \beta_1$	$\displaystyle =$	$\displaystyle \frac{(1 - g_{\pi})+ (1 + g_{\pi})\alpha_1}{2},$

and

$\begin{displaymath} \alpha_1 = \left\{ \begin{array}{ll} 0, & \quad g_{\pi} = 1 ... ...\frac{1}{2}}}, & \quad g_{\pi} \neq 1. \\ \end{array}\right. \end{displaymath}$

To form the coefficients of a prototype shelf filter with a transition frequency of $\pi/2$

$\displaystyle \eta = \frac{(g_{\pi} + 1)}{(g_{\pi} - 1)}, \quad g_{\pi} \neq 1.$

The parameter

$\displaystyle \rho = \frac{\sin(\pi f_t/2 - \pi/4)}{\sin(\pi f_t/2 + \pi/4)}$

is the coefficient of the first-order allpass transformation warping the prototype filter to the proper transition frequency.

An efficient way to model the frequency response is to use a cascade of minimum-phase first-order shelf filters $\sigma_i(z)$ ,

$\displaystyle \hat{\lambda}(z) = \prod_{i = 1}^{N} \sigma_i(z;{f_t}(i),g_{\pi}(i)).$

Unlike with other methods, the shelf filter coefficients are easily computed in real time.

Since the shelf filters are first order, they have real poles and zeros, and are relatively free of artifacts when made time varying.

In designing the prototype filters for different orders it was discovered that a cascade of shelf filters with band-edge gains (in units of amplitude) and transition frequencies (in radians $/\pi$ ) given by

$\displaystyle g_{\pi}(i)$	$\displaystyle =$	$\displaystyle \exp\left\{ \frac{[(i-\frac{1}{2})/N]^{\frac{1}{2}}}{\sum_{k=1}^N... ...\frac{1}{2})/N]^{\frac{1}{2}}} \cdot \ln \vert\lambda(2\pi f_s/2)\vert\right\},$
$\displaystyle f_t(i)$	$\displaystyle =$	$\displaystyle [(i-\frac{1}{2}) / N]^3,$

has a transfer function which is an excellent approximation to $\lambda (\omega )$ for a wide range of filter orders

, tube lengths

, tube radii

, and sampling rates