First order digital equalizers

Tunable digital equalizers are used to adjust gains at specific frequencies whiles maintaining the gains at all other frequencies. First order equalizers can be used to real time signal enhancement procedures to improve speech quality.

Consider a first order analog transfer function in Laplace domain:

$H(s) = \frac{s + \alpha \beta}{s + \alpha} = \frac{s}{s + \alpha} + \beta \frac{\alpha}{s + \alpha}$

The transfer function can be rewritten as:

$H(s) = \frac{1}{2} (1+\gamma(s)) + \frac{\beta}{2} (1-\gamma(s))$

where $\gamma(s) = \frac{s-\alpha}{s + \alpha}$ . Then, using the bi-linear transformation of $s = \frac{2}{T} \frac{z-1}{z+1}$ , the realized digital filter becomes:

$H(z) = \frac{1}{2} (1+\gamma(z)) + \frac{\beta}{2} (1-\gamma(z))$

$\gamma(z) = - \frac{z^{-1} -\hat{\alpha}}{1-\hat{\alpha} z^{-1}}$

where $\hat{\alpha} = \frac{\frac{2}{T}-\alpha}{\frac{2}{T}+\alpha}$ where $T$ is the integration step of the trapezoid rule. Figure 1 below illustrates the gains at different frequencies using $\alpha = 0.995$ on the left and the gains at different frequencies using $\beta = 0.2$ on the right.

Figure 1: Low frequency gain adjusted by varying $\beta$ with $\alpha = 0.995$ on the left and varying $\alpha$ with $\beta = 0.2$ on the right

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