IIR Filter Design

Approximation of the Differentiation Operator

The approximation of the differentiation operator method defines an IIR filter by replacing the derivative ($s$ in the Laplace domain) by its discrete-time, z-transform approximation $(1/T_s)(1-z^{-1})$, where $T_s$ is the sampling period of the signal. Hence, this approach takes the desired Laplace transform transfer function $H_d(s)$ and maps $$s \rightarrow \frac{1}{T_s}\left( 1 - z^{-1} \right)$$ to get the Z-transform transfer function $H(z)$ approximation.

Example

Consider the ideal, desired Laplace transfer function for an integrator $$H_d(s) = \frac{1}{s} \; .$$ Use the approximation of the differentiation operator method to approximate the IIR filter coefficients $h[n]$ for the filter transfer function $H_d(s)$. Assume $T_s = 0.1$.

So the approximation Z-transform transfer function is $$H(z) = \frac{1}{(1/T_s)(1 - z^{-1})} = \frac{1}{10(1 - z^{-1})} \; .$$ In the magnitude response, the blue curve represents the desired response and the red curve represents the approximate response.

Impulse Invariance

The impulse invariance method creates a discrete-time impulse response that mirrors a continuous-time impulse response. This effectively recognizes that the partial fraction decomposition of a continuous-time signal $$H(s) = \sum_{m=1}^{M-1} \frac{A_m}{s-\alpha_m}$$ corresponds to a continuous-time impulse response $$h(t) = \sum_{m=1}^{M-1} A_m e^{-\alpha_m t} u(t)$$ for some amplitude $A_m$ and some decay term $\alpha_m$.

Hence, the corresponding Z-domain transfer function is $$ \begin{align*} H(z) &= T_s \sum_{m=1}^{M-1} \frac{A_m}{1-e^{\alpha_m T_s} z^{-1}} \; . \end{align*} $$ Note that the $T_s$ constant mulitplier is necessary to get the correct amplitude. The discrete-time impulse response then becomes $$ \begin{align*} h[n] &= T_s \sum_{m=1}^{M-1} A_m (e^{\alpha_m T_s})^n u[n] \\ &= T_s \sum_{m=1}^{M-1} A_m e^{\alpha_m T_s n} u[n] \end{align*} $$ where $t \approx T_s n$.

Hence, this approach effectively takes the desired Laplace transform poles $s^*$ and maps them into z-transform poles $z^*$ such that $$z^* = e^{s^* T_s}$$ to get the Z-transform transfer function $H(z)$ approximation. Note that the Laplace transform must be represented as a sum of single pole systems (i.e., a partial fraction decomposition) for this method to be applicable.

Example

Consider the ideal, desired Laplace transfer function for an integrator $$H_d(s) = \frac{1}{s} \; .$$ Use impulse invariance to approximate the IIR filter coefficients $h[n]$ for the filter transfer function $H_d(s)$. Assume $T_s = 0.1$.

The Laplace domain is $s^* = 0$. The corresponding z-domain pole is $z^* = e^{(0) T_s} = 1$. Hence, the z-domain transfer function is $$H(z) = T_s \frac{1}{1 - e^{(0) T_s} z^{-1}} = \frac{0.1}{1 - z^{-1}} \; .$$ In the magnitude response, the blue curve represents the desired response and the red curve represents the approximate response.

s

Bilinear Transform

The bilinear transform defines an IIR filter by replacing the derivative ($s$ in the Laplace domain) by a discrete-time, z-transform approximation. Hence, this approach takes the desired Laplace transform transfer function $H_d(s)$ and maps $$s \rightarrow \frac{2}{T_s} \frac{1 - z^{-1}}{1 + z^{-1}}$$ to get the Z-transform transfer function $H(z)$ approximation.

Example

Consider the ideal, desired Laplace transfer function for an integrator $$H_d(s) = \frac{1}{s} \; .$$ Use the bilinear transform to approximate the IIR filter coefficients $h[n]$ for the filter transfer function $H_d(s)$. Assume $T_s = 0.1$.

So the approximation Z-transform transfer function is $$H(z) = \frac{T_s}{2} \frac{1 + z^{-1}}{1 - z^{-1}} = \frac{1 + z^{-1}}{5(1 - z^{-1})} \; .$$ In the magnitude response, the blue curve represents the desired response and the red curve represents the approximate response.