Bilinear Transform IIR Filter Design and Frequency Warping

1. The Bilinear Transform: Solving the Aliasing Problem

The fundamental issue with impulse invariance was its underlying s-to-z mapping, $z=e^{s{T}_s}$. This mapping takes infinitely many horizontal strips of the s-plane and maps them to the same unit circle in the z-plane (a many-to-one mapping), causing aliasing.

The bilinear transform proposes a different mapping. It is a one-to-one transformation that maps the entire left half of the s-plane uniquely into the interior of the unit circle in the z-plane. This single property is what makes it so powerful: by avoiding the many-to-one mapping, it completely eliminates aliasing.

The bilinear transform is defined by the following substitution:

where $T_s$ is the sampling period.

To see how we arrive at this specific formula, we will explore its mathematical origins. You've expressed an interest in deeper mathematical derivations, and the origin of the bilinear transform is a clever piece of numerical approximation theory that connects continuous and discrete-time systems.

As the video shows, the bilinear transform isn't just an arbitrary algebraic trick. It arises from approximating the fundamental calculus operation of integration using a simple numerical method (the trapezoidal rule) and then finding the transfer function of that discrete-time system. This connection between differential/integral equations and difference equations is a cornerstone of digital signal processing.

2. The Catch: Frequency Warping

The bilinear transform elegantly solves aliasing, but this comes at a price. The mapping of the continuous frequency axis ($s=j\Omega$) to the discrete frequency axis ($z=e^{j\omega }$) is highly non-linear. This phenomenon is called frequency warping.

Let's derive this relationship formally. We start with the bilinear transform equation and substitute $s=j\Omega$ and $z=e^{j\omega }$:

We can simplify the right-hand side using Euler's identities:

Canceling $j$ from both sides gives the crucial frequency warping relationship:

This equation tells us that the analog frequency $\Omega$ is not linearly proportional to the digital frequency $\omega$. Instead, they are related by a tangent function. The entire infinite analog frequency axis ($-\infty <\Omega <\infty$) is compressed to fit into the finite digital frequency range ($-\pi /T_s<\omega <\pi /T_s$).

The following video provides an excellent visual and intuitive explanation of why this warping occurs, contrasting it with the mapping from the impulse invariance method.

The key consequence of frequency warping is that if we take an analog filter and blindly apply the bilinear transform, the critical frequencies (like the cutoff frequency) of the resulting digital filter will be in the wrong place.

3. The Solution: Pre-Warping in Filter Design

To design a digital filter that meets a specific set of frequency specifications (e.g., a low-pass filter with a cutoff at ${\omega }_c$), we must compensate for the frequency warping effect. The technique used is called pre-warping.

The logic is simple: if we know our target digital frequency ${\omega }_c$ will be warped by the transformation, we can pre-distort the analog filter's cutoff frequency ${\Omega }_c$ so that after warping, it lands exactly at our desired ${\omega }_c$.

We find the required analog frequency by inverting the warping formula:

This pre-warping step is an essential part of the design process.

4. Complete Design Procedure for IIR Filters

We can now outline the complete procedure for designing an IIR filter using the bilinear transform.

Let's summarize the design flow based on the resource:

1. Start with Digital Specifications: Define the desired properties of your digital filter, such as passband edge frequency (${\omega }_p$), stopband edge frequency (${\omega }_s$), passband ripple, and stopband attenuation.
2. Pre-warp Frequencies: Use the pre-warping formula to convert your critical digital frequencies (${\omega }_p,{\omega }_s$) into the corresponding analog frequencies (${\Omega }_p,{\Omega }_s$) that the analog prototype must have.
3. Design the Analog Prototype Filter: Using the pre-warped analog specifications (${\Omega }_p,{\Omega }_s$), design a suitable continuous-time filter (e.g., Butterworth, Chebyshev). This involves calculating the required filter order $N$ and the exact analog cutoff frequency ${\Omega }_c$.
4. Find the Analog Transfer Function $H_c(s)$: Based on the order and cutoff frequency, determine the transfer function of the stable and causal analog filter.
5. Apply the Bilinear Transform: Substitute $s=\frac{2}{T_s}\frac{1-z^{-1}}{1+z^{-1}}$ into your analog transfer function $H_c(s)$ and simplify the expression to obtain the final digital filter transfer function, $H(z)$.

To see how pre-warping visually corrects the filter, the animation in the Brian Douglas video is invaluable.

Conclusion

Today we have explored the bilinear transform, the workhorse method for designing IIR filters. You have seen that it is a powerful technique motivated by the need to solve the aliasing problem inherent in the simpler impulse invariance method.

Key Takeaways:

• Principle: The bilinear transform is an algebraic substitution $s=\frac{2}{T_s}\frac{1-z^{-1}}{1+z^{-1}}$ that maps an analog transfer function $H_c(s)$ to a digital one $H(z)$.
• No Aliasing: It provides a one-to-one mapping from the s-plane's imaginary axis to the z-plane's unit circle, which completely eliminates aliasing.
• Frequency Warping: This mapping is non-linear, described by $\Omega =\frac{2}{T_s}\tan (\frac{\omega }{2})$. This "warps" the frequency axis, misaligning critical frequencies if uncorrected.
• Pre-Warping: To achieve an accurate design, one must pre-warp the digital frequency specifications into the analog domain before designing the analog prototype filter. This ensures the critical frequencies of the final digital filter land in the correct places.

Preview of the Next Lesson:

Now that you have learned the fundamental design process, including the crucial step of pre-warping, our next lesson will build upon this. We will focus more deeply on the application of pre-warping and explore how to use it in conjunction with frequency transformations. This will allow us to take a standard low-pass analog prototype, apply pre-warping, and then transform it into other filter types like high-pass, band-pass, and band-stop digital filters, giving us a complete toolkit for IIR filter design.