Frequency Transformations for Filter Design

1. The Core Idea: Mapping the Complex Plane

At its heart, a frequency transformation is a mapping from one complex frequency plane to another. Let's say we have the transfer function of our normalized lowpass prototype, $H_{LP}(S)$, where $S=\sigma +j{\Omega }_{LP}$ is the complex frequency variable for the prototype. We want to find the transfer function of our target filter, $H_{target}(s)$, where $s={\sigma }^{\prime }+j\Omega$ is the complex frequency variable for the target (e.g., highpass) filter.

The entire method rests on finding a suitable transformation function $S=g(s)$ that maps the frequency response characteristics of the prototype onto the desired characteristics of the target filter.

The design workflow is as follows:

1. Begin with the specifications for your target filter (e.g., a highpass filter with cutoff ${\Omega }_c$).
2. Use the transformation to map these target specifications onto an equivalent set of specifications for a normalized lowpass prototype.
3. Design the lowpass prototype $H_{LP}(S)$ to meet these new specifications (using the methods for calculating order and pole locations you've already learned).
4. Obtain the final transfer function by substituting the transformation $S=g(s)$ into your prototype's transfer function: $H_{target}(s)=H_{LP}(g(s))$.

This process allows us to reuse all our hard work in deriving lowpass filters to create any other frequency-selective type. Let's examine the specific transformations.

2. Lowpass-to-Highpass Transformation

To convert a lowpass filter into a highpass filter, we need a transformation that maps low frequencies to high frequencies and vice versa. This is achieved with a frequency inversion.

The transformation is:
$S=\frac{{\Omega }_p}{s}$
where:

• $S$ is the complex frequency variable of the lowpass prototype.
• $s$ is the complex frequency variable of the target highpass filter.
• ${\Omega }_p$ is the desired passband edge (cutoff frequency) of the highpass filter.

Let's analyze the effect of this transformation on the imaginary axis by setting $S=j{\Omega }_{LP}$ and $s=j\Omega$:
$j{\Omega }_{LP}=\frac{{\Omega }_p}{j\Omega }=-j\frac{{\Omega }_p}{\Omega }⟹{\Omega }_{LP}=-\frac{{\Omega }_p}{\Omega }$

This equation reveals the mapping:

• The DC point of the prototype (${\Omega }_{LP}=0$) is mapped to infinity in the highpass domain ($\Omega \to \infty$).
• The cutoff frequency of the prototype (${\Omega }_{LP}=\pm 1$) is mapped to the highpass cutoff ($\Omega =\mp {\Omega }_p$).
• Frequencies near zero in the highpass domain ($\Omega \to 0$) are mapped to infinity in the prototype domain (${\Omega }_{LP}\to \mp \infty$).

This effectively swaps the passband and stopband. The passband of the prototype, $|{\Omega }_{LP}|\le 1$, is mapped to the passband of the highpass filter, $|\Omega |\ge {\Omega }_p$.

The following reading walks through a complete example of designing a highpass Butterworth filter using this method.

As you saw in the example, the process is systematic. Notice that the order of the filter, $N$, remains the same after the transformation. A 4th-order lowpass prototype becomes a 4th-order highpass filter.

3. Lowpass-to-Bandpass Transformation

To create a bandpass filter, we need a transformation that maps the single passband of the lowpass prototype to two passbands centered around a specific frequency. This requires a second-order transformation.

The transformation is:
$S=\frac{s^2+{\Omega }_0^2}{Bs}$
where:

• $s$ is the bandpass filter's frequency variable.
• $S$ is the lowpass prototype's frequency variable.
• The passband is defined by ${\Omega }_{p1}\le |\Omega |\le {\Omega }_{p2}$.
• ${\Omega }_0=\sqrt{{\Omega }_{p1}{\Omega }_{p2}}$ is the geometric center frequency.
• $B={\Omega }_{p2}-{\Omega }_{p1}$ is the passband bandwidth.

This transformation maps the prototype's passband, $-1\le {\Omega }_{LP}\le 1$, to the two bandpass regions, $[{\Omega }_{p1},{\Omega }_{p2}]$ and $[-{\Omega }_{p2},-{\Omega }_{p1}]$.

A crucial consequence of this quadratic transformation is that it doubles the filter order. Each pole in the lowpass prototype $H_{LP}(S)$ at $S=p_k$ leads to a quadratic equation in $s$:
$p_k=\frac{s^2+{\Omega }_0^2}{Bs}⟹s^2-(B{p}_k)s+{\Omega }_0^2=0$
Solving this gives two poles in the $s$-plane for every single pole in the $S$-plane. Therefore, an $N$-th order lowpass prototype transforms into a $2N$-th order bandpass filter.

The following resource details this transformation and provides another excellent worked example.

4. Lowpass-to-Bandstop Transformation

The bandstop (or notch) filter is the complement of the bandpass filter. It's therefore not surprising that its frequency transformation is the inverse of the bandpass transformation.

The transformation is:
$S=\frac{Bs}{s^2+{\Omega }_0^2}$
where $B$ and ${\Omega }_0$ are defined similarly to the bandpass case, but based on the passband edges of the bandstop filter.

This transformation maps the prototype's passband to the two passbands of the bandstop filter, and the prototype's stopband to the rejection band of the target filter.

• DC in the prototype ($S=0$) maps to the notch frequencies $s=\pm j{\Omega }_0$, creating the perfect nulls.
• Infinity in the prototype ($S\to \infty$) maps to DC and infinity in the bandstop filter, creating the surrounding passbands.

Just like the bandpass case, this is a quadratic mapping that doubles the filter order. An $N$-th order lowpass prototype becomes a $2N$-th order bandstop filter.

Conclusion

Today you have learned a powerful and efficient technique for analog filter design. By mastering a single class of filters—the lowpass prototypes—we can generate all other standard frequency-selective types through simple algebraic substitution.

Key Takeaways:

• Frequency transformation is a design strategy that maps a normalized lowpass prototype's transfer function $H_{LP}(S)$ to a target filter's transfer function $H_{target}(s)$ via a substitution $S=g(s)$.
• Lowpass to Highpass: $S={\Omega }_p/s$. This is a 1-to-1 mapping that preserves the filter order.
• Lowpass to Bandpass: $S=(s^2+{\Omega }_0^2)/(Bs)$. This is a 2-to-1 mapping that doubles the filter order.
• Lowpass to Bandstop: $S=Bs/(s^2+{\Omega }_0^2)$. This is also a 2-to-1 mapping that doubles the filter order.

Preview of the Next Lesson:
We have now assembled all the conceptual tools needed for analog filter design. In the next lesson, we will synthesize everything from this module. You will learn to tackle a complete design problem from start to finish: given a set of real-world specifications, you will design an analog filter by determining the required filter type, order, and pole-zero locations, using the frequency transformations you've mastered today.