Question

Derive the prototype transfer function for a fitthorder high-pass Butterworth filter by first writing the transfer function for a fifth-order prototype low- pass Butterworth filter and then replacing \(s\) by \(1 / s\) in the low-pass expression.

Short answer

The high-pass Butterworth transfer function is \( H_{HP}(s) = \frac{s^5}{s^5 + 5s^4 + 10s^3 + 10s^2 + 5s + 1} \).

Step-by-step solution

Write the Standard Low-Pass Butterworth Transfer Function

For a fifth-order Butterworth filter, the standard low-pass transfer function is given by:\[ H_{LP}(s) = \frac{1}{s^5 + 5s^4 + 10s^3 + 10s^2 + 5s + 1} \] The coefficients can be found using the binomial expansion for \((1+s)^n\) where \(n\) is the order of the filter.

Convert Low-Pass to High-Pass by Substituting

To convert the low-pass filter to a high-pass filter, replace \(s\) in the transfer function with \(\frac{1}{s}\). This yields the prototype high-pass Butterworth transfer function:\[ H_{HP}(s) = \frac{1}{\left(\frac{1}{s}\right)^5 + 5\left(\frac{1}{s}\right)^4 + 10\left(\frac{1}{s}\right)^3 + 10\left(\frac{1}{s}\right)^2 + 5\left(\frac{1}{s}\right) + 1} \]

Simplify the High-Pass Transfer Function

Simplify the expression by multiplying both the numerator and denominator by \(s^5\):\[ H_{HP}(s) = \frac{s^5}{1 + 5s + 10s^2 + 10s^3 + 5s^4 + s^5} \] This yields the desired high-pass Butterworth transfer function.

Key concepts

Transfer Function

A transfer function represents the relationship between the input and output of a linear time-invariant system in the Laplace domain. It's a mathematical model helpful in analyzing system behavior. The transfer function is typically denoted as:\[ H(s) = \frac{Y(s)}{X(s)} \]where:

• \( Y(s) \) is the Laplace transform of the output signal.
• \( X(s) \) is the Laplace transform of the input signal.

This function is crucial in determining system characteristics like stability, frequency response, and gain. By looking at the poles and zeros of the transfer function, we get insights into the system's behavior such as resonance and bandwidth.In filter design, transfer functions help craft frequency-selective filters, which allow or block specific frequency ranges. For Butterworth filters, the transfer function ensures a maximally flat response in the passband, which is ideal for certain applications where distortion needs to be minimal.

High-Pass Filter

A high-pass filter (HPF) permits high-frequency signals to pass while attenuating low-frequency signals. This is useful in processes like noise reduction and signal processing where low-frequency noise needs to be removed.The transfer function of a high-pass filter can be viewed as the inverse of a low-pass filter. In our example of a fifth-order Butterworth filter:- The transfer function was converted by replacing \( s \) with \( \frac{1}{s} \).- This transforms the originally low-pass characteristics to high-pass, allowing us to control the frequency cutoff.High-pass filters are essential in:

• Radio communications to block unwanted frequencies.
• Audio electronics to remove rumble or low-frequency drone.
• Image processing to enhance edges.

Low-Pass Filter

Low-pass filters (LPF) allow signals with a frequency lower than a certain cutoff frequency to pass through, while attenuating signals with frequencies higher than that cutoff. They are fundamental in applications like signal smoothing and noise reduction.In the context of a Butterworth filter, a low-pass filter ensures a flat frequency response in the passband. For a fifth-order low-pass Butterworth filter, as provided in the exercise, the transfer function is:\[ H_{LP}(s) = \frac{1}{s^5 + 5s^4 + 10s^3 + 10s^2 + 5s + 1} \]The fifth-order implies it has a steeper roll-off and better attenuation of high frequencies compared to a lower-order filter.LPFs are utilized in:

• Audio applications to provide smoother sound by reducing high-frequency noise.
• Data processing for reducing noise from input signals.
• Communication systems to limit bandwidth.

Filter Design

Designing filters involves choosing the appropriate type and order of a filter to meet specific requirements. The design process begins with defining the filter's intended application, such as audio processing, radio transmission, or noise reduction. Key considerations in filter design include:

• Type of filter: Low-pass, high-pass, band-pass, or band-stop.
• Order of the filter: Higher orders mean a steeper roll-off but also more complexity.
• Passband and stopband frequencies.
• Attenuation levels and ripple in the passband.

Butterworth filters are specifically designed to have a maximally flat response in the passband, which is particularly important for applications requiring minimal signal distortion. The ease of transitioning between low-pass and high-pass filter designs highlights the flexibility within filter design when using transfer functions.

Signal Processing

Signal processing involves manipulating signals to improve their quality or to extract useful information. It's a broad field encompassing a wide range of applications from audio enhancement to medical imaging. Filters are a crucial component of signal processing. They allow selective frequency manipulation, which enhances or reduces certain characteristics. In signal processing:

• High-pass filters can enhance details and remove baseline noise.
• Low-pass filters can smooth signals and reduce random noise.
• Complex filter designs like Butterworth are used for applications requiring precise frequency control.

The goal in signal processing is to improve signal fidelity, increase signal-to-noise ratio, and ensure the processed signal serves its intended purpose, whether in communication systems, audio devices, or scientific instruments.