Butterworth filter
The Butterworth filter is a type of signal processing filter designed to have a maximally flat frequency response in the passband, providing uniform gain across the desired frequency range without ripples. This characteristic makes it ideal for applications requiring smooth attenuation from the passband to the stopband, with the transition sharpness controlled by the filter order N.[1] First described in 1930 by British engineer Stephen Butterworth in his seminal paper "On the Theory of Filter Amplifiers," published in Experimental Wireless and the Wireless Engineer, the filter derives its name from its inventor and emphasizes optimal flatness in magnitude response for analog amplifier circuits.[1] The magnitude squared of the transfer function for a low-pass Butterworth filter is given by |H(\omega)|^2 = \frac{1}{1 + (\omega / \omega_c)^{2N}}, where \omega_c is the cutoff frequency (typically defined at -3 dB attenuation) and N is the order, which determines the roll-off rate of -20N dB per decade beyond \omega_c.[1] Unlike filters with ripples, such as Chebyshev types, the Butterworth response is monotonic, ensuring no overshoot in the passband but a gentler transition to the stopband.[2] These filters can be realized in analog form using passive components like resistors and capacitors or active circuits with operational amplifiers, and in digital form as infinite impulse response (IIR) filters via bilinear transformation. Butterworth filters are widely applied in audio processing for equalization and noise reduction, communications for band-limiting signals, biomedical signal analysis to smooth physiological data, and control systems for stable frequency selection due to their predictable phase response and ease of design.[3] Variants include high-pass, band-pass, and band-stop configurations, all sharing the core maximally flat property, making them a foundational tool in electrical engineering despite trade-offs in steeper roll-off compared to elliptic filters.History
Invention and Original Publication
The Butterworth filter was invented by British physicist and engineer Stephen Butterworth, who introduced the concept in his seminal 1930 paper titled "On the Theory of Filter Amplifiers," published in the journal Experimental Wireless and the Wireless Engineer (volume 7, pages 536–541).[4] Working at the Admiralty Research Laboratory, Butterworth addressed the limitations of existing electrical wave filter theories, which were primarily focused on sharp cutoffs but often resulted in irregular responses unsuitable for practical applications.[4] The primary motivation for Butterworth's work stemmed from the demands of early radio engineering, where amplifiers required frequency responses that were as smooth and uniform as possible to avoid distortion in signal processing for wireless communication systems. At the time, radio technology was rapidly advancing, and filter amplifiers needed to maintain consistent gain across the passband to ensure reliable transmission and reception of signals without unwanted variations.[1] Butterworth's approach prioritized this uniformity over abrupt transitions, marking a shift toward filters optimized for amplifier integration in radio circuits.[4] In the paper, Butterworth derived the foundational squared magnitude response of the filter, given by |H(j\omega)|^2 = \frac{1}{1 + \left( \frac{\omega}{\omega_c} \right)^{2N}} where \omega is the angular frequency, \omega_c is the cutoff angular frequency, and N is the filter order.[4] This expression ensures a maximally flat response in the passband, with the filter attenuating higher frequencies gradually based on the order N. Butterworth illustrated this with plots for various orders, demonstrating how increasing N steepens the roll-off while preserving passband flatness, a concept he explicitly tied to practical amplifier design needs in wireless engineering.[4]Development and Naming
Following its introduction in 1930, the Butterworth filter saw widespread adoption in telephony and electronics during the ensuing decades. Butterworth's original work on filter amplifiers for radio applications provided a foundational basis for subsequent developments in filter design.[5] The filter became known as the "Butterworth filter" in recognition of Stephen Butterworth's pioneering contribution, with the nomenclature formalized and consistently applied in engineering literature by the mid-20th century. This evolution extended to its integration into network synthesis techniques during the 1940s, which embedded the Butterworth approach within systematic filter design methodologies, shaping subsequent theoretical progress in the field.Fundamental Characteristics
Overview of Filter Behavior
The Butterworth filter is a linear time-invariant signal processing filter designed to provide a maximally flat frequency response in the passband, ensuring minimal distortion of signals within the desired frequency range.[6][7] This characteristic makes it particularly suitable for applications requiring smooth amplitude preservation without ripples or oscillations in the passband.[1] Invented by British engineer Stephen Butterworth in 1930, it serves as a foundational prototype in analog and digital filtering.[1] Butterworth filters are available in several configurations, including low-pass, high-pass, band-pass, and band-stop types, each derived from transformations of the fundamental low-pass prototype.[8] The low-pass variant, which attenuates frequencies above a specified cutoff while passing lower frequencies, is the most commonly referenced and serves as the basis for designing the other forms.[7] These configurations allow for versatile frequency shaping in various systems. In operation, the Butterworth filter exhibits a smooth, monotonic transition from the passband to the stopband, avoiding abrupt changes or ripples that could introduce unwanted artifacts in the output signal.[6] This gradual roll-off provides an intuitive balance between selectivity and simplicity, making it ideal for audio processing—such as equalizers and noise reduction—and general signal processing tasks where preserving the natural waveform integrity is essential, particularly in environments intolerant to passband variations.[9][7]Maximal Flatness Property
The maximal flatness property of the Butterworth low-pass filter refers to the condition where the squared magnitude response |H(j\omega)|^2 and its first $2N-1 derivatives with respect to the normalized angular frequency \omega are all zero at \omega = 0 for an Nth-order filter.[10] This ensures that the passband response remains as constant as possible near direct current (DC), with no ripples or variations in gain, distinguishing it from other filter approximations like Chebyshev types that introduce equiripple behavior.[11] The property arises from the filter's design criterion, originally proposed by Stephen Butterworth, to maximize the number of derivatives that vanish at the origin, thereby providing the "flattest" possible monotonic response in the passband.[12] This flatness approximates the ideal brick-wall low-pass filter, which has a perfectly constant magnitude of unity for \omega < 1 and zero otherwise, but the Butterworth achieves it through a rational all-pole transfer function without finite zeros, leading to a smooth, gradual roll-off beyond the cutoff.[11] Unlike filters with zeros that can introduce sharper transitions but potential ripples, the all-pole structure of the Butterworth prioritizes passband uniformity, making it suitable for applications requiring minimal distortion in the signal's low-frequency components, such as audio processing or anti-aliasing.[13] Conceptually, the maximal flatness can be illustrated through the Taylor series expansion of the squared magnitude response around \omega = 0: |H(j\omega)|^2 = \frac{1}{1 + \omega^{2N}} = 1 - \omega^{2N} + O(\omega^{4N}), where the expansion shows that all terms up to order $2N-1 match those of the ideal constant response (unity), with the first deviation occurring at the $2Nth power of \omega.[13] This structure maximizes the order of contact between the actual and ideal responses at DC, enhancing flatness as N increases, though at the cost of a slower transition to the stopband compared to sharper approximations.[10]Mathematical Description
Transfer Function
The transfer function for a continuous-time low-pass Butterworth filter of order N and cutoff angular frequency \omega_c takes the form H(s) = \frac{1}{B_N\left( \frac{s}{\omega_c} \right)}, where B_N(s) denotes the Nth-order Butterworth polynomial with B_N(0) = 1. This polynomial is constructed such that its roots (the poles of H(s)) lie in the left half of the complex s-plane, ensuring stability and the characteristic maximally flat frequency response near \omega = 0.[4] The squared magnitude of the frequency response, |H(j\omega)|^2, is given by |H(j\omega)|^2 = \frac{1}{1 + \left( \frac{\omega}{\omega_c} \right)^{2N}}. This form arises from the design requirement for maximal flatness in the passband, where the first $2N-1 derivatives of |H(j\omega)|^2 vanish at \omega = 0. To derive the transfer function H(s), consider the analytic continuation via H(s)H(-s) = 1 / [1 + (-s^2 / \omega_c^2)^N] for the normalized case (\omega_c = 1). The poles of this expression are the $2N roots of $1 + (-1)^N s^{2N} = 0, located at s_k = e^{j \left( \frac{\pi}{2} + \frac{(2k-1)\pi}{2N} \right)} for k = 1, \dots, 2N, which are equally spaced around the unit circle in the s-plane due to the geometric symmetry of the roots of unity. Stability dictates selecting only the N poles in the left half-plane for H(s), with the constant adjusted for unity DC gain.[14] High-pass Butterworth filters are obtained by applying the frequency transformation s \to \omega_c / s to the low-pass prototype transfer function, inverting the frequency axis while preserving the magnitude-squared response shape but shifting the cutoff to high frequencies. This substitution maps low frequencies to high frequencies and ensures the resulting H(s) retains the same order N and flatness properties.[15]Normalized Polynomials and Poles
The normalized Butterworth low-pass filter is defined with a cutoff frequency of ω_c = 1 rad/s, where the transfer function takes the form H(s) = 1 / B_N(s), and B_N(s) is the Nth-order Butterworth polynomial with all coefficients real and positive. These polynomials are derived from the requirement of maximally flat magnitude response at ω = 0 and are monic (leading coefficient 1). For practical design, they are often expressed in factored form using first- and second-order factors corresponding to real and complex-conjugate poles, respectively.[15] The normalized polynomials for orders N = 1 to 5 are as follows:- N = 1: B_1(s) = s + 1
- N = 2: B_2(s) = s^2 + \sqrt{2} s + 1
- N = 3: B_3(s) = (s + 1)(s^2 + s + 1) = s^3 + 2s^2 + 2s + 1
- N = 4: B_4(s) = (s^2 + 0.765s + 1)(s^2 + 1.848s + 1) = s^4 + 2.613s^3 + 3.414s^2 + 2.613s + 1
- N = 5: B_5(s) = (s + 1)(s^2 + 0.618s + 1)(s^2 + 1.618s + 1) = s^5 + 3.236s^4 + 5.236s^3 + 5.236s^2 + 3.236s + 1