A linear response function describes the input-output relationship of a signal transducer, such as a radio turning electromagnetic waves into music or a neuron turning synaptic input into a response. Because of its many applications in information theory, physics and engineering there exist alternative names for specific linear response functions such as susceptibility, impulse response or impedance; see also transfer function. The concept of a Green's function or fundamental solution of an ordinary differential equation is closely related.
Denote the input of a system by (e.g. a force), and the response of the system by (e.g. a position). Generally, the value of will depend not only on the present value of , but also on past values. Approximately is a weighted sum of the previous values of , with the weights given by the linear response function :
The explicit term on the right-hand side is the leading order term of a Volterra expansion for the full nonlinear response. If the system in question is highly non-linear, higher order terms in the expansion, denoted by the dots, become important and the signal transducer cannot adequately be described just by its linear response function.
The complex-valued Fourier transform of the linear response function is very useful as it describes the output of the system if the input is a sine wave with frequency . The output reads
with amplitude gain and phase shift .
Consider a damped harmonic oscillator with input given by an external driving force ,
The complex-valued Fourier transform of the linear response function is given by
The amplitude gain is given by the magnitude of the complex number and the phase shift by the arctan of the imaginary part of the function divided by the real one.
From this representation, we see that for small the Fourier transform of the linear response function yields a pronounced maximum ("Resonance") at the frequency . The linear response function for a harmonic oscillator is mathematically identical to that of an RLC circuit.
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.
Le but de ce cours est d'apporter les connaissances et les expériences fondamentales pour comprendre les systèmes électriques et électroniques de base.
This lecture is oriented towards the study of audio engineering, with a special focus on room acoustics applications. The learning outcomes will be the techniques for microphones and loudspeaker desig
An RLC circuit is an electrical circuit consisting of a resistor (R), an inductor (L), and a capacitor (C), connected in series or in parallel. The name of the circuit is derived from the letters that are used to denote the constituent components of this circuit, where the sequence of the components may vary from RLC. The circuit forms a harmonic oscillator for current, and resonates in a manner similar to an LC circuit. Introducing the resistor increases the decay of these oscillations, which is also known as damping.
In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions. This means that if is the linear differential operator, then the Green's function is the solution of the equation , where is Dirac's delta function; the solution of the initial-value problem is the convolution ().
In physics and engineering, the quality factor or Q factor is a dimensionless parameter that describes how underdamped an oscillator or resonator is. It is defined as the ratio of the initial energy stored in the resonator to the energy lost in one radian of the cycle of oscillation. Q factor is alternatively defined as the ratio of a resonator's centre frequency to its bandwidth when subject to an oscillating driving force. These two definitions give numerically similar, but not identical, results.
Explores the fundamentals of cavities and resonators, covering topics such as damped harmonic oscillators, cavity frequency response, and coupling to transmission lines.
Resonant sensors based on micro- and nano-electro mechanical systems (M/NEMS) are ubiquitous in many sensing applications due to their outstanding performance capabilities, which are directly proportional to the quality factor (Q) of the devices. We addres ...
2020
DC-DC converters based on Application Specific Integrated Circuits (ASICs) have been developed in this doctoral work for the High-Luminosity Large Hadron Collider (HL-LHC) experiments at CERN. They step down the voltage from a 2.5 V line and supply a load ...
We investigate both theoretically and experimentally a laser-based controlled tuning of the nonlinear behaviors of a single mechanical resonator. Thanks to localized three-dimensional modifications induced by femtosecond-laser irradiation, a Duffing-like o ...