signal processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, Scalar potential, potential fields, Seismic tomograph ...

and

electronics Electronics is a scientific and engineering discipline that studies and applies the principles of physics to design, create, and operate devices that manipulate electrons and other Electric charge, electrically charged particles. It is a subfield ...

, the frequency response of a system is the quantitative measure of the magnitude and phase of the output as a function of input frequency. The frequency response is widely used in the design and analysis of systems, such as

audio Audio most commonly refers to sound, as it is transmitted in signal form. It may also refer to: Sound *Audio signal, an electrical representation of sound *Audio frequency, a frequency in the audio spectrum *Digital audio, representation of sound ...

and

control system A control system manages, commands, directs, or regulates the behavior of other devices or systems using control loops. It can range from a single home heating controller using a thermostat controlling a domestic boiler to large industrial ...

s, where they simplify mathematical analysis by converting governing differential equations into

algebraic equations In mathematics, an algebraic equation or polynomial equation is an equation of the form P = 0, where ''P'' is a polynomial with coefficients in some field, often the field of the rational numbers. For example, x^5-3x+1=0 is an algebraic equation ...

. In an audio system, it may be used to minimize audible

distortion In signal processing, distortion is the alteration of the original shape (or other characteristic) of a signal. In communications and electronics it means the alteration of the waveform of an information-bearing signal, such as an audio signal ...

by designing components (such as microphones,

amplifiers An amplifier, electronic amplifier or (informally) amp is an electronic device that can increase the magnitude of a signal (a time-varying voltage or current). It is a two-port electronic circuit that uses electric power from a power suppl ...

and

loudspeakers A loudspeaker (commonly referred to as a speaker or, more fully, a speaker system) is a combination of one or more speaker drivers, an enclosure, and electrical connections (possibly including a crossover network). The speaker driver is an ...

) so that the overall response is as flat (uniform) as possible across the system's bandwidth. In control systems, such as a vehicle's cruise control, it may be used to assess system

stability Stability may refer to: Mathematics *Stability theory, the study of the stability of solutions to differential equations and dynamical systems ** Asymptotic stability ** Exponential stability ** Linear stability **Lyapunov stability ** Marginal s ...

, often through the use of

Bode plot In electrical engineering and control theory, a Bode plot is a graph of the frequency response of a system. It is usually a combination of a Bode magnitude plot, expressing the magnitude (usually in decibels) of the frequency response, and a B ...

s. Systems with a specific frequency response can be designed using analog and

digital filter In signal processing, a digital filter is a system that performs mathematical operations on a Sampling (signal processing), sampled, discrete-time signal to reduce or enhance certain aspects of that signal. This is in contrast to the other ma ...

s. The frequency response characterizes systems in the

frequency domain In mathematics, physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency (and possibly phase), rather than time, as in time ser ...

, just as the

impulse response In signal processing and control theory, the impulse response, or impulse response function (IRF), of a dynamic system is its output when presented with a brief input signal, called an impulse (). More generally, an impulse response is the reac ...

characterizes systems in the

time domain In mathematics and signal processing, the time domain is a representation of how a signal, function, or data set varies with time. It is used for the analysis of mathematical functions, physical signals or time series of economic or environmental ...

. In

linear system In systems theory, a linear system is a mathematical model of a system based on the use of a linear operator. Linear systems typically exhibit features and properties that are much simpler than the nonlinear case. As a mathematical abstractio ...

s (or as an approximation to a real system neglecting second order non-linear properties), either response completely describes the system and thus there is a one-to-one correspondence: the frequency response is the

Fourier transform In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...

of the impulse response. The frequency response allows simpler analysis of cascaded systems such as multistage amplifiers, as the response of the overall system can be found through multiplication of the individual stages' frequency responses (as opposed to

convolution In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two ...

of the impulse response in the time domain). The frequency response is closely related to the

transfer function In engineering, a transfer function (also known as system function or network function) of a system, sub-system, or component is a function (mathematics), mathematical function that mathematical model, models the system's output for each possible ...

in linear systems, which is the

Laplace transform In mathematics, the Laplace transform, named after Pierre-Simon Laplace (), is an integral transform that converts a Function (mathematics), function of a Real number, real Variable (mathematics), variable (usually t, in the ''time domain'') to a f ...

of the impulse response. They are equivalent when the real part

\sigma

of the transfer function's complex variable

s = \sigma + j\omega

is zero.

Measurement and plotting

Measuring the frequency response typically involves exciting the system with an input signal and measuring the resulting output signal, calculating the frequency spectra of the two signals (for example, using the

fast Fourier transform A fast Fourier transform (FFT) is an algorithm that computes the discrete Fourier transform (DFT) of a sequence, or its inverse (IDFT). A Fourier transform converts a signal from its original domain (often time or space) to a representation in ...

for discrete signals), and comparing the spectra to isolate the effect of the system. In linear systems, the frequency range of the input signal should cover the frequency range of interest. Several methods using different input signals may be used to measure the frequency response of a system, including: * Applying constant amplitude sinusoids stepped through a range of frequencies and comparing the amplitude and phase shift of the output relative to the input. The frequency sweep must be slow enough for the system to reach its

steady-state In systems theory, a system or a process is in a steady state if the variables (called state variables) which define the behavior of the system or the process are unchanging in time. In continuous time, this means that for those properties ''p'' ...

at each point of interest * Applying an impulse signal and taking the Fourier transform of the system's response * Applying a wide-sense stationary

white noise In signal processing, white noise is a random signal having equal intensity at different frequencies, giving it a constant power spectral density. The term is used with this or similar meanings in many scientific and technical disciplines, i ...

signal over a long period of time and taking the Fourier transform of the system's response. With this method, the cross-spectral density (rather than the power spectral density) should be used if phase information is required The frequency response is characterized by the ''magnitude'', typically in

decibel The decibel (symbol: dB) is a relative unit of measurement equal to one tenth of a bel (B). It expresses the ratio of two values of a Power, root-power, and field quantities, power or root-power quantity on a logarithmic scale. Two signals whos ...

s (dB) or as a generic

amplitude The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of am ...

of the dependent variable, and the '' phase'', in

radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. It is defined such that one radian is the angle subtended at ...

s or degrees, measured against frequency, in radian/s,

Hertz The hertz (symbol: Hz) is the unit of frequency in the International System of Units (SI), often described as being equivalent to one event (or Cycle per second, cycle) per second. The hertz is an SI derived unit whose formal expression in ter ...

(Hz) or as a fraction of the sampling frequency. There are three common ways of plotting response measurements: *

s graph magnitude and phase against frequency on two rectangular plots *

Nyquist plot In control theory and stability theory, the Nyquist stability criterion or Strecker–Nyquist stability criterion, independently discovered by the German electrical engineer at Siemens in 1930 and the Swedish-American electrical engineer Harry ...

s graph magnitude and phase parametrically against frequency in polar form * Nichols plots graph magnitude and phase parametrically against frequency in rectangular form For the design of control systems, any of the three types of plots may be used to infer closed-loop stability and stability margins from the open-loop frequency response. In many frequency domain applications, the phase response is relatively unimportant and the magnitude response of the Bode plot may be all that is required. In digital systems (such as digital filters), the frequency response often contains a main lobe with multiple periodic sidelobes, due to

spectral leakage The Fourier transform of a function of time, s(t), is a complex-valued function of frequency, S(f), often referred to as a frequency spectrum. Any LTI system theory, linear time-invariant operation on s(t) produces a new spectrum of the form H(f)� ...

caused by digital processes such as sampling and windowing.

Nonlinear frequency response

If the system under investigation is nonlinear, linear frequency domain analysis will not reveal all the nonlinear characteristics. To overcome these limitations, generalized frequency response functions and nonlinear output frequency response functions have been defined to analyze nonlinear dynamic effects.Billings S.A. "Nonlinear System Identification: NARMAX Methods in the Time, Frequency, and Spatio-Temporal Domains". Wiley, 2013 Nonlinear frequency response methods may reveal effects such as

resonance Resonance is a phenomenon that occurs when an object or system is subjected to an external force or vibration whose frequency matches a resonant frequency (or resonance frequency) of the system, defined as a frequency that generates a maximu ...

intermodulation Intermodulation (IM) or intermodulation distortion (IMD) is the amplitude modulation of Signal (electrical engineering), signals containing two or more different frequencies, caused by non-linear, nonlinearities or time variance in a system. ...

, and energy transfer.

Applications

In the audible range frequency response is usually referred to in connection with

electronic amplifier An amplifier, electronic amplifier or (informally) amp is an electronic device that can increase the magnitude of a Signal (information theory), signal (a time-varying voltage or Electric current, current). It is a two-port network, two-port ...

microphone A microphone, colloquially called a mic (), or mike, is a transducer that converts sound into an electrical signal. Microphones are used in many applications such as telephones, hearing aids, public address systems for concert halls and publi ...

s and

. Radio spectrum frequency response can refer to measurements of

coaxial cable Coaxial cable, or coax (pronounced ), is a type of electrical cable consisting of an inner Electrical conductor, conductor surrounded by a concentric conducting Electromagnetic shielding, shield, with the two separated by a dielectric (Insulat ...

, twisted-pair cable, video switching equipment,

wireless Wireless communication (or just wireless, when the context allows) is the transfer of information (''telecommunication'') between two or more points without the use of an electrical conductor, optical fiber or other continuous guided transm ...

communications devices, and antenna systems. Infrasonic frequency response measurements include

earthquakes An earthquakealso called a quake, tremor, or tembloris the shaking of the Earth's surface resulting from a sudden release of energy in the lithosphere that creates seismic waves. Earthquakes can range in intensity, from those so weak they c ...

and

electroencephalography Electroencephalography (EEG) is a method to record an electrogram of the spontaneous electrical activity of the brain. The biosignal, bio signals detected by EEG have been shown to represent the postsynaptic potentials of pyramidal neurons in ...

(brain waves). Frequency response curves are often used to indicate the accuracy of electronic components or systems.Stark, 2002, p. 51. When a system or component reproduces all desired input signals with no emphasis or attenuation of a particular frequency band, the system or component is said to be "flat", or to have a flat frequency response curve. In other cases, 3D-form of frequency response graphs are sometimes used. Frequency response requirements differ depending on the application.Luther, 1999, p. 141. In

high fidelity High fidelity (hi-fi or, rarely, HiFi) is the high-quality reproduction of sound. It is popular with audiophiles and home audio enthusiasts. Ideally, high-fidelity equipment has inaudible noise and distortion, and a flat (neutral, uncolored) ...

audio, an amplifier requires a flat frequency response of at least 20–20,000 Hz, with a tolerance as tight as ±0.1 dB in the mid-range frequencies around 1000 Hz; however, in

telephony Telephony ( ) is the field of technology involving the development, application, and deployment of telecommunications services for the purpose of electronic transmission of voice, fax, or data, between distant parties. The history of telephony is ...

, a frequency response of 400–4,000 Hz, with a tolerance of ±1 dB is sufficient for intelligibility of speech. Once a frequency response has been measured (e.g., as an impulse response), provided the system is linear and time-invariant, its characteristic can be approximated with arbitrary accuracy by a

. Similarly, if a system is demonstrated to have a poor frequency response, a digital or

analog filter Analogue Filter (signal processing), filters are a basic building block of signal processing much used in electronics. Amongst their many applications are the separation of an audio signal before application to bass (music), bass, mid-range sp ...

can be applied to the signals prior to their reproduction to compensate for these deficiencies. The form of a frequency response curve is very important for anti-jamming protection of radars, communications and other systems. Frequency response analysis can also be applied to biological domains, such as the detection of hormesis in repeated behaviors with opponent process dynamics, or in the optimization of drug treatment regimens.

References

;Notes ;Bibliography *Luther, Arch C.; Inglis, Andrew F
''Video engineering''
McGraw-Hill, 1999. *Stark, Scott Hunter
''Live Sound Reinforcement''
Vallejo, California, Artistpro.com, 1996–2002. * L. R. Rabiner and B. Gold. Theory and Application of Digital Signal Processing. – Englewood Cliffs, NJ: Prentice-Hall, 1975. – 720 pp

External links

University of Michigan The University of Michigan (U-M, U of M, or Michigan) is a public university, public research university in Ann Arbor, Michigan, United States. Founded in 1817, it is the oldest institution of higher education in the state. The University of Mi ...

Frequency Response Analysis and Design Tutorial
*Smith, Julius O. III
Introduction to Digital Filters with Audio Applications
has a nice chapter o

{{Authority control Signal processing Control theory Audio amplifier specifications

Measurement and plotting

Nonlinear frequency response

Applications

See also

References

External links