signal processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as sound, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, ...

and

electronics The field of electronics is a branch of physics and electrical engineering that deals with the emission, behaviour and effects of electrons using electronic devices. Electronics uses active devices to control electron flow by amplification ...

, the frequency response of a system is the quantitative measure of the magnitude and

phase Phase or phases may refer to: Science *State of matter, or phase, one of the distinct forms in which matter can exist *Phase (matter), a region of space throughout which all physical properties are essentially uniform * Phase space, a mathematic ...

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 sou ...

and control systems, 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 many authors, the term ''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 signa ...

by designing components (such as

microphones 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 public ...

, amplifiers and

loudspeakers A loudspeaker (commonly referred to as a speaker or speaker driver) is an electroacoustic transducer that converts an electrical audio signal into a corresponding sound. A ''speaker system'', also often simply referred to as a "speaker" or "l ...

) so that the overall response is as flat (uniform) as possible across the system's

bandwidth Bandwidth commonly refers to: * Bandwidth (signal processing) or ''analog bandwidth'', ''frequency bandwidth'', or ''radio bandwidth'', a measure of the width of a frequency range * Bandwidth (computing), the rate of data transfer, bit rate or thr ...

. In control systems, such as a vehicle's

cruise control Cruise control (also known as speed control, cruise command, autocruise, or tempomat) is a system that automatically controls the speed of a motor vehicle. The system is a servomechanism that takes over the throttle of the car to maintain a s ...

, 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 ** Linear stability ** Lyapunov stability ** Orbital stability ** Structural sta ...

, often through the use of Bode plots. Systems with a specific frequency response can be designed using

analog Analog or analogue may refer to: Computing and electronics * Analog signal, in which information is encoded in a continuous variable ** Analog device, an apparatus that operates on analog signals *** Analog electronics, circuits which use analog ...

and

digital filter In signal processing, a digital filter is a system that performs mathematical operations on a sampled, discrete-time signal to reduce or enhance certain aspects of that signal. This is in contrast to the other major type of electronic filter, t ...

s. The frequency response characterizes systems in the

frequency domain In physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency, rather than time. Put simply, a time-domain graph shows how a s ...

, just as the impulse response characterizes systems in the

time domain Time domain refers to the analysis of mathematical functions, physical signals or time series of economic or environmental data, with respect to time. In the time domain, the signal or function's value is known for all real numbers, for the c ...

. 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, either response completely describes the system and thus have one-to-one correspondence: the frequency response is the Fourier transform of the impulse response. The frequency response allows simpler analysis of cascaded systems such as

multistage amplifier A multistage amplifier is an electronic amplifier consisting of two or more single-stage amplifiers connected together. In this context, a single stage is an amplifier containing only a single transistor (sometimes a pair of transistors) or ot ...

s, 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 mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' ...

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 mathematical function that theoretically models the system's output for each possible input. They are widely used ...

in linear systems, which is the

Laplace transform In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable (usually t, in the '' time domain'') to a function of a complex variable s (in the ...

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 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 at each point of interest * Applying an

impulse Impulse or Impulsive may refer to: Science * Impulse (physics), in mechanics, the change of momentum of an object; the integral of a force with respect to time * Impulse noise (disambiguation) * Specific impulse, the change in momentum per uni ...

signal and taking the Fourier transform of the system's response * Applying a

wide-sense stationary In mathematics and statistics, a stationary process (or a strict/strictly stationary process or strong/strongly stationary process) is a stochastic process whose unconditional joint probability distribution does not change when shifted in time. Con ...

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 ...

signal over a long period of time and taking the Fourier transform of the system's response. With this method, the

cross-spectral density The power spectrum S_(f) of a time series x(t) describes the distribution of power into frequency components composing that signal. According to Fourier analysis, any physical signal can be decomposed into a number of discrete frequencies, ...

(rather than the

power spectral density The power spectrum S_(f) of a time series x(t) describes the distribution of power into frequency components composing that signal. According to Fourier analysis, any physical signal can be decomposed into a number of discrete frequencies, ...

) should be used if phase information is required The frequency response is characterized by the ''magnitude'', typically in decibels (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 ''

'', 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. The unit was formerly an SI supplementary unit (before tha ...

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), equivalent to one event (or cycle) per second. The hertz is an SI derived unit whose expression in terms of SI base units is s−1, meaning that o ...

(Hz) or as a fraction of the

sampling frequency In signal processing, sampling is the reduction of a continuous-time signal to a discrete-time signal. A common example is the conversion of a sound wave to a sequence of "samples". A sample is a value of the signal at a point in time and/or s ...

. There are three common ways of plotting response measurements: * Bode plots 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 Har ...

s graph magnitude and phase

parametrically A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...

against frequency in polar form *

Nichols plot The Nichols plot is a plot used in signal processing and control design, named after American engineer Nathaniel B. Nichols.Allen Stubberud, Ivan Williams, and Joseph DeStefano, ''Shaums Outline Feedback and Control Systems'', McGraw-Hill, 1995 ...

s 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 (such as audio systems), 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 linear time-invariant operation on s(t) produces a new spectrum of the form H(f)•S(f), which cha ...

caused by digital processes such as sampling and windowing.L. R. Rabiner and B. Gold. Theory and Application of Digital Signal Processing. – Englewood Cliffs, NJ: Prentice-Hall, 1975. – 720 pp

Nonlinear frequency response

If the system under investigation is

nonlinear In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many othe ...

, 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 describes the phenomenon of increased amplitude that occurs when the frequency of an applied Periodic function, periodic force (or a Fourier analysis, Fourier component of it) is equal or close to a natural frequency of the system ...

intermodulation Intermodulation (IM) or intermodulation distortion (IMD) is the amplitude modulation of signals containing two or more different frequencies, caused by nonlinearities or time variance in a system. The intermodulation between frequency comp ...

, and

energy transfer In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of heat ...

Applications

In electronics this stimulus would be an input signal.Stark, 2002, p. 51. In the audible range it is usually referred to in connection with electronic amplifiers,

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 publ ...

s and

. Radio spectrum frequency response can refer to measurements of coaxial cable, twisted-pair cable, video switching equipment,

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

communications devices, and antenna systems. Infrasonic frequency response measurements include earthquakes and electroencephalography (brain waves). Frequency response requirements differ depending on the application.Luther, 1999, p. 141. In high fidelity audio, an amplifier requires a 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 telecommunication services for the purpose of electronic transmission of voice, fax, or data, between distant parties. The history of telephony is i ...

, a frequency response of 400–4,000 Hz, with a tolerance of ±1 dB is sufficient for intelligibility of speech. Frequency response curves are often used to indicate the accuracy of electronic components or systems. 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 case, we can be use 3D-form of frequency response surface. 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 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, mid-range, and tweeter loudspeakers; the combining and ...

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.

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 , mottoeng = "Arts, Knowledge, Truth" , former_names = Catholepistemiad, or University of Michigania (1817–1821) , budget = $10.3 billion (2021) , endowment = $17 billion (2021)As o ...

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