Frequency response

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In signal processing and electronics, the frequency response of a system is the quantitative measure of the magnitude and phase of the output as a function of input frequency. [1] The frequency response is widely used in the design and analysis of systems, such as audio and control systems, where they simplify mathematical analysis by converting governing differential equations into algebraic equations. In an audio system, it may be used to minimize audible distortion by designing components (such as microphones, amplifiers and loudspeakers) 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, often through the use of Bode plots. Systems with a specific frequency response can be designed using analog and digital filters.

Contents

The frequency response characterizes systems in the frequency domain, just as the impulse response characterizes systems in the time domain. In linear systems (or as an approximation to a real system neglecting second order non-linear properties), 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 amplifiers, as the response of the overall system can be found through multiplication of the individual stages' frequency responses (as opposed to convolution of the impulse response in the time domain). The frequency response is closely related to the transfer function in linear systems, which is the Laplace transform of the impulse response. They are equivalent when the real part of the transfer function's complex variable is zero. [2]

Measurement and plotting

Magnitude response of a low pass filter with 6 dB per octave or 20 dB per decade roll-off Butterworth response.svg
Magnitude response of a low pass filter with 6 dB per octave or 20 dB per decade roll-off

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:

The frequency response is characterized by the magnitude, typically in decibels (dB) or as a generic amplitude of the dependent variable, and the phase , in radians or degrees, measured against frequency, in radian/s, Hertz (Hz) or as a fraction of the sampling frequency.

There are three common ways of plotting response measurements:

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 caused by digital processes such as sampling and windowing. [3]

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. [4] Nonlinear frequency response methods may reveal effects such as resonance, intermodulation, and energy transfer.

Applications

In the audible range frequency response is usually referred to in connection with electronic amplifiers, microphones and loudspeakers. Radio spectrum frequency response can refer to measurements of coaxial cable, twisted-pair cable, video switching equipment, wireless communications devices, and antenna systems. Infrasonic frequency response measurements include earthquakes and electroencephalography (brain waves).

Frequency response curves are often used to indicate the accuracy of electronic components or systems. [5] 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. [5] In other case, we can be use 3D-form of frequency response surface.

Frequency response requirements differ depending on the application. [6] In high fidelity 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, a frequency response of 400–4,000 Hz, with a tolerance of ±1 dB is sufficient for intelligibility of speech. [6]

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 digital filter. Similarly, if a system is demonstrated to have a poor frequency response, a digital or analog filter 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, [7] or in the optimization of drug treatment regimens. [8]

See also

Related Research Articles

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In digital signal processing, upsampling, expansion, and interpolation are terms associated with the process of resampling in a multi-rate digital signal processing system. Upsampling can be synonymous with expansion, or it can describe an entire process of expansion and filtering (interpolation). When upsampling is performed on a sequence of samples of a signal or other continuous function, it produces an approximation of the sequence that would have been obtained by sampling the signal at a higher rate. For example, if compact disc audio at 44,100 samples/second is upsampled by a factor of 5/4, the resulting sample-rate is 55,125.

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<span class="mw-page-title-main">Ringing artifacts</span> Form of error in digital signals; spurious signals near sharp transitions

In signal processing, particularly digital image processing, ringing artifacts are artifacts that appear as spurious signals near sharp transitions in a signal. Visually, they appear as bands or "ghosts" near edges; audibly, they appear as "echos" near transients, particularly sounds from percussion instruments; most noticeable are the pre-echos. The term "ringing" is because the output signal oscillates at a fading rate around a sharp transition in the input, similar to a bell after being struck. As with other artifacts, their minimization is a criterion in filter design.

In signal processing, a filter is a device or process that removes some unwanted components or features from a signal. Filtering is a class of signal processing, the defining feature of filters being the complete or partial suppression of some aspect of the signal. Most often, this means removing some frequencies or frequency bands. However, filters do not exclusively act in the frequency domain; especially in the field of image processing many other targets for filtering exist. Correlations can be removed for certain frequency components and not for others without having to act in the frequency domain. Filters are widely used in electronics and telecommunication, in radio, television, audio recording, radar, control systems, music synthesis, image processing, computer graphics, and structural dynamics.

<span class="mw-page-title-main">Audio analyzer</span> Test and measurement instrument

An audio analyzer is a test and measurement instrument used to objectively quantify the audio performance of electronic and electro-acoustical devices. Audio quality metrics cover a wide variety of parameters, including level, gain, noise, harmonic and intermodulation distortion, frequency response, relative phase of signals, interchannel crosstalk, and more. In addition, many manufacturers have requirements for behavior and connectivity of audio devices that require specific tests and confirmations.

References

Notes
  1. ↑ Smith, Steven W. (1997). The Scientist and Engineer's Guide to Digital Signal Processing. California Technical Pub. pp. 177–180. ISBN   978-0966017632.
  2. ↑ Dennis L. Feucht (1990). Handbook of Analog Circuit Design. Elsevier Science. p. 192. ISBN   978-1-4832-5938-3.
  3. ↑ L. R. Rabiner and B. Gold. Theory and Application of Digital Signal Processing. – Englewood Cliffs, NJ: Prentice-Hall, 1975. – 720 pp
  4. ↑ Billings S.A. "Nonlinear System Identification: NARMAX Methods in the Time, Frequency, and Spatio-Temporal Domains". Wiley, 2013
  5. 1 2 Stark, 2002, p. 51.
  6. 1 2 Luther, 1999, p. 141.
  7. ↑ Henry, N.; Pedersen, M.; Williams, M.; Donkin, L. (2023-07-03). "Behavioral Posology: A Novel Paradigm for Modeling the Healthy Limits of Behaviors". Advanced Theory and Simulations. 6 (9). doi: 10.1002/adts.202300214 . ISSN   2513-0390.
  8. ↑ Schulthess, Pascal; Post, Teun M.; Yates, James; van der Graaf, Piet H. (February 2018). "Frequency-Domain Response Analysis for Quantitative Systems Pharmacology Models: Frequency-domain response analysis for QSP models". CPT: Pharmacometrics & Systems Pharmacology. 7 (2): 111–123. doi:10.1002/psp4.12266. PMC   5824121 . PMID   29193852.
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