# Frequency response

**Frequency response** is the quantitative measure of the output spectrum of a system or device in response to a stimulus, and is used to characterize the dynamics of the system. It is a measure of magnitude and phase of the output as a function of frequency, in comparison to the input. In simplest terms, if a sine wave is injected into a system at a given frequency, a linear system will respond at that same frequency with a certain magnitude and a certain phase angle relative to the input. Also for a linear system, doubling the amplitude of the input will double the amplitude of the output. In addition, if the system is time-invariant (so LTI), then the frequency response also will not vary with time. Thus for LTI systems, the frequency response can be seen as applying the system's transfer function to a purely imaginary number argument representing the frequency of the sinusoidal excitation.^{[1]}

Two applications of frequency response analysis are related but have different objectives. For an audio system, the objective may be to reproduce the input signal with no distortion. That would require a uniform (flat) magnitude of response up to the bandwidth limitation of the system, with the signal delayed by precisely the same amount of time at all frequencies. That amount of time could be seconds, or weeks or months in the case of recorded media. In contrast, for a feedback apparatus used to control a dynamic system, the objective is to give the closed-loop system improved response as compared to the uncompensated system. The feedback generally needs to respond to system dynamics within a very small number of cycles of oscillation (usually less than one full cycle), and with a definite phase angle relative to the commanded control input. For feedback of sufficient amplification, getting the phase angle wrong can lead to instability for an open-loop stable system, or failure to stabilize a system that is open-loop unstable. Digital filters may be used for both audio systems and feedback control systems, but since the objectives are different, generally the phase characteristics of the filters will be significantly different for the two applications.

## Estimation and plotting

Estimating the frequency response for a physical system generally involves exciting the system with an input signal, measuring both input and output time histories, and comparing the two through a process such as the Fast Fourier Transform (FFT). One thing to keep in mind for the analysis is that the frequency content of the input signal must cover the frequency range of interest or the results will not be valid for the portion of the frequency range not covered.

The frequency response of a system can be measured by applying a *test signal*, for example:

- applying an impulse to the system and measuring its response (see impulse response)
- sweeping a constant-amplitude pure tone through the bandwidth of interest and measuring the output level and phase shift relative to the input
- applying a signal with a wide frequency spectrum (for example digitally-generated maximum length sequence noise, or analog filtered white noise equivalent, like pink noise), and calculating the impulse response by deconvolution of this input signal and the output signal of the system.

The frequency response is characterized by the *magnitude* of the system's response, typically measured in decibels (dB) or as a decimal, and the *phase*, measured in radians or degrees, versus frequency in radians/sec or Hertz (Hz).

These response measurements can be plotted in three ways: by plotting the magnitude and phase measurements on two rectangular plots as functions of frequency to obtain a Bode plot; by plotting the magnitude and phase angle on a single polar plot with frequency as a parameter to obtain a Nyquist plot; or by plotting magnitude and phase on a single rectangular plot with frequency as a parameter to obtain a Nichols plot.

For audio systems with nearly uniform time delay at all frequencies, the magnitude versus frequency portion of the Bode plot may be all that is of interest. For design of control systems, any of the three types of plots [Bode, Nyquist, Nichols] can be used to infer closed-loop stability and stability margins (gain and phase margins) from the open-loop frequency response, provided that for the Bode analysis the phase-versus-frequency plot is included.

### Nonlinear frequency response

If the system under investigation is nonlinear then applying purely 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 that allow the user to analyze complex nonlinear dynamic effects.^{[2]} The nonlinear frequency response methods reveal complex resonance, inter modulation, and energy transfer effects that cannot be seen using a purely linear analysis and are becoming increasingly important in a nonlinear world.

## Applications

In electronics this stimulus would be an input signal.^{[3]} In the audible range it 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 requirements differ depending on the application.^{[4]} 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, a frequency response of 400–4,000 Hz, with a tolerance of ±1 dB is sufficient for intelligibility of speech.^{[4]}

Frequency response curves are often used to indicate the accuracy of electronic components or systems.^{[3]} 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.^{[3]}

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.

## See also

- Impulse response
- Bode plot
- Bandwidth (signal processing)
- Audio system measurements
- Transient response & steady-state response
- Spectral sensitivity

## References

- Notes

- Bibliography

- Luther, Arch C.; Inglis, Andrew F.
*Video engineering*, McGraw-Hill, 1999. ISBN 0-07-135017-9 - Stark, Scott Hunter.
*Live Sound Reinforcement*, Vallejo, California, Artistpro.com, 1996–2002. ISBN 0-918371-07-4

## External links

- University of Michigan: Frequency Response Analysis and Design Tutorial
- Smith, Julius O. III: Introduction to Digital Filters with Audio Applications has a nice chapter on Frequency Response