Frequency response refers to the phenomenon where the sound pressure produced by a speaker increases or decreases as the frequency of an audio signal—fed into the system with a constant voltage output—changes, while the phase also shifts with frequency. This interrelated variation between sound pressure, phase, and frequency is precisely what we call the frequency response. It also describes the range of frequencies that an audio system can accurately reproduce within acceptable amplitude limits, and the degree of variation in the signal across this range is similarly referred to as the frequency response, or frequency characteristics. Within the specified frequency range, the ratio of the maximum to minimum output voltage amplitude is measured and expressed as decibels (dB) to quantify its unevenness. In the context of power quality, frequency response typically refers to how the impedance of a system or measuring sensor changes with frequency.
Frequency Response
The system's steady-state response characteristics to a sinusoidal signal. Steady state refers to the system's behavior after the transient process has ended. The system's frequency response consists of two key components: the magnitude-frequency characteristic and the phase-frequency characteristic. The magnitude-frequency characteristic illustrates how the gain varies with changes in signal frequency, while the phase-frequency characteristic reveals the relationship between phase distortion and different signal frequencies. By examining the frequency response, one can intuitively assess the system's ability to accurately reproduce signals as well as its capability to filter out noise. In control theory, the frequency response provides a convenient framework for analyzing system stability and other dynamic behaviors. Moreover, the concept of frequency response plays a crucial role in system design. Introducing appropriately designed compensators (refer to methods for controlling and correcting systems) allows engineers to fine-tune the frequency response, thereby enhancing overall system performance. Analysis and design techniques that rely heavily on frequency response are collectively known as the frequency-domain method, which remains one of the fundamental approaches in classical control theory.

In control engineering, it is also known as the frequency response—it represents the system's steady-state response to sinusoidal signals of varying frequencies.
Determination method
Analytical Method
The physics-based theoretical calculation method is only suitable for cases where the system's structural composition can be easily determined. Once the system's structure is specified, the corresponding physical laws can be applied to derive and compute the system's frequency response. The accuracy of the analysis, however, depends on how precisely the system's structure is understood. For complex systems, this analytical approach often involves a significant amount of computational effort.

Experimental method
The method of direct measurement using instruments is particularly useful when the system structure is difficult to determine. A common experimental approach involves applying a sinusoidal signal as the test input and selecting several frequency values within the range of interest. At each selected frequency, the amplitude and phase angle of both the input sinusoidal signal and the steady-state output are measured separately. The ratio of the output amplitude to the input amplitude as a function of frequency defines the magnitude-frequency response, while the difference in phase angle between the output and input signals, also plotted against frequency, reveals the phase-frequency response.
Response Chart
When analyzing and designing control systems using the frequency response method, engineers often start their analysis by examining the curve plots of the frequency response. The most common types of frequency response graphs are the Nyquist plot, Bode plot, and Nichols plot.
Nyquist Plot
Also known as a polar plot, it is a graphical representation of the frequency response \( G(j\omega) \), showing how its magnitude \( |G(j\omega)| \) and phase angle \( \angle G(j\omega) \) vary as the frequency \( \omega \) changes from zero to infinity. One key advantage of the polar plot is that it clearly reveals the distribution of frequencies across the response curve. To construct the polar plot, you must calculate the magnitude \( |G(j\omega)| \) and phase angle \( \angle G(j\omega) \) for each selected value of \( \omega \); these values then define a vector \( G(j\omega) \) on the polar plot. The Nyquist plot, in particular, is the trajectory traced by the terminal point of the vector \( G(j\omega) \) as \( \omega \) varies continuously from zero to infinity.

Bode Plot
Also known as a logarithmic coordinate plot, the Bode plot consists of two graphs: the logarithmic magnitude plot and the phase-angle plot of the frequency response \( G(j\omega) \). In the logarithmic magnitude plot, the frequency axis is scaled logarithmically, while the magnitude axis is labeled in \( 20\log|G(j\omega)| \), with the unit expressed in decibels (dB) and using a linear scale. Similarly, in the phase-angle plot, the frequency axis is also logarithmically scaled, whereas the angle axis remains linear, measured in degrees. One key advantage of the Bode plot is that it simplifies the multiplication of magnitudes into the addition of their logarithmic values. Moreover, when only rough information about the frequency response is needed, the Bode plot can often be approximated by drawing asymptotic lines composed of straight segments—making the plotting process remarkably straightforward. If a more precise curve is required, minor adjustments can easily be made based on the asymptotic approximation, keeping the overall process simple and efficient. 
 

Nichols Chart
Also known as the logarithmic magnitude-phase plot, it is a graphical representation on rectangular coordinates where the logarithmic magnitude \(20 \log|G(j\omega)|\) and the phase angle \(\angle G(j\omega)\) are plotted as functions of frequency \(\omega\). The logarithmic magnitude-phase plot can be easily constructed using the magnitude and phase characteristics derived from the Bode diagram. One of the key advantages of the Nichols plot is that it allows for a straightforward assessment of a control system's relative stability. 

Performance
The system's transient response has a definite relationship with its frequency response, which can be determined using mathematical methods. However, aside from first- and second-order systems, this approach often requires significant time and, in many cases, offers limited practical value. A more common method involves directly estimating the performance of the system's transient response based on key characteristics of the frequency response. These primary features include gain margin and phase margin, resonant peak and resonant frequency, bandwidth, and cutoff frequency.
Gain Margin and Phase Margin
It can provide information on whether the control system is stable and how much stability margin it has.
Resonance peak Mr and resonance frequency ωr
Mr and ωr are defined as the maximum value of the magnitude-frequency response |G(jω)| and the corresponding frequency, respectively. For high-order linear time-invariant systems with a pair of conjugate complex dominant poles (refer to the root-locus method), when the Mr value falls within the range of (1.0 to 1.4)M0, the system can achieve relatively satisfactory transient performance. Here, M0 represents the magnitude of the frequency response at ω = 0. Meanwhile, the magnitude of ωr indicates the speed of the system's transient response: the larger the ωr value, the faster the system’s output response will be under a unit-step input.
Bandwidth and Cutoff Frequency
The cutoff frequency ωc is defined as the critical frequency at which the magnitude of the frequency response |G(jω)| drops to 0.7M0 and continues to decrease thereafter. The corresponding frequency range, 0 ≤ ω ≤ ωc, is referred to as the bandwidth. The significance of the cutoff frequency is that the system effectively filters out signal components with frequencies higher than ωc, while allowing lower-frequency components to pass through with minimal attenuation—or even slightly less attenuation. From the perspective of accurately reproducing the input signal, it’s often desirable to have a wider bandwidth, as this corresponds to shorter rise times and faster response speeds. However, from the standpoint of suppressing high-frequency noise, an excessively wide bandwidth may not be ideal. Therefore, determining the appropriate bandwidth requires a balanced, comprehensive consideration.
Scope
The frequency range refers to the span between the lowest and highest effective playback frequencies that an audio system can reproduce. Frequency response, on the other hand, describes how the sound pressure produced by a speaker changes—either increasing or decreasing—as the frequency varies when the speaker is connected to an audio signal output at a constant voltage. Additionally, the phase of the sound also shifts with frequency. This interrelated relationship between sound pressure, phase, and frequency (measured as the degree of change) is known as the frequency response, typically expressed in decibels (dB). Sometimes, the concepts of frequency range and frequency response are used interchangeably, collectively referred to as "frequency response."

The frequency response of an audio system is typically depicted using a frequency-response curve, where the vertical axis—measured in decibels—represents power, and the horizontal axis—displayed on a logarithmic scale—shows frequency. When the sound power drops by 3 dB below the normal level, this point is referred to as the high-frequency cutoff and low-frequency cutoff of the frequency response. The range of frequencies between these two cutoff points defines the device’s overall frequency response. Meanwhile, the curves illustrating how sound pressure and phase lag vary with frequency are known as the "magnitude-frequency characteristic" and "phase-frequency characteristic," respectively—collectively referred to as the "frequency characteristic." This is a critical metric for evaluating the quality of a speaker system, as it directly correlates with both performance and price: the lower the decibel value, the flatter the speaker’s frequency response curve, the less distortion, and ultimately, the higher the speaker’s overall performance.
Theoretically, a frequency response range of 20–20,000 Hz is sufficient. While sounds below 20 Hz are inaudible to the human ear, they can still be detected by other sensory organs—specifically, the subtle perception of bass intensity. To faithfully reproduce the full range of musical instruments and speech signals, amplifiers must strive for high-fidelity performance, ensuring that all harmonic components of the audio signal are accurately reproduced. Therefore, the amplifier’s frequency bandwidth should ideally extend downward well below 20 Hz and upward beyond 20,000 Hz. It’s worth noting that different signal sources—such as radio tuners, tape recorders, and laser disc players—may use varying methods to specify their frequency responses. For instance, the European Broadcasting Union mandates an FM stereo broadcast frequency response of 40–15,000 Hz with tolerances of ±2 dB. Meanwhile, the International Electrotechnical Commission sets a minimum frequency response standard for tape recorders: 40–12,500 Hz, with tolerances of ±2.5 dB at the lower end and ±4.5 dB at the upper end—though actual performance typically exceeds these specifications significantly. Notably, CD players boast an impressive frequency response extending up to 20,000 Hz, with exceptionally low-frequency reproduction capabilities—often down to just a few hertz. This remarkable ability to handle both ultra-high and extremely low frequencies is one of the key factors contributing to the superior sound quality of CD playback systems.
Amplifier circuit
Frequency response is a technical metric that measures an amplifier circuit's ability to handle input signals of different frequencies.
Due to the inherent inter-electrode capacitances of the amplifying devices themselves—whether bipolar junction transistors or field-effect transistors—and, in addition, the presence of reactive components in the amplifier circuit, the circuit's gain varies depending on the frequency of the input sinusoidal signal, effectively becoming a function of frequency. This functional relationship is known as the amplifier circuit's frequency response or frequency characteristics.