Definition
Frequency response is the quantitative description of a system’s output magnitude and phase as a function of input signal frequency. For linear time‑invariant (LTI) systems, it is commonly expressed as the complex-valued transfer function $H(j\omega)$, where $\omega$ is angular frequency, and its magnitude $|H(j\omega)|$ and phase $\angle H(j\omega)$ together constitute the frequency response.
Overview
The concept is fundamental in electrical engineering, acoustics, control theory, and signal processing. It characterizes how filters, amplifiers, loudspeakers, control loops, and other devices modify sinusoidal inputs of varying frequencies. Frequency response is typically visualized with magnitude (often in decibels) and phase plots versus frequency, known collectively as Bode plots. In audio engineering, the term often refers to the measured sound pressure level output of a loudspeaker or headphones across the audible spectrum (≈20 Hz–20 kHz). In control systems, the frequency response of the open‑loop transfer function determines stability margins such as gain and phase margins.
Etymology/Origin
The word “frequency” derives from the Latin frequentia meaning “a frequent occurrence.” “Response” comes from the Latin respondere meaning “to answer.” The compound term “frequency response” began to appear in scientific literature in the early to mid‑20th century alongside the development of frequency‑domain analysis methods for electrical circuits and control systems. Its use became widespread with the introduction of Bode’s graphical techniques in the 1930s.
Characteristics
| Aspect | Description |
|---|---|
| Magnitude response | Shows how the amplitude of the output varies with frequency; expressed as a ratio, decibels (dB), or percent. |
| Phase response | Indicates the phase shift introduced by the system at each frequency; measured in degrees or radians. |
| Bandwidth | Frequency interval over which the magnitude remains within a specified range (commonly within –3 dB of the peak). |
| Roll‑off | Rate at which magnitude declines outside the passband, expressed in dB per octave or decade. |
| Resonant peaks | Frequencies where magnitude rises sharply due to system resonances. |
| Group delay | Derivative of phase with respect to angular frequency; measures the delay of amplitude envelopes. |
| Linearity | For LTI systems, frequency response fully characterizes behavior; non‑linear systems do not have a unique frequency response. |
| Measurement methods | Sine‑wave sweep, chirp signals, or fast Fourier transform (FFT) analysis of impulse responses. |
Related Topics
- Transfer function
- Bode plot
- Phase response
- Amplitude (magnitude) response
- Filter design (low‑pass, high‑pass, band‑pass, etc.)
- Impulse response (time‑domain counterpart)
- Nyquist plot
- Control system stability margins (gain margin, phase margin)
- Audio equalization
- Signal‑to‑noise ratio (SNR) considerations in frequency‑domain analysis
These topics share the common principle of analyzing or shaping system behavior with respect to frequency.