Frequency Response Basics
A system's frequency response is how it scales and phase-shifts sinusoids of each frequency, the basis of Bode and Nyquist analysis.
How Systems Treat Sinusoids
A defining property of a linear time-invariant system is that a sinusoidal input produces a sinusoidal output at the same frequency, changed only in amplitude and phase. The frequency response is the full record of how the amplitude scaling and phase shift depend on frequency. It is found by evaluating the transfer function along the imaginary axis, setting s equal to j times the frequency.
Magnitude and phase
- Magnitude: the ratio of output amplitude to input amplitude at each frequency, how much the system amplifies or attenuates.
- Phase: the shift in timing between output and input sinusoids at each frequency, measured in degrees.
- Together they completely characterize an LTI system's steady-state response to any sinusoid, and by superposition to any signal.
Why it is so useful
The frequency response separates a system's behavior by frequency band, which matches how requirements are naturally stated. Tracking and disturbance rejection are low-frequency goals; noise rejection and robustness are high-frequency goals. Viewing the system across frequency lets each goal be addressed in its own band, which is the essence of loop shaping.
Bandwidth
A central single-number summary is the bandwidth: the frequency up to which the system responds effectively, often defined where the closed-loop magnitude drops to about 70 percent (minus 3 dB) of its low-frequency value. Bandwidth correlates with speed of response: a wider bandwidth means a faster system, though it also admits more noise.
Measured directly
Frequency response has the practical virtue of being measurable without a model. Drive the real plant with sinusoids at a range of frequencies, record the amplitude ratio and phase shift at each, and the frequency response is obtained empirically. This measured data feeds directly into Nyquist and Bode analysis, letting design proceed even when a first-principles model is unavailable.
From the frequency response flow all the classical frequency-domain tools: Bode plots for loop shaping, the Nyquist plot and criterion for stability, and gain and phase margins for robustness.