Frequency Response - Estimation and Plotting

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 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.