The most informative frequency domain plot, by far, is the Nyquist plot, Named after Harry Nyquist of Bell Telephone Laboratories, famed for his work on feedback circuit analysis, the Nyquist plot makes visual simultaneously the following viewpoints of the signal:

1. The real part
2. The imaginary part
3. The magnitude
4. The phase angle between the real and imaginary parts
5. The polarity of the system
6. The causality or non-causality of the time behavior
7. The presence of resonant systems
8. Non-signal synchronization by frequency

The Nyquist plot can be used in sound system work, where in acoustics the real part is particle pressure and the imaginary part is particle velocity (near a boundary or in a standing wave) and in impedance measurements where the real part is the resistance and the imaginary part is the reactance.

Fig. 1 - The analytic signal di5played as a Heyser Spiral

Let’s label some key observations to be made about the Nyquist plot of a pair of loudspeakers. Dennis Gabor proposed the “analytic signal” which Richard Heyser developed into what has since been named the Heyser Spiral. The Nyquist plot is the “end view” shadow of the complex analytic signal

Fig. 2 - The “end view” of the Heyser Spiral (Nyquist)

An Interesting Anomaly

Apparent non-causal’ signals are being encountered more frequently these days. A non-causal signal is one that arrives before it is sent. Impossible you say? Yes, in the real world, to the best of our knowledge that is true. But in the world of measurements, thanks to digital technology, we find our measurements telling us a signal is not causal Let’s look at an example:

Fig. 3 - ETC of a two-way loudspeaker. Using the woofer arrival to set the reference time and measuring the full-range response causes the Nyquist to rotate counterclockwise above crossover.

We have a two-way loudspeaker with a crossover frequency of 2 kHz. The energy-time curve (ETC) shows the tweeter arriving before the woofer (Fig. 3). We now choose to call the woofer arrival the time point of reference, while attempting to measure the full-range response. When we do this our measurement instruments will see the tweeter energy as arriving before we sent the signal and the Nyquist will rotate counter clockwise above crossover, indicating non-causality on the instrument, which is not aware that the woofer arrival was chosen as the time reference. Note that in the diagram the cursor is set at the point where the reverse rotation begins, the cursor coordinates being indicated to the lower left of the plot. This situation can also occur with a digital crossover net work where one frequency range has one delay associated with it and the other frequency range still another delay. In measuring phase response, choice of a correct origin and time is paramount if you want a meaningful measurement of the phase response of a device rather than a delayed measurement of the signal path.

For a quick overview of the acoustic signature from a loudspeaker array I know of no tool that is the peer of the Nyquist plot. When large complex acoustic arrays require a rapid overview that will allow analysis of what problems, if any, are present, the Nyquist instantly tells which component relationships are incorrect, be it signal synchronization, digital crossover or other delays not obvious. Also obvious are inverted polaritys, unexpected resonances or even measurement mistakes in choosing the correct signal origin.

The Nyquist plot, as embodied in contemporary acoustic analyzers, usually computes at the bottom of the screen the values for frequency, real part, imaginary part, phase, and magnitude for a given cursor position on the screen. Moving the cursor allows clockwise or counter clockwise frequency changes to be observed.

Ron Bennett, a number of years ago, provided me with a program that allows acquisition of the real and imaginary part separately, i.e., acoustically as pressure and velocity: electrically as voltage and current; impedance as real and reactive. After acquisition it then forms a complex analytic signal from those two parts for display.

One innovative use is to see the complex signal formed by the left ear/right ear signals using ITE microphones.

What in 1928 was a truly laborious process, is in 1998 a major convenience and an important tool. dbd

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