The definition of the word response – “a reaction to a specific stimulus” -happens to offer a valuable lesson in system testing. The circumstances ofa test can scew the results, sometimes in a more favorable light than isdeserved; that’s one of the dangers of blindly accepting equipmentspecifications. With system testing, if you don’t understand the varioustypes of system response, you might not be using your testing to fulladvantage. Therefore, the question “What is its response?” in regard to aloudspeaker system should always elicit the return question, “to what?”
Loudspeakers can have time response (transient behavior), steady-statefrequency response, directional response (polar), nonlinear response tolinear input signals (distortion), and dynamic response (compression), justto name a few. Let’s look at three useful perspectives: the time domain,the frequency domain and the joint domain.
The time domainThe term “response” was originally applied to how a loudspeaker reproduceda time-limited voltage impulse applied to its terminals. A well-designedloudspeaker will reproduce the actual impulse shape remarkably well plus anovershoot with exponentially decaying oscillations. An impulse stimuli thatis infinite in amplitude and instantaneous in time contains allfrequencies. An impulse in the time domain will always be broadband inenergy content.
Is it possible to have a single frequency impulse? A good example is DonKeele’s test signal. It is 6.5 cycles at specific frequencies, making thespectral content about 1/3 octave wide. The impulse response shown inFigure 1 is of a 6 inch (152 mm) loudspeaker excited by a full-rangefrequency sweep inverse Fourier transformed into the time domain.
We can easily see that the loudspeaker is in polarity because a positivepulse at the microphone produced a positive pulse at the loudspeaker. Wecan also see that it has a reasonably well-damped transient responsebecause its exponential decay is brief.
We can construct a viewpoint based on the Hilbert Transform – a 90 degreesthe view of what is called the analytic signal. (See Figure 2.)Mathematically, the impulse response is called the real part and thedoublet response is called the imaginary part.
The analytic signalRichard C Heyser (1931-1987) was the first to apply Dennis Gabor’s(1900-1979) analytic signal concept to audio and acoustics, and thereforeits full display has become known to a generation of engineers as theHeyser spiral. (See Figure 3.) As we will see, there are two Heyserspirals, one in the time domain, shown here, and one in the frequencydomain.
The Heyser spiral, in the time domain shown here, is for the identicalmeasure of the loudspeaker with the impulse and doublets previously shownand which can be seen to be the “shadows” of the three dimensional complexanalytic signal shown spread over the time scale T. Still another shadow ofthe complex analytic signal is the end view of the complex signal calledthe polar.
Log squared impulse responseBy placing the squared amplitude of the analytic signal on the logarithmicscale (i.e., in decibels), it becomes easier to detect significantreflection energies. (See Figure 4.) Heyser thought that squaring only thereal part of the response missed subtleties inherent in the complexinterweaving of the doublet energies with the real parts’ energies. Toobserve these subtleties, he developed the energy time curves. (See Figure5.) Once again, remember that all these views are of the same loudspeakermeasurement and simply represent differing viewpoints of the same event.
The frequency domainMost of us were taught that frequency response was the logarithmicfrequency scale view of magnitude vs. frequency. There is also a phaseresponse depicting the time behaviors frequency by frequency – thedifference in travel time from the source to the receiver, frequency byfrequency, within the constraints of fRxTR = 1.0, where fR is frequencyresolution and TR is time resolution in seconds. (See Figure 6.)
The frequency domain Heyser spiralThe Heyser spiral in the frequency domain shows the complex analytic signalalong the frequency line F. The real part shadow is the coincidentresponse; the imaginary part is the quadrature response. The end view isthe polar or classic Nyquist response, developed by Harry Nyquist(1889-1976). (See Figure 7.)
The magnitude of the frequency response is, for all points on both curves,calculated with the following equation:
10 log (real2 + imag2)= mag
and the phase response is found by
for all points on both curves.
The Nyquist plot is the magnitude as the distance from the crossing of thecoordinates to the plot; the phase angle is the angle between the linedrawn from the crossing of the coordinates to the right horizontalcoordinate.
Joint domain displaysOur final viewpoint is obtained by slightly “smearing” both frequency andtime domain information to gain a perspective on the behavior of eachfrequency area over increasing time to a full-frequency range impulse (i.e.an impulse capable of exciting all frequencies equally). (See Figure 8.)
One of the great insights this format allows the engineer to see and studyis the role peaks in response have in the sound system ringing and how, forexample, equalization of peaks back to uniform response also removesringing in time.
A cautionPerhaps the most overlooked basic in measurement is the failure to realizethat fRxTR = 1.0 applies to every form of measurement known. Although manyanalyzers will display what appears to be higher resolutions, keep in mindthat fR = 1/TR and TR = 1/fR, no matter what the screen display says.