Understanding Phase Response Curves from Different Perspectives

It can also be found from Figure 13 that the peak of the pulse is slightly offset and located on the right side of 0 ms. The pulse response depicted in the form of image points is shown in Figure 14. It can be seen that the initially arriving pulse signal is accurately aligned at 0 ms.

Most analyzers do not have the function of converting continuous curves into image points. Therefore, it is very important to observe the transfer function (amplitude response and phase response) in the frequency domain and the impulse response in the time domain at the same time. What happens when the automatic delay search function is used? By observing Figure 15, we find that:

• When the impulse response is periodically shifted, there will be no change in the frequency response;

• The impulse response peak was moved to 0 ms;

• The phase response becomes flatter, but note that it is positive before 10,000 Hz. The positive phase value indicates that the impulse response is slightly over-shifted, especially given the flat midrange frequency response and the rolling off at high frequencies (above 8,000 Hz).

Figure 16 shows an example of a positive phase response. A 2-inch loudspeaker unit is installed in a small closed box, and the test microphone is 1 meter away. This measurement is performed without any electronic filter intervention, and is a minimum phase system. The following results can be obtained from Figure 16:

• The red dashed line is the minimum phase curve calculated in ARTA based on the amplitude response. The result is basically consistent with the actual measured phase curve (black solid line), which means that the transmission delay removed from the measured data is correct.

• The phase values ​​are positive up to 20,000 Hz, which is due to the increasing amplitude response.

Figure 17 is a comparison of Figure 16 and Figure 12. Note the two green rectangular areas: below 200 Hz, the frequency response is rising, so the phase response starts to show positive values ​​as shown in Figure 12A; in the high frequency range, around 10,000 Hz, how the measured and calculated values ​​match those in Figure 12C.

By periodically shifting the impulse response, it is easy to change the phase response or find the position where the phase response is the flattest. However, in actual work, one must consider one question: Is the obtained phase response meaningful?

2.3 Correct interpretation of phase response

In this article, "correct interpretation" means finding a phase response that is valuable for engineering applications. Take a look at the two sets of phase response curves in Figure 18 from the same measurement file:

• The red curve looks like the better looking (flatter) phase response curve;

• The black curve has many bends, especially above 1 000 Hz;

• The black curve has the propagation delay correctly removed, the red curve is the result of placing the pulse peak at 0 ms (using the automatic delay finder function), and the polarity of the input signal is inverted.

There are many ways to make the phase response look flatter, but this does not mean that a flat curve is valid for engineering applications. Therefore, more discussion will be focused on the black curve.

Let's discuss another example of phase response for comparison. Figure 19 shows two sets of measurement data for a 12-inch two-way horn speaker. The black curve is measured using the analog crossover circuit inside the speaker (passive crossover), without DSP intervention, and only requires a power amplifier to drive; the red curve is measured using DSP to divide the tweeter and woofer (active crossover), and each unit requires a power amplifier to drive.

When the propagation delay of the impulse response is properly removed, the curve can be correctly interpreted. The following observations and discussions focus only on topics related to phase.

(1) The calculated minimum phase response is different from the actual measured phase response

This is normal. This is because this two-way speaker is not a minimum phase system, whether it uses passive or active crossover. A single speaker unit may be a minimum phase system, but when the tweeter and woofer are put together as a speaker system, the introduction of the crossover destroys the possibility of it being a minimum phase system, and naturally the phase response cannot be calculated through the frequency response.

(2) The black curve has a significant phase drop/fold in the high frequency range, especially above 1000 Hz.

The black curve is measured when the speaker is using a passive crossover. If you look at the impulse response, you can see that there is a small bump before the impulse peak, which should be the impulse of the woofer, which is weaker than the higher impulse peak. This is because the high-frequency energy in the impulse response is dominant; when a low-pass filter is applied to the woofer (cutoff frequency below 2000 Hz), the height of the impulse will be significantly reduced. The higher impulse peak should be the tweeter, which arrives at the test microphone later (about 0.7 ms later).

Since the speaker uses a passive crossover, we can surmise that the tweeter voice coil is located further back than the woofer voice coil, which is usually due to the depth of the high-frequency horn; the direct sound from the woofer arrives at the microphone first, and the tweeter sound arrives 0.7 ms later. As discussed earlier in Figure 5, the additional transmission delay caused by the tweeter's lag compared to the woofer's first arrival results in a high-frequency phase drop/fold and a higher group delay value (discussed later).

(1) The black curve has a higher group delay value in the high frequency band (above 1000 Hz).

Continuing from the previous discussion, a positive group delay of about 0.7 ms can be observed above 1000 Hz due to the difference in arrival time between the tweeter and woofer. This clearly shows that the tweeter arrives 0.7 ms later than the woofer.

The red group delay curve has a value of 0 ms above 1 000 Hz. By comparing the red and black group delay curves and the impulse response graph, it can be inferred that during the measurement of the red curve, the DSP not only performed active crossover, but also deliberately added a little extra delay to the woofer to adjust the time difference of the direct sound arrival of the high and low unit.

(2) The red group delay curve has a higher group delay value at low frequencies (below 100 Hz)

If you look at the frequency response, the red curve has a higher amplitude than the black curve. This means that in addition to frequency division and adjusting the time difference of the speaker units, the DSP also performs a small gain increase at around 65 Hz on the woofer.

The intervention of this parametric equalization will increase the group delay. Looking at the red curve in Figure 18, after moving the peak of the impulse response of this passive speaker to 0 ms, the phase response becomes flatter. However, this flat phase curve is meaningless for engineering applications, and it even reverses the polarity of the input signal. Maybe it can play a role in the marketing field.

3 Conclusion

The conclusion discussed in this article can be summarized in one sentence: Only by correctly judging and removing the transmission delay can a phase response that is valuable for engineering applications be obtained.