Understanding Phase Response Curves from Different Perspectives

Nowadays, in the digital audio world, audio analyzers based on Fast Fourier Transform (FFT) and Time Delay Spectrum (TDS) functions can easily reveal the amplitude response and phase response of the speaker. These software analyzers are not expensive and are widely used in live sound reinforcement, fixed installation and speaker research and development. Familiar software includes ARTA, Smaart, Systune and EASERA, etc.

Phase response is a frequently discussed topic. Many practitioners use the delay catcher, automatic delay catcher or phase compensation functions in the software to check the phase response. However, most of them do not understand the basic principles behind this function. This article is aimed at readers who have experience in loudspeaker measurements and discusses how to set a correct time reference value for interpreting the phase response.

1 Basic Principles of Phase Response

1.1 Impulse Response and Digital Audio

First, let’s discuss the impulse response graph, as shown in Figure 1, which shows an ideal impulse response (Dirac pulse) with a peak at 1 ms. The information displayed in this time domain graph includes:

• One pulse of energy occurs at a time point of 1 ms;

• The transmission delay of this signal is 1 ms;

• A value of 1 on the Y-axis means that the signal amplitude is +1.

A careful analysis of Figure 1 shows that the impulse response starts a little earlier than 1 ms. The question that follows is, does the actual time of the signal arrival really start a little earlier than 1 ms?

It helps to visualize the impulse response if you connect the dots. When Figure 1 is plotted with only the plot points, as shown in Figure 2, it is clear that the pulse consists of only one energy spike that arrives at exactly 1 ms. Understanding where the impulse response begins is the first step in reading the phase response. Not all analyzers are capable of displaying plot points, but it is important to understand how the curve is formed in digital equipment.

Note that in the digital domain, all values ​​are discrete (non-continuous) and the accuracy of the graph is determined by the sampling rate. A Dirac pulse consists of only one sample.

1.2 Transmission Delay and Phase Response

Why do we need to consider the starting point of the pulse? Let's first verify the effect of transmission delay on phase response. Transmission delay may be caused by the following factors.

(1) Sound propagation time

Sound propagation time refers to the time it takes for direct sound to reach the microphone. Sound travels in air at a speed of approximately 344 m/s (20 °C). It is important to understand how long it takes for sound to travel in air.

(2) Processing delay

Digital signal processing, D/A conversion and A/D conversion usually require a certain amount of processing time. The latency introduced by the A/D and D/A conversion of a digital loudspeaker management system is typically around 2 ms. If the processing is heavy, the processor will need more processing time and the signal transmission delay will increase accordingly.

1.3 Finite Impulse Response Filter

Using a finite impulse response linear phase filter may result in an increase in propagation delay. Depending on the application, the processing delay introduced by the filter may be as little as 1 ms or greater than 500 ms. In Figure 3, a 0.5 ms propagation delay can be observed by measuring a "perfect" loudspeaker using a dual-channel fast Fourier transform. The impulse response is the blue pulse in the figure (assuming a "perfect" test microphone is about 17 cm from the loudspeaker). The black pulse shows the case when the impulse peak is exactly at 0 ms after the propagation delay is removed. This result is obtained by periodically shifting the impulse response (which will be explained later). The red pulse shows the case when the propagation delay is removed by an additional 0.5 ms. The red pulse is a non-causal response, that is, the output is not caused by the input. The pulse before 0 ms can be considered as the output obtained before the input signal enters the system.

Figure 4 shows the phase response of the loudspeaker system when the propagation delay has been correctly removed, with the peak of the impulse at 0 ms. Note that the phase response is flat at 0°, and the group delay is also 0°.

The group delay is the negative value of the slope of the phase curve. The group delay can be expressed as follows:

Where ω is the angular frequency or the frequency times 2π; φ is the phase; and τ is the group delay.

FIG5 shows the phase response of the loudspeaker when the 0.5 ms transmission delay is not removed. In order to observe the phase response more clearly, FIG5(d) shows the phase response of the unfolded part after magnification.

By using the formula 1 = T × f (T is the period, in seconds; f is the frequency, in Hz), we can calculate that 0.5 ms is one period of 2 000 Hz. The period of a sound wave is the time it takes to move 360°. Therefore, the phase of 2 000 Hz is -360° (a negative phase angle means that the phase of that frequency is lagging). For 1 000 Hz, it takes 1 ms to complete one period, so the phase it presents is -180°. In other words, with a 0.5 ms transmission delay, 1 000 Hz is slowed down by half a period, while 2 000 Hz is slowed down by one period. Without removing the transmission delay, all frequency components of the input signal are shifted by the same amount of time. This is not phase distortion. If the X-axis (frequency axis) in Figure 5 is changed from a logarithmic scale to a linear scale, a straight line can be seen, as shown in Figure 6. Note: A negative phase angle means that the phase of that frequency is lagging.

The group delay can also be observed in Figure 5(d), corresponding to a 0.5 ms transmission delay. What happens when the pulse is shifted by an excess of 0.5 ms? The different directions of the phase response can be observed by overlaying the phase responses in Figure 5 (0.5 ms transmission delay - blue curve) and Figure 7 (0.5 ms transmission delay removed in excess - red curve) in Figure 8. Excessive removal of transmission delay results in positive phase and negative group delay values.

It is important to understand that even a small propagation delay can have a significant effect on the phase response curve at high frequencies.

2 Correct interpretation of phase response and removal of transmission delay

2.1 Remove transmission delay

Since propagation delays cause a linear phase shift, they can be easily predicted and calculated. A common method to remove propagation delays is to periodically shift the signal so that the peak of the impulse response is at 0 ms. This function is often named Auto Delay Finder, Auto Peak Finder, Nomalize Max to Zero, etc. in measurement software. However, this method can give a misleading reading of the phase response, for example, when a low-pass filter is used, resulting in weak energy above 1 000 Hz.

Different software uses different methods to remove transmission delay (see Figure 9-11). Some software is equipped with a function to directly align the pulse peak to 0 ms, while other software requires manual setting of the cursor position. The cursor position will define a left rectangular window or cutoff position.

Figure 10 Systune removes transmission delay by manually entering the delay amount or using the “Peak” button

It is important to remember that the impulse response of a loudspeaker begins at the moment the sound reaches the test microphone. It is important to understand that the peak of the impulse response does not always coincide exactly with the initial peak reaching the microphone. Using the peak of the impulse response as a reference point can be misleading in the interpretation and calculation of the phase response.

2.2 Introduction to Minimum Phase

Minimum phase means that the relationship between the amplitude response and the phase response can predict each other. A change in the amplitude response will bring about a change in the phase response. This relationship can be calculated using the Hilbert transform. Figure 12 shows the relationship between the amplitude response and the corresponding minimum phase response.

Most loudspeaker systems are not minimum phase systems, but most speaker drivers are. Another example of a minimum phase system is an infinite impulse response filter, except for the all-pass filter.

When the phase value is positive, it means that the phase of that frequency is advanced. In a minimum phase system, if the frequency response (from low frequency to high frequency) is rising, the phase value is positive. For example, when a high-pass filter is added to a speaker unit, this speaker unit will usually have this situation in the frequency range where the filter attenuation goes from maximum to 0 (no attenuation). Another example is when using a low-pass filter, the frequency response (from low frequency to high frequency) is falling in the frequency range where the filter attenuation goes from 0 (no attenuation) to maximum. In a minimum phase system, this will result in a negative phase value (phase lag).

Figure 13 shows the response curve of a speaker without a crossover network after being processed by an IIR filter. The cutoff frequency of the high-pass filter is 60 Hz, and the cutoff frequency of the low-pass filter is 12,500 Hz. By observation, it can be found that:

• In the low frequency region (below 200 Hz), the phase value is positive because of the effect of the high-pass filter (the frequency response shows an upward trend);

• In the high frequency region (above 8 000 Hz), the phase value is negative because of the effect of the low-pass filter (the frequency response shows a decreasing trend);

• Around 1000 Hz, the phase value is close to 0. This is a result of the fact that there is no change in the frequency response in the mid-frequency band.