In order to get the most out of your subwoofer, it is critical that it is correctly integrated with the rest of the system. Phase alignment is an important aspect of integration. For any speaker type, this is important. For sealed main speakers, it is simple. Vented main speakers which are common, are a little more difficult, however in this article we present a method of aligning them correctly that is actually quite simple.
Phase and delay
Phase and delay are two concepts related to each other. A complete cycle time is defined as 360 degrees. They have increments in a fashion similar to a clock. It goes back to 0 phase after it reaches 360 degrees. Filter theorists discovered that the phase lag from low pass filters (the filter used on the subwoofer to crossover to front speakers) has a phase relation resembling a delay in the signal. A class of so called "linear phase" filters are designed based on perfecting this relationship. For linear phase filter, it has a phase delay that is proportional to frequency and therefore the delay D can be calculated as D=(phase shift)/frequency. Consequently, they are also called constant delay filters as the time delay is same for all frequencies.
On the other hand, the phase shift caused by the high pass filters (those used on front speaker to crossover to subwoofer) is a phase lead which is in the opposite direction of phase lag. One can imagine the phase lag as clockwise movement and phase lead is counter clockwise movement. As with clock, a clock arm move clockwise by x o'clock will coincide with counterclockwise move by y o'clock if x+y=12*m. (m is integer). And in phase that means x+y=360*m.
How to adjust phase with a home theatre receiver
The simple method is to compensate by changing the speaker distance setting on your receiver. Bass management in HT receivers has a speaker distance adjustment which process the signal on digital domain. If one puts distance of the sub x feet further away than its physical distance relative to other speakers, the HT receiver will put out the signal to the sub x/1000 sec before it puts out signals to other channels. That essentially puts a negative delay on the sub which can be used to reduces the "phase lag" on the sub and therefore reduces the phase difference between the sub and the front speakers.
This trick enables us to use the speaker distance as a tool for phase adjustment between subwoofer and front speakers. Since it is a constant delay adjustment, the phase adjustment is not fixed for all frequencies, instead it is proportional to frequency. That does not affect its effectiveness as our objective is to get correct phase alignment at the crossover frequency.
Read more about calculating the delay required with a worked example 
Linkwitz Riley filters
The filter setup in today's HT receiver is based on Linkwitz-Riley filters. Linkwitz and Riley published a AES paper describing a family of filters boasting excellent phase alignment and therefore dispersion pattern. They found if one uses a 4th order HP built with two cascading Butterworth 2nd order filters (with Q=0.7), and a 4th order LP built with also two cascading Butterworth 2nd order filter (with Q=0.7), the two sides of the crossover will be in phase.
Arguable the most significant contribution is the concept of phase alignment over a wide bandwidth. For bass frequencies in particular, it's not merely about amplitude response. To implement a L-R filter, the HT receiver uses a 2nd order (12db/octave) high pass filter, and a 4th order (24db/octave) low pass filter. A high pass filter cuts frequencies below a chosen corner frequency, and a low pass filters above.
You may notice that they are not symmetrical. The reason is that we must also consider the acoustic roll-off of the main speakers. In this case, it is assumed that they are sealed, and as a result will have a 2nd order roll-off. If the speakers are in fact vented, the roll-off will have a steeper 4th order slope. In order to achieve a correct integration, the summed roll-off slopes of the main speakers and subwoofer should be symmetrical once the acoustic and crossover filters are added.
Consider the example of an 80 Hz crossover point. If the main speakers have a sealed box which are -3db at 80 Hz, the acoustic roll-off will be 2nd order. When this speaker is set to "small," in the receiver settings, it will have a 2nd order high pass filter at the crossover point of 80 Hz. The summed roll-off is therefore 4th order. The subwoofer should then have its crossover set to 80 Hz and use a 4th order low pass filter. This will yield a correctly implemented 4th order crossover. At 80 Hz, the response of the subwoofer and the mains will each be -6db.
Continue to measured examples
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