Minimum Phase and Accurate Crossover Simulation

Minimum phase for a speaker can be described of the phase of the frequency response of the speaker at it’s acoustic centre, or the phase without any excess phase or “time of flight” included. It can be calculated from the speaker’s frequency response using Hilbert-bode transform, or HBT. When comparing multiple minimum phase responses, we can say that the responses occur at the same point in space, ie. there is no physical distance between them. When measuring speakers and processing the impulse response to a frequency response using FFT, the resulting phase is dependent on the start time of the FFT window.

For single channel measurements using a USB mic, there is no precise start time for the measurement to set t=0, so the timing reference is often determined by peak detection of the impulse response, however can be user-adjusted in most software. The resulting phase may not be precisely minimum phase as a result. Additionally, due to the lacking timing reference, capturing delay between drivers in the measurement is not possible without extra steps. Minimum phase is one way to ensure that multiple measurements have a common point in space with respect to phase.

For dual channel measurements, the reference feedback loop includes a common point in time, so t=0 can remain absolute across many measurements. This allows for determination of time of flight between the speaker and mic, and accurate recording of differences in phase (delay) between multiple measurements. The window start time can be arbitrarily set near the start of the impulse, and the phase result of the FFT will include the excess phase (time of flight) from window start time to the impulse. Keeping a constant distance from microphone to speaker baffle, and a constant window start time allows for accurate delay between multiple measurements to be captured.

For either single or dual channel measurement processes, determining minimum phase is rarely needed, and is a step of response processing that can be skipped without any detrimental effects. Additionally, HBT processing usually requires user selection of response slopes at either end of the measurement, so there is potential for user error in the process. HBT overall will affect phase response at the frequency extents, when completed correctly is not detrimental to crossover results, however it is damaging to the measured phase and should be avoided where possible.

Delay Comparison

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For comparison of minimum phase to measured phase, a 3 measurements process will be completed in order to capture the delay between multiple measurements. The mic will remain stationary for all 3 measurements. A measurement of each speaker individually is recorded, followed by a measurement of both speakers in parallel. Delay (phase) is adjusted until the sum of the two individual measurements is equal to the recorded sum that was measured.

To compare, this 3 measurement process is completed by simply using the measured results directly, then again processing the measurements for minimum phase. This process can be used to determine the delay between any two audio sources. It doesn’t need to be a tweeter and a woofer, for this example I used a pair of small bookshelf speakers stacked on top of each other, so delay between the top speaker and bottom speaker will be determined. The mic was located on axis with the tweeter of the top speaker, at 1m distance.

Basic Speaker Test Setup

Dual Channel
A 2-channel measurement was completed for top speaker, bottom speaker, then both in parallel. The resulting delay determined using VituixCAD time alignment tool was nearly zero. This is expected because the delay was captured in the measured phase information.

Dual channel time alignment

Single Channel with Measured Phase
Next, a single channel measurement was completed. This time, the alignment error is about the same, however with a delay value of 32.7mm. You can see the small error between the measured sum and the individual measurement with delay is identical to the 2-channel results above. This is because the measured phase has not changed, only the excess time of flight (32.9mm) is no longer included in the single channel measurement.

Single channel time alignment

Single Channel with Minimum Phase
Now, compare the same results but with calculated minimum phase instead of using the measured data directly. Here you can see the measured phase (brown), and minimum phase (grey). The HBT tails chosen are shown at the right, and measured response is red, HBT response is blue.

Top speaker minimum phase

Bottom speaker minimum phase

Returning to the time alignment tool with the minimum phase responses, the delay determined is 36.9mm, however there is a greater error due to the HBT modified phase. What should be gleaned from this is that there was nothing wrong with the measured phase, it matched the acoustic sum nearly perfectly, and HBT processing only made it worse.

Minimum phase time alignment

The conclusion that can be made from this simple test, is that minimum phase and HBT did not provide any useful improvement or correction to the measured data, rather it has damaged the measured phase information.

It’s also important that if a minimum phase response is desired, that delay determination must be completed after the minimum phase calculation is completed.

Crossover Comparison

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With the delay between the two speakers determined, the comparison of crossover simulation vs reality can be easy. A simple active crossover was used for this comparison, the lower speaker has a low pass filter applied, and the upper speaker has a high pass filter applied. The resulting frequency response of the filter is not very important for this comparison, as the comparison is how well each simulation matches each other, and the real world measurement with the filter applied. A crossover in the digital domain ensures accuracy so that the measured result doesn’t stray from the simulation due to component part tolerances. APO EQ was used as the DSP engine using impulse response convolution to ensure the filter transfer function matches the simulation as closely as possible.

The following crossover was used for the comparison. Since the mic remained stationary for all measurements, no x,y,z coordinates for the drivers were used, only the delay value adjusted for the bottom driver using the delay values previously determined.

Simple Crossover Schematic

Using the above schematic, the dual channel measurements, single channel measurements, and minimum phase data was loaded, and previously determined delay values were entered for each. The resulting responses are overlaid, dual channel and single channel results overlay perfectly as both used measured phase information with only different delay values. The minimum phase result shows a slight change from expected result.

Crossover Simulation Comparison

Next, the crossover filter was applied, and measured result compared against the above simulation. Unsurprisingly, the results using measured phase information agree with the measured response using measured phase, not the minimum phase result.

Crossover Measurement Comparison

The conclusion made from this comparison, is that crossover simulation accuracy is achieved by using the measured phase information, avoiding minimum phase and HBT processing.

Merging Comparison

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Merging near field and far field responses is a crucial aspect of measuring loudspeakers indoors and creating a full spectrum response that is free of room interaction. There are many software solutions to merge responses together, each having different effects on the phase response of the result. Using the right tool and using it correctly is critical to maintaining the intended phase response of a driver. Here is a summary of the capabilities of the various methods of response merging.

FRD Response Blender and Minimum Phase Extractor
This mouthful is an old spreadsheet that will merge two responses together along with applying diffraction response to the near-field response. It only allows for a single near-field response, so merging driver and port near-field responses is not possible, as well it will only provide minimum phase of the merged result. If maintaining measured far field phase was a goal, this is not the tool to use. Due to the minimum phase only result, it will be critical to complete the merging result prior to determining delay between drivers.

Pros

• Relatively easy to use
• Includes diffraction modeller
• Merged result is blended over user selectable range

Cons

• Result is minimum phase only
• Requires Excel
• Single response only for low frequency portion so combining port output is not possible
• HBT slopes chosen manually can result in bad result due to user error
• Order of operations is critical for correct delay determination

SoundEasy
Soundeasy will merge responses via the Easylab post-processing tab. The merge is quite crude, providing a hard cut at the merge frequency and no adjustment to phase near or far is completed. Either near-field phase can be manually adjusted to align with far field data, or HBT can be applied to generate minimum phase, then delay added back in to restore the measured time of flight. Very clunky and unintuitive user interface.

Pros

• Measured phase can be maintained
• Multiple responses can be combined
• Includes diffraction modeller

Cons

• Complicated and unintuitive user interface
• Merged result is hard cut, no blending option is available
• Clunky manual process to align low frequency phase with high frequency response
• HBT plus delay seems counter-intuitive process to maintain measured phase/delay.

Omnimic
Omnimic allows response splicing via the added curves menu. Splicing retains phase of both near and far field result, so near-field phase can be manually adjusted to align with far field phase. Omnimic software does not provide a simple method to include diffraction, as well the diffraction model must be included externally which limits functionality.

Pros

• Delay is easily adjusted to align low frequency and high frequency responses
• Merged response is “blended”, not a hard cut.

Cons

• Diffraction is not included, additional software required to complete the process correctly
• phase/delay adjustment is a manual process

REW
REW allows for merging of two responses together via trace arithmetic merge function. It appears to maintain phase of the far field response quite well, and including diffraction response is possible using the trace arithmetic division process, however a bit clunky through manual import of response and arithmetic functions. It’s functional but clearly not purpose built for this processes.

Pros

• Multiple responses can be combined
• Measured phase is maintained
• Low frequency response phase is automatically adjusted
• Merged result can be blended

Cons

• Diffraction response requires additional software
• Process is not simple / straight forward

ARTA
ARTA provides the ability to merge responses using overlays. Similar to SoundEasy, the merge is a hard cut with no modification to the phase of either response. Overlay cannot be imported FRD text file, limiting functionality. ARTA is a very capable measurement software, but the operation here is quite clunky and lacking any ability to process the near-field measurement for diffraction effects.

Pros

• Measured phase is maintained

Cons

• Overlay cannot be text based frequency response
• Diffraction response requires additional software
• Merge is not blended
• Only two responses can be combined
• Adjusting phase/delay to match is complicated

Holmimpulse
Holmimpulse provides the merging capabilities via the manipulation menu. Diffraction response will require external software, however combining with near-field response can be completed using response division. Merging is completed using the “stitching” operation. This is quite functional, as it includes an option to match phase before stitching, allowing for automatic delay adjustment of the low frequency portion, keeping the high frequency far-field phase in tact.

Pros

• Measured phase is retained
• Low frequency response phase can be automatically adjusted
• Merged result can be blended

Cons

• Only two responses can be combined
• Diffraction response requires additional software

VituixCAD
VituixCAD provides merging of multiple near-field responses with multiple far field responses, allowing for bulk processing of merge process for on and off-axis measurements in bulk. Diffraction can be provided via a response file, or a basic spherical calculation completed for basic results. It will maintain the phase of the far field response as it was provided, and automatically adjust the delay of the near-field phase to align with the far field data. This appears to be the most featureful and functional merger tool available.

Pros

• Easy to use
• Multiple measurements can be combined, including multiple high frequency off-axis responses
• Includes diffraction modeller
• Low frequency phase is automatically adjusted to match high frequency portion
• Includes calculations for automatic level adjustment
• Merged result is blended

Cons

• ...

Minimum Phase Use Cases

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There are a few use cases where Minimum phase can be useful. One example would be software simulated results, such as that from enclosure modelling tools, or diffraction modelling tools, will be minimum phase. It’s important to understand this when incorporating simulated data into measured data, such as the near field and far field merging process.

Another use for minimum phase is cases where measured phase is unavailable, and only a frequency response is known, such as when tracing manufacturer response curves. A Minimum phase result can be extracted from any known frequency response , providing assumptions are made on the slope of the response beyond the frequency range available (ie. Below 20Hz and above 20kHz).

For near-field measurements, combining response of the driver cone as well as the port output requires moving the mic to a different location on the speaker. For the best combined result, a minimum phase response of these measurement is useful, as the timing information of near-field measurements is not important, the phase should be adjusted to match the far field phase regardless of whether the near-field phase is minimum phase or not.