Faster Non-Integer Sample Rate Conversion
A couple months ago, I was doing some research into emulating bucket-brigade devices for creating analog-style delay-lines, when I ran into a rather interesting paper, written by Martin Holters and Julian Parker.
Although bucket-brigade devices (BBDs) are made up of analog circuits, in concept they behave like a fixed-length digital delay-line with a variable sample rate which determines the length of the delay created by the device. One of the primary challenges of emulating BBDs is performing fast sample rate conversion (SRC) to match the effective sample rate of the BBD.
Holters and Parker found a rather brilliant solution to this problem. They noted that BBDs typically use filters at their input and output, to reduce imaging and aliasing artifacts, and figured out how to implement these filters in the digital domain as “multi-rate” filters, so that the filter can use the audio sample rate on one side of the filter, and the BBD sample rate on the other side.
While I was eventually able to implement a BBD emulation using Holters and Parker’s method, I came to realize that their algorithm can be used much more generally for any task that requires sample rate conversion. (If anyone is interested in the results of the BBD emulation, it is now impemented in the Surge Synthesizer as part of their “Ensemble” effect.)
Non-Integer Sample Rate Conversion
There are two fundamental kinds of sample rate conversions: Integer Sample Rate Conversion and Non-Integer Sample Rate Conversion.
With Integer SRC, the sample rate is being changed by an integer factor. In this case, the block of audio can simply be truncated or padded with zeros, and then filtered, again to avoid any imaging or aliasing artifacts in the reconstructed signal.
It is possible to use two stages of Integer SRC to perform Non-Integer SRC. For example, to upsample a signal by a factor of 1.5, one option would be to upsample by a factor of 3, and then downsample by a factor of 2. However, this approach can become very costly in cases where a very large upsampling factor is required.
A better option that works well for a broad range of SRC factors is to use interpolation, for example Sinc interpolation, or Lanczos interpolation. One of the most commonly used sample rate conversion libraries, libsamplerate (AKA "Secret Rabbit Code") uses Sinc interpolation, with a couple different options, to trade off quality for performance.
Non-Integer sample rate conversion is a very common issue in audio signal processing. For example, two commonly used audio sample rates are 44.1 kHz and 48 kHz, which differ by a factor 0f 1.088. If an algorithm is designed to run at one sample rate, but a user wants to use the algorithm at a different sample rate, then real-time non-integer SRC is often required. In these cases, we want to be able to do the SRC as quickly as possible, so that the majority of the time spent in our processing code can be spent actually doing the signal processing that we’re interested in, rather than getting to and from our target sample rate.
Holters-Parker Resampling
Here I’d like to introduce Holters-Parker resampling as a potential alternative to interpolation-based resampling. Holters and Parker start with a filter in the analog domain, to be used for anti-aliasing or anti-imaging similar to the filters used in integer sample rate conversion. Next, Holters and Parker mention the “impulse invariance” method for discretizing analog filters. The idea behind impulse invariance is to derive an expression for the impulse response of the analog system, and then essentially “sample” the impulse response at the sample rate of the digital system.
However, Holters and Parker instead use what they refer to as a “modified impulse-invariant transform”, so that the input and output signals for the transformed filters may be at different sample rates. When seen in the context of sample rate conversion, this is a pretty neat insight: rather than resampling and then filtering, the filtering and resampling happen in the process!
I won’t delve too deeply into the implementation of the Holters-Parker resampler here, except to note that while it can be used with any choice of filters, I would recommend choosing a filter order that is a multiple of four. The reason for this is that my implementation uses SIMD registers for computing the filter stages in parallel. For CPUs with the SSE instruction set (most CPUs these days), a single SIMD register can contain 4 32-bit floating point numbers, enabling us to compute a fourth-order filter for the same computational cost as a first-order filter. In my implementation, I use a fourth-order Butterworth filter, but it should be possible to produce a higher-quality output using a filter with a steeper rolloff, or with a higher-order filter.
Results
Finally, let’s see how the Holters-Parker resampler compares to the commonly used Sinc interpolation algorithm in libsamplerate. Since I'm primarily interested in speed rather than quality, I've set libsamplerate to use the SRC_SINC_FASTEST option. As it turns out, the results are not even close! On my Linux system running an Intel i7 CPU, the Holters-Parker resampler measures 10-40x faster than libsamplerate depending on the sample rate conversion factor! Further, I don't think my implementation of the Holters-Parker resampler is fully optimal just yet. With a couple more rounds of optimisations, I bet I could have it running even faster!
So does this mean that you should immediately abandon libsamplerate and interpolation-based resampling altogether? Certainly not! Sinc interpolation is still the most ideal sample rate conversion scheme in most cases, and libsamplerate has a thoroughly tested implementation, with well understood trade-offs between quality and speed. Further, there are other interpolation-based methods that give decent quality results and can also out-perform libsamplerate.
My goal here is simply to introduce the Holters-Parker resampler as a potential option that may be useful for some cases. Eventually, I’ll need to see how my implementation stacks up in both speed and quality, to libsamplerate, and other interpolation-based SRC implementations. In particular, I think choosing the right filter design for the Holters-Parker resampler will be an important part of getting the highest quality results.
Conclusion
I hope this discussion of non-integer sample rate conversion and Holters-Parker resampling has been interesting and useful! If you’d like to take a look at the source code for my implementation of the resampler, the code can be found on GitHub. I would greatly appreciate any suggestions on how to improve the quality or performance of the implementation!
Thanks to the Surge Synthesizer Team for piquing my interest about all this stuff in the first place, and especially to Paul Walker for showing me how to use SIMD to vectorize the filter computations!
Update: Since writing this post, I’ve found an implementation of resampling with Lanczos interpolation that performs comparably to the Holters-Parker resampler, but with better accuracy. For more details, see the repository linked above.
