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How do you implement fractional delay filters in the Z-domain?

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Fractional delay filters are useful for applications that require precise control over the phase or delay of a signal, such as audio processing, beamforming, or interpolation. However, designing and implementing these filters in the Z-domain, which is the discrete-time equivalent of the Laplace domain, can be challenging. In this article, you will learn how to use different methods and tools to design and implement fractional delay filters in the Z-domain.

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1 Basic concept

A fractional delay filter is a filter that introduces a delay that is not an integer multiple of the sampling period. For example, if the sampling period is T, a fractional delay filter can delay the signal by T/2, T/3, or any other fraction. The fractional delay can be expressed as D, where D = N + d, and N is an integer and d is a fraction. The Z-transform of a fractional delay filter is H(z) = z^-D, which can be rewritten as H(z) = z^-N z^-d.

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Robert Bristow-Johnson

audio and music synthesis signal processing algorithm architecting/design/implementation
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Here is a Stack Exchange answer that has the theory and math spelled out for implementing fractional-precision delay, derived straight forwardly from the fundamental sampling and reconstruction theorem we get from Nyquist, Shannon, Whittaker, Kotelnikov. https://dsp.stackexchange.com/questions/17978/comb-filter-attenuation/18389#18389

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2 Approximation methods

Since z^-d is not a rational function, it cannot be implemented directly using finite impulse response (FIR) or infinite impulse response (IIR) filters, thus necessitating the use of approximation methods. Lagrange interpolation is a common method that uses a polynomial of degree N-1 to approximate the fractional delay filter. The coefficients of the polynomial can be found by solving a system of linear equations. Alternatively, the Thiran approximation method uses an allpass IIR filter of order N to approximate the fractional delay filter, with coefficients obtainable via a recursive formula. Lastly, the Farrow structure method utilizes a cascade of FIR filters to approximate the fractional delay filter, with coefficients obtainable through a Taylor series expansion.

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Dan Boschen

Technical Staff Engineer at Microchip
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I use polyphase interpolation filter design which results in bank of M filters, with each filter providing a fractional delay evenly spaced over the sample period, given in fractions of a sample as 1/M, 2/M, ... up to (M-1)/M. From that, any one of the filters can be chosen when a fixed fractional delay filter is needed. M is increased to the precision desired. I first design the prototype lowpass FIR filter for an interpolate by M filter using least squares (using `firls` in MATLAB,and Python), which is superior to using a truncated windowed Sinc impulse response, especially when we can also take advantage of multiband filtering to concentrate rejection just in the interpolation images. See https://dsp.stackexchange.com/questions/43344

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3 Design tools

There are several tools that can help you design fractional delay filters in the Z-domain, such as MATLAB, Python, or online calculators. For example, in MATLAB, you can use the function fdesign.fracdelay to specify the desired fractional delay and the approximation method, and then use the function design to obtain the filter coefficients. Alternatively, you can use the function fracdelay to directly compute the filter output for a given input and fractional delay.

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4 Implementation tips

Once you have designed your fractional delay filter, you need to implement it in your DSP system. It is important to choose the appropriate filter order and approximation method for your application, as higher order filters can achieve better accuracy but also require more resources. You should also test your filter performance using simulations and measurements, using tools such as freqz, grpdelay, or impz in MATLAB to analyze the frequency response, group delay, or impulse response of your filter. Additionally, you can optimize your filter code for speed and efficiency by using techniques such as loop unrolling, coefficient quantization, or fixed-point arithmetic to reduce the execution time and memory usage of your filter.

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5 Applications and examples

Fractional delay filters have many uses in DSP, such as audio processing, beamforming, and interpolation. In audio processing, they can create effects like chorus, flanger, or pitch shifting. They can also compensate for speaker or microphone delays or to align multiple audio channels. In beamforming, they can steer the direction of a beamformer array by applying different delays to each element and help synthesize virtual microphone arrays or improve spatial resolution. As for interpolation, they can be used to interpolate a signal at arbitrary points or resample a signal at a different rate. Additionally, they can implement fractional sample rate conversion or oversampling. Examples of fractional delay filters in the Z-domain include a Lagrange interpolator of order 3 with a fractional delay of 0.25, a Thiran allpass filter of order 2 with a fractional delay of 0.5, and a Farrow structure of order 1 with a fractional delay of d.

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Robert Bristow-Johnson

audio and music synthesis signal processing algorithm architecting/design/implementation
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The way I do fractional delays is the most straight-forward application of the Nyquist-Shannon-Whittaker-Kotelnikov-whomever sampling and reconstruction theorem. It's just evaluating the sinc() reconstruction for a given argument of fractional delay. The sinc() function is multiplied by a good (say, Kaiser) window to make it finite in length. (Or use firpm() or firls().) For a finite upsampling ratio (let's say 512x oversampling) the finite-length windowed sinc() function can be stored in a table. A decently large, but manageable table. For any fractional delay that's not a multiple of 1/512th sample period (which are all of them), I compute the two adjacent fractional delays that *are* multiples of 1/512 and linearly interpolate these.

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James D. (jj) Johnston

Immersive Audio Processing, Codec, and loudness management researcher.
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I would suggest that people who need to solve this for a time-varying delay go back to Proakis' textbook on signal processing. Yes, he has several, but I'm not at my office to cite the right one. None the less, a tricky, but very general solution is put forth to provide any desired level of accuracy.

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6 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?

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How do you implement fractional delay filters in the Z-domain?

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1 Basic concept

2 Approximation methods

3 Design tools

4 Implementation tips

5 Applications and examples

6 Here’s what else to consider

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