Arbitrary Resampling Filter

Arbitrary Resampling Filter Design (XAPP1373)

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Consider the signal waveform shown in Figure 3 where input samples are evenly distributed at an interval T. One desirable output sample is located between xn-3 and xn-2 with a timing offset u, which can be any real number between 0 and 1. For all finite impulse response (FIR) filters, the output can be written as a linear combination of the input samples as:

Figure 1. FIR Output Linear Combination of the Input Sample
where L is the number of taps and {ck} is the set of coefficients. In the case of ARF, to account for dynamic timing offsets of output samples, ck becomes a function of the timing offset u:

where f(·) generalizes the discrete coefficients of a low-pass filter to a continuous function in the domain of real numbers. One method of implementing the continuous function f(·) is to prestore an array {Fk } in a memory where

Then, approximate ck(u) by a linear interpolation of two nearest prestored values

Figure 2. Linear Interpolation of Two Nearest Prestored Values

where [x] is the floor function that gives the largest integer less than or equal to x and

{Fk } and {Gk } can be implemented by two look-up tables addressed by s. The example in the following figure has L = 6 and P = 4.

Figure 3. Arbitrary Resampling Filter (L = 6, P = 4)

For an ARF with a maximum interpolation ratio K, at most ceil(K) new output samples can be computed from one input. The following figure shows the case when K = 2 and either one or two outputs are computed from every new input sample.

Figure 4. Number of Outputs for Interpolation Ratio up to K = 2

The complexity of an ARF is dominated by the computation of Figure 1 and Figure 2. The former needs two L real-to-real multiplications for each output sample, and the latter needs an additional L real-to-real multiplications. Because the output sample rate is K times that of input, the total number of real-to-real multiplications is 3·L·K·Input_Sample_Rate.

This gives an estimate for the minimum number of AI Engines required by the ARF.