Sample Rate Conversion

Sample rate conversion (resampling) changes a signal from one sample rate to another. This is essential when combining audio from different sources, preparing data for systems that expect a specific rate, or reducing storage by downsampling.

miniDSP provides a high-quality offline polyphase sinc resampler that handles arbitrary rate ratios (e.g., 44100 Hz to 48000 Hz) with >100 dB stopband attenuation using default parameters.

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Mathematical building blocks

Bessel I₀

The zeroth-order modified Bessel function of the first kind appears in the Kaiser window formula. It is computed via the convergent power series:

\[I_0(x) = \sum_{k=0}^{\infty} \left[\frac{(x/2)^k}{k!}\right]^2 \]

Reading the formula in C:

// x -> input value, sum -> I₀(x), term -> (x/2)^k / k!
double sum = 1.0;
double term = 1.0;
double half_x = x / 2.0;
 
for (unsigned k = 1; k < 300; k++) {
    term *= (half_x / (double)k);
    double term_sq = term * term;
    sum += term_sq;
    if (term_sq < 1e-15 * sum) break;  // converged
}
// sum now holds I₀(x)

API:

double MD_bessel_i0(double x);

Normalized sinc

The sinc function is the ideal lowpass interpolation kernel. The normalized form is:

\[\mathrm{sinc}(x) = \begin{cases} 1 & \text{if } |x| < 10^{-12} \\ \dfrac{\sin(\pi x)}{\pi x} & \text{otherwise} \end{cases} \]

Reading the formula in C:

// x -> input value, result -> sinc(x)
double result;
if (fabs(x) < 1e-12) {
    result = 1.0;
} else {
    double px = M_PI * x;
    result = sin(px) / px;
}

API:

double MD_sinc(double x);

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The polyphase sinc resampler

How it works

The resampler converts a signal from sample rate \(f_{\mathrm{in}}\) to \(f_{\mathrm{out}}\) by treating each output sample as a fractional-position lookup into the input signal, filtered through a windowed sinc kernel.

For each output sample \(n\):

1. Compute the fractional input position: \(p = n \cdot f_{\mathrm{in}} / f_{\mathrm{out}}\)
2. Split into integer index \(\lfloor p \rfloor\) and fractional offset
3. Select two adjacent filter sub-phases from a precomputed table
4. Linearly interpolate the sub-phase coefficients
5. Dot product with surrounding input samples

The filter table contains 512 sub-phases, each with \(2 \times \mathrm{num\_zero\_crossings}\) taps of a Kaiser-windowed sinc. Anti-aliasing is handled automatically: for downsampling, the sinc cutoff is scaled to \(\min(f_{\mathrm{in}}, f_{\mathrm{out}})/2\).

Output buffer sizing

\[N_{\mathrm{out}} = \left\lceil N_{\mathrm{in}} \cdot \frac{f_{\mathrm{out}}}{f_{\mathrm{in}}} \right\rceil \]

Reading the formula in C:

// input_len -> N_in, in_rate -> f_in, out_rate -> f_out
unsigned out_len = (unsigned)ceil((double)input_len * out_rate / in_rate);

API:

unsigned MD_resample_output_len(unsigned input_len,
                                double in_rate, double out_rate);

Resampling a signal

API:

unsigned MD_resample(const double *input, unsigned input_len,
                     double *output, unsigned max_output_len,
                     double in_rate, double out_rate,
                     unsigned num_zero_crossings, double kaiser_beta);

Parameters:

• num_zero_crossings controls filter quality. Range: 8 (fast) to 64 (high quality). Default recommendation: 32.
• kaiser_beta controls stopband attenuation. Recommendation: 10.0 for ~100 dB.
• Returns the number of samples written to output.

Quick example:

    const double out_rate = 48000.0;
    unsigned N_out = MD_resample_output_len(N_in, in_rate, out_rate);
    double *output = malloc(N_out * sizeof(double));
 
    unsigned n_written = MD_resample(input, N_in, output, N_out,
                                     in_rate, out_rate, 32, 10.0);

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Common rate pairs

Conversion	Ratio	Use case
44100 to 48000	160/147	CD audio to professional video/DAW
48000 to 44100	147/160	Professional to CD quality
48000 to 16000	1/3	Wideband to narrowband speech
44100 to 22050	1/2	Half-rate for reduced storage
16000 to 8000	1/2	Wideband to telephone bandwidth

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Further reading

• Sample-rate conversion – Wikipedia
• Sinc filter – Wikipedia
• Polyphase filter – Wikipedia
• Kaiser window – Wikipedia

API reference

• MD_bessel_i0() – zeroth-order modified Bessel function I₀
• MD_sinc() – normalized sinc function
• MD_resample_output_len() – compute required output buffer size
• MD_resample() – polyphase sinc resampler