This post unpacks what these terms mean, why they matter for low-power and real-time systems, and how you can implement 10-digit efficient transforms without floating-point hardware. | Term | Meaning | |------|---------| | FP | Fixed-point arithmetic (integer scaling, no FPU) | | CAT | Category theory — composable, structure-preserving transforms | | ET | Efficient transforms (FFT, DCT, wavelet, etc.) | | 10dig | 10 significant decimal digits of precision (~33–34 bits) |
// Categorical fixed-point FFT stage void fft_stage_fixpt(q31_t *x, q31_t *w, int n, int stage) // morphism composition from FixPt category for (int i = 0; i < n/2; i++) q63_t sum = (q63_t)x[i] + ((q63_t)x[i+n/2] * w[i] >> 31); q63_t diff = (q63_t)x[i] - ((q63_t)x[i+n/2] * w[i] >> 31); x[i] = saturate_q31(sum >> scale[stage]); x[i+n/2] = saturate_q31(diff >> scale[stage]); fp cat et 10dig
| Transform | Precision (digits) | Cycles/sample (FP) | Cycles/sample (10dig fixed) | |-----------|-------------------|--------------------|-------------------------------| | 256-FFT | 7.2 (float) | 142 | 38 | | 256-FFT | 10.1 (10dig fixed) | — | 41 | | DCT (128) | 9.8 (float) | 98 | 29 | This post unpacks what these terms mean, why