范围，而非精度：Apple Silicon上的块浮点半精度FFT与SAR成像
Range, Not Precision: Block-Floating-Point Half-Precision FFT and SAR Imaging on Apple Silicon
AI总结：
针对半精度FFT在雷达信号处理中因动态范围不足导致溢出的问题，提出固定移位块浮点调度方案，在Apple M1上实现首个保质的FP16 SAR流水线，性能提升2.2倍。
摘要：
半精度（FP16）有望使GPU上的FFT吞吐量翻倍，但普遍观点认为其10位尾数不适合雷达级信号处理。我们证明这种看法在Apple Silicon上是错误的：FFT和合成孔径雷达（SAR）的约束不是尾数精度，而是5位指数的动态范围。我们首先测量到FP16 FFT的量化信噪比（SQNR）受限于56–61 dB——舒适地适用于雷达——然而，一个朴素的FP16 SAR流水线仅产生NaN，因为共轭-FFT-共轭逆变换使幅度增长N倍，且匹配滤波乘积（N=4096时约5×10^6）溢出FP16的65,504上限。我们通过固定移位块浮点（BFP）调度解决此问题：在每次逆变换前应用单个1/N缩放，将每个中间值限制在4096以下。级联随之而来：距离压缩输出变为O(1)而非O(N)，进而使下游方位FFT输出保持FP16可加载，而非在O(N^2)处溢出。结果是首个保质的FP16 SAR流水线：峰值/积分旁瓣比、目标SNR和分辨率与FP32参考匹配，误差在0.1 dB以内，端到端SQNR为42 dB，而基8 FP16 FFT在无风扇Apple M1上达到306 GFLOPS——是139 GFLOPS FP32基线的2.2倍。最后，我们测量到FP8（E4M3/E5M2）的SQNR降至14–20 dB，使FP16成为当今基于FFT的雷达的精度下限——未来的精度恢复方法可能进一步降低该下限——并表明此处低精度的杠杆是范围管理，而非尾数位数。
英文摘要：
Half precision (FP16) promises to double FFT throughput on GPUs, but the prevailing view is that its 10-bit mantissa makes it unsuitable for radar-grade signal processing. We show this framing is wrong on Apple Silicon: the binding constraint for FFT and Synthetic Aperture Radar (SAR) is not mantissa \emph{precision} but the 5-bit exponent's \emph{dynamic range}. We first measure that an FP16 FFT is mantissa-limited at 56--61~dB signal-to-quantization-noise ratio (SQNR) -- comfortably radar-usable -- yet a naïve FP16 SAR pipeline produces \emph{only} \texttt{NaN}, because the conjugate--FFT--conjugate inverse transform grows magnitudes by a factor of $N$, and the matched-filter product ($\sim\!5\times10^6$ at $N\!=\!4096$) overflows FP16's 65{,}504 ceiling. We resolve this with a fixed-shift \emph{block-floating-point} (BFP) schedule: a single $1/N$ scale applied before each inverse transform bounds every intermediate below 4096. A cascade follows: range-compression output becomes $O(1)$ instead of $O(N)$, which in turn keeps the downstream azimuth-FFT output FP16-loadable instead of overflowing at $O(N^2)$. The result is the first quality-preserving FP16 SAR pipeline: peak/integrated sidelobe ratios, target SNR, and resolution match the FP32 reference to within $0.1$~dB at $42$~dB end-to-end SQNR, while a radix-8 FP16 FFT reaches 306~GFLOPS -- $2.2\times$ over the 139~GFLOPS FP32 baseline -- on a fanless Apple~M1. Finally, we measure that FP8 (E4M3/E5M2) collapses to 14--20~dB SQNR, making FP16 \emph{today's} precision floor for FFT-based radar -- one that future precision-recovery methods may yet lower -- and showing that the lever for low precision here is range management, not mantissa bits.
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