<p>The Fourier transform is a powerful tool to analyse the frequency characteristics of signals. Discrete Fourier transform hardware typically implements Cooley–Tukey-based algorithms for reduced operational complexity. However, such schemes bring a sequential window schedule and separate real and imaginary computations, and their hardware implementations struggle to support runtime arbitrary radix and non-uniform discrete Fourier transform. Here we report a first-principles hetero-integrated Fourier transform system based on volatile and non-volatile memristors. Uniform vanadium oxide volatile memristor arrays provide oscillatory waves for arbitrary radix, and together with compact shaping and phase alignment circuits, runtime-calibratable frequency spectra can be generated, recording a maximum frequency of up to 1.74 MHz and a resolution down to 50 Hz. Non-volatile multilevel tantalum oxide/hafnium oxide memristor arrays are incorporated with bipolar differential conductance mapping for parallel signed discrete Fourier transform in-memory computing. Our hetero-integrated Fourier transform system can support arbitrary radix values up to 2,048, uniform or non-uniform 1D/2D discrete Fourier transform with cross-window parallelism, as well as unified real and imaginary computations, with a discrete Fourier transform accuracy up to 99.2% and O(<i>N</i>) operational complexity. The system can reach a throughput of 504.3 GS s⁻¹, outperforming existing hardware by up to 96.98 times and reduce memory cost.</p>