Design algorithms and simulation results are presented for vector quantizers for Fourier transformed data. Transforming the data prior to quantization has two potential advantages. First, each sample in the transform domain depends on many samples in the original domain. Thus, even scalar quantization in the transform domain is a form of vector quantization or block source coding in the original waveform domain and the basic coding theorems of information theory show that such block codes can provide better performance than scalar codes, even for memoryless sources. Second, vector quantization of Fourier transformed speech waveforms provides distinctly better subjective quality than ordinary vector quantization of the waveform using codes of comparable complexity. While the system is, of course, more complicated due to the need to take Fourier transforms, its envisioned application is as a coder for the output of FFT chips currently available or under development.The proposed implementation of a Fourier transform vector quantizer (FTVQ) uses a product code structure, providing different codes for different coefficient vectors corresponding to different frequency bands. This is a form of subband coding and yields a simple means of optimizing bit allocations among the subcodes. Two coding structures with corresponding distortion measures are considered: those that quantize vectors of pairs of real and imaginary coefficients and those that quantize separate vectors of magnitude and phase coefficients. Both structures yield good performance for the given complexity in comparison to waveform vector quantizers. For speech coding, a magnitude-phase FTVQ yields better subjective quality than a real-imaginary FTVQ when the rate allocation is properly chosen.