Analysis of Quantization Noise in Fixed-Point HDFT Algorithms

The Discrete Fourier Transform (DFT) algorithm is widely used in signal processing and communication systems to transform the signal to the frequency-domain. As real-time signal analysis is required for fast processing, several recursive algorithms were proposed to perform the calculation with overlapping sequences in a sliding manner. One Sliding DFT (SDFT) method is the Hopping DFT (HDFT), where the DFT calculations are not evaluated sample-by-sample but with longer steps, thus further reducing the computational complexity compared to the other SDFT algorithms. This letter analyses the effect of fixed-point roundoff error in the HDFT algorithm, including the Updating Vector Transform (UVT) block. A closed-form expression for the resulting quantization noise power at the output of the HDFT algorithm is provided, which is validated through simulations. The results show that the roundoff error can be determined based on the number and size of the hops, the window size, and the number of fractional bits used in the quantization process.