### Abstract

This paper presents a low-complexity algorithm and architecture to compute power spectral density (PSD) using the Welch method. The Welch algorithm provides a good estimate of the spectral power at the cost of high computational complexity. We propose a new modified approach to reduce the computational complexity of the Welch PSD computation for a 50% overlap. In the proposed approach, an N/2-point FFT is computed, where N is the length of the window and is merged with the FFT of the previous N/2-point to generate an N-point FFT of the overlapped segment. This requires replacing the windowing operation as a convolution in the frequency domain. Fortunately, the frequency-domain filtering requires a symmetric 3-tap or 5-tap FIR filter for raised cosine windows. The proposed method needs to compute (L+1) N/2-point FFTs instead of L N-point FFTs, where L is the number of overlapping segments. In the proposed novel merged FFT approach, the even samples are computed exactly, while the odd samples require a shift by a half-sample delay and are estimated using a bidirectional fractional-delay filter. The complexity reduction comes at the cost of slight performance loss due to the approximation used for the implementation of the fractional-delay filter. The performance loss is about 8% using fractional-delay filter with 2 multipliers. A novel architecture is presented based on the proposed algorithm. The proposed architecture not only consumes 33% less energy compared to the original method but also reduces the latency by about 44% for 8 overlapping segments. Further a low-complexity architecture is presented to compute a special case of the short-time Fourier transform based on the proposed PSD computation algorithm.

Original language | English (US) |
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Article number | 6525419 |

Pages (from-to) | 172-182 |

Number of pages | 11 |

Journal | IEEE Transactions on Circuits and Systems I: Regular Papers |

Volume | 61 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2014 |

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### Keywords

- FFT
- Welch method
- fractional-delay filter
- frequency-domain convolution
- low-complexity
- low-power
- power spectral density
- windowing

### Cite this

*IEEE Transactions on Circuits and Systems I: Regular Papers*,

*61*(1), 172-182. [6525419]. https://doi.org/10.1109/TCSI.2013.2264711