### Abstract

Canonic real-valued fast Fourier transform (RFFT) has been proposed to reduce the arithmetic complexity by eliminating redundancies. In a canonic N-point RFFT, the number of signal values at each stage is canonic with respect to the number of signal values, i.e., N. The major advantage of the canonic RFFTs is that these require the least number of butterfly operations and only real datapaths when mapped to architectures. In this paper, we consider the FFT computation whose inputs are not only real but also even/odd symmetric, which indeed lead to the well-known discrete cosine and sine transforms (DCTs and DSTs). Novel algorithms for generating the flow graphs of canonic RFFTs with even/odd symmetric inputs are proposed. It is shown that the proposed algorithms lead to canonic structures with N2+1 signal values at each stage for an N-point real even symmetric FFT (REFFT) or N2−1 signal values at each stage for an N-point RFFT real odd symmetric FFT (ROFFT). In order to remove butterfly operations, several twiddle factor transformations are proposed in this paper. We also discuss the design of canonic REFFT for any composite length. Performances of the canonic REFFT/ROFFT are also discussed. It is shown that the flow graph of canonic REFFT/ROFFT has less number of interconnections, less butterfly operations, and less twiddle factor operations, compared to prior works.

Original language | English (US) |
---|---|

Article number | 45 |

Journal | Eurasip Journal on Advances in Signal Processing |

Volume | 2017 |

Issue number | 1 |

DOIs | |

State | Published - Dec 1 2017 |

### Fingerprint

### Keywords

- Canonic flow graph
- Discrete cosine transform (DCT)
- Discrete sine transform (DSTs)
- Even symmetric inputs
- Fast Fourier transform (FFT)
- Odd symmetric inputs
- Real-valued FFT (RFFT)
- Twiddle factor transformation

### Cite this

**Canonic FFT flow graphs for real-valued even/odd symmetric inputs.** / Lao, Yingjie; Parhi, Keshab K.

Research output: Contribution to journal › Article

*Eurasip Journal on Advances in Signal Processing*, vol. 2017, no. 1, 45. https://doi.org/10.1186/s13634-017-0477-9

}

TY - JOUR

T1 - Canonic FFT flow graphs for real-valued even/odd symmetric inputs

AU - Lao, Yingjie

AU - Parhi, Keshab K

PY - 2017/12/1

Y1 - 2017/12/1

N2 - Canonic real-valued fast Fourier transform (RFFT) has been proposed to reduce the arithmetic complexity by eliminating redundancies. In a canonic N-point RFFT, the number of signal values at each stage is canonic with respect to the number of signal values, i.e., N. The major advantage of the canonic RFFTs is that these require the least number of butterfly operations and only real datapaths when mapped to architectures. In this paper, we consider the FFT computation whose inputs are not only real but also even/odd symmetric, which indeed lead to the well-known discrete cosine and sine transforms (DCTs and DSTs). Novel algorithms for generating the flow graphs of canonic RFFTs with even/odd symmetric inputs are proposed. It is shown that the proposed algorithms lead to canonic structures with N2+1 signal values at each stage for an N-point real even symmetric FFT (REFFT) or N2−1 signal values at each stage for an N-point RFFT real odd symmetric FFT (ROFFT). In order to remove butterfly operations, several twiddle factor transformations are proposed in this paper. We also discuss the design of canonic REFFT for any composite length. Performances of the canonic REFFT/ROFFT are also discussed. It is shown that the flow graph of canonic REFFT/ROFFT has less number of interconnections, less butterfly operations, and less twiddle factor operations, compared to prior works.

AB - Canonic real-valued fast Fourier transform (RFFT) has been proposed to reduce the arithmetic complexity by eliminating redundancies. In a canonic N-point RFFT, the number of signal values at each stage is canonic with respect to the number of signal values, i.e., N. The major advantage of the canonic RFFTs is that these require the least number of butterfly operations and only real datapaths when mapped to architectures. In this paper, we consider the FFT computation whose inputs are not only real but also even/odd symmetric, which indeed lead to the well-known discrete cosine and sine transforms (DCTs and DSTs). Novel algorithms for generating the flow graphs of canonic RFFTs with even/odd symmetric inputs are proposed. It is shown that the proposed algorithms lead to canonic structures with N2+1 signal values at each stage for an N-point real even symmetric FFT (REFFT) or N2−1 signal values at each stage for an N-point RFFT real odd symmetric FFT (ROFFT). In order to remove butterfly operations, several twiddle factor transformations are proposed in this paper. We also discuss the design of canonic REFFT for any composite length. Performances of the canonic REFFT/ROFFT are also discussed. It is shown that the flow graph of canonic REFFT/ROFFT has less number of interconnections, less butterfly operations, and less twiddle factor operations, compared to prior works.

KW - Canonic flow graph

KW - Discrete cosine transform (DCT)

KW - Discrete sine transform (DSTs)

KW - Even symmetric inputs

KW - Fast Fourier transform (FFT)

KW - Odd symmetric inputs

KW - Real-valued FFT (RFFT)

KW - Twiddle factor transformation

UR - http://www.scopus.com/inward/record.url?scp=85020500878&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85020500878&partnerID=8YFLogxK

U2 - 10.1186/s13634-017-0477-9

DO - 10.1186/s13634-017-0477-9

M3 - Article

VL - 2017

JO - Eurasip Journal on Advances in Signal Processing

JF - Eurasip Journal on Advances in Signal Processing

SN - 1687-6172

IS - 1

M1 - 45

ER -