Canonic FFT flow graphs for real-valued even/odd symmetric inputs
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 t...
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2017-06-01
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| Series: | EURASIP Journal on Advances in Signal Processing |
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| Online Access: | http://link.springer.com/article/10.1186/s13634-017-0477-9 |
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doaj-art-8f2d9f70c4d64905b21c7c6f3bf33c432018-08-16T00:56:33ZengSpringerEURASIP Journal on Advances in Signal Processing1687-61802017-06-012017112310.1186/s13634-017-0477-9Canonic FFT flow graphs for real-valued even/odd symmetric inputsYingjie Lao0Keshab K. Parhi1Department of Electrical and Computer Engineering, Clemson UniversityDepartment of Electrical and Computer Engineering, University of MinnesotaAbstract 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 N 2 + 1 $\frac {N}{2}+1$ signal values at each stage for an N-point real even symmetric FFT (REFFT) or N 2 − 1 $\frac {N}{2}-1$ 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.http://link.springer.com/article/10.1186/s13634-017-0477-9Fast Fourier transform (FFT)Real-valued FFT (RFFT)Canonic flow graphEven symmetric inputsOdd symmetric inputsTwiddle factor transformation |
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| collection |
Open Access Journals |
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| language |
English |
| format |
Article |
| author |
Yingjie Lao Keshab K. Parhi |
| spellingShingle |
Yingjie Lao Keshab K. Parhi Canonic FFT flow graphs for real-valued even/odd symmetric inputs EURASIP Journal on Advances in Signal Processing Fast Fourier transform (FFT) Real-valued FFT (RFFT) Canonic flow graph Even symmetric inputs Odd symmetric inputs Twiddle factor transformation |
| author_facet |
Yingjie Lao Keshab K. Parhi |
| author_sort |
Yingjie Lao |
| title |
Canonic FFT flow graphs for real-valued even/odd symmetric inputs |
| title_short |
Canonic FFT flow graphs for real-valued even/odd symmetric inputs |
| title_full |
Canonic FFT flow graphs for real-valued even/odd symmetric inputs |
| title_fullStr |
Canonic FFT flow graphs for real-valued even/odd symmetric inputs |
| title_full_unstemmed |
Canonic FFT flow graphs for real-valued even/odd symmetric inputs |
| title_sort |
canonic fft flow graphs for real-valued even/odd symmetric inputs |
| publisher |
Springer |
| series |
EURASIP Journal on Advances in Signal Processing |
| issn |
1687-6180 |
| publishDate |
2017-06-01 |
| description |
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 N 2 + 1 $\frac {N}{2}+1$ signal values at each stage for an N-point real even symmetric FFT (REFFT) or N 2 − 1 $\frac {N}{2}-1$ 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. |
| topic |
Fast Fourier transform (FFT) Real-valued FFT (RFFT) Canonic flow graph Even symmetric inputs Odd symmetric inputs Twiddle factor transformation |
| url |
http://link.springer.com/article/10.1186/s13634-017-0477-9 |
| _version_ |
1612696167767867392 |