Fast algorithms for polynomial time frequency transform

Wei, Yongmei; Bi, Guoan

doi:10.1016/j.sigpro.2006.07.010

Cited by 9 publications

(10 citation statements)

References 19 publications

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“…polynomial time frequency transforms was used to estimate the phase for a special class of signal. It may be evaluated by the use of FFT and this transform reduces the computational complexity significantly as compared to the computational complexity involved by the FFT [11]. In the year 2008, Bi and Ju proposed an efficient algorithm to reduce the computational complexity for polynomial time frequency transform [12].…”

Section: Introductionmentioning

confidence: 99%

A novel approach for DFT computation

Arif

2015

2015 International Conference on Circuits, Power and Computing Technologies [ICCPCT-2015]

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Author has explored an idea to develop a novel procedure to compute the DFT by utilizing the concept of radix-4 decimation in time FFT algorithm instead of a divide and conquer method for the same computation. This novel method is used to compute the DFT in a systematic manner. The number of computation by this proposed method is same as those required by divide and conquer method. This method is an effective, easy and systematic way of computing DFT compared to the same computation performed by traditional divide and conquer method.

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Section: Introductionmentioning

confidence: 99%

A novel approach for DFT computation

Arif

2015

2015 International Conference on Circuits, Power and Computing Technologies [ICCPCT-2015]

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“…The disadvantage of the LPFT is that it is computationally demanding because its calculation involves the estimation of extra parameters using the maximum likelihood estimator polynomial time frequency transform (PTFT) [39,40]. Luckily, various fast algorithms [39,[41][42][43][44][45][46][47] have been proposed to reduce the computational complexity of the PTFT.…”

Section: Motivationmentioning

confidence: 99%

“…It was further generalized to support an arbitrary order PTFT in [42], based on the decimation-in-time (DIT) decomposition technique, to reduce the overall computational complexity. For example, the numbers of complex multiplications and additions are reduced by a factor of 2 M −1 log 2 N for the M th-order PTFT of length-N input sequence, compared with the algorithm that directly uses the 1D FFTs.…”

Section: Fast Algorithms For the Ptftmentioning

confidence: 99%

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Signal processing methods based on the local polynomial fourier transform

Li¹

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The majority of signals encountered in real applications, such as radar, sonar, speech, and communications, are often characterized by time-varying spectral contents. For this type of signals whose frequency contents evolve with time, signal representation in time or frequency domain alone cannot fully describe its time-varying characteristics. It would be far more useful to describe the signals with the time-frequency representations (TFRs). The TFRs for processing signals with time-varying frequencies can be generally categorized as linear and nonlinear transforms. The widely used linear transform is the short-time Fourier transform (STFT). The nonlinear transforms include the Wigner-Ville distribution (WVD) and various classes of quadratic time-frequency transforms. During the studies of various signal processing methods, it is found that the local polynomial Fourier transform (LPFT) is an important and effective processing tool for many practical applications, mainly because the LPFT is a linear transform and free from the cross terms that exist in the WVD. Furthermore, the LPFT uses extra parameters to approximate the phase of the signal into a polynomial form to describe time-varying signals with a much better accuracy than the STFT. This thesis focuses on the theoretical analysis of the LPFT, such as its uncertainty principle and SNR analysis, followed by applications to demonstrate its advantages and verify the theoretical analysis of the LPFT. Moreover, the ATTENTION: The Singapore Copyright Act applies to the use of this document. Nanyang Technological University Library Contents viii reassignment technique is employed to further increase the concentration of the local polynomial periodogram (LPP), which is the square of the LPFT. First, the uncertainty principle of the LPFT is investigated. It is shown that the uncertainty product of the LPFT of an arbitrary order is related to the signal parameters, the window function, and the errors of estimating the polynomial coefficients. The uncertainty principle of the LPFT becomes time-independent when the Gaussian window is used to segment the signal and the parameters of the LPFT kernel are estimated correctly. Factors that affect resolutions of signal representation, such as the window width, the length of overlap between signal segments, order mismatch and estimation errors of polynomial coefficients, are also discussed. Examples in speech and bat sound processing are demonstrated to show the advantage of the LPFT, compared with the STFT and the WVD. Second, the quantitative signal-to-noise ratio (SNR) analysis of the LPFT is derived based on the relationship between the LPFT and the WVD. The quantitative SNR analysis of the pseudo WVD (PWVD) in continuous-time form is presented as well. Comparisons are made among the SNRs achieved by using the LPFT, the FT, the STFT and the PWVD. Both the theoretical analysis and simulations have shown that the LPFT can provide higher SNR improvement than the FT, the STFT and the PWVD. Third, applications in radar imaging...

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“…The work in[3] was extended for the third order PTFT and was reported in[99]. It was further generalized to support arbitrary order PTFT in[100] which exploits the "quasi-periodic property" of the PTFT to reduce the overall computational co-No for j > i > 0 to achieve a satisfactory accuracy of the parameter estimation. Thus, it is assumed that Pi = NdN o is a positive integer for any i > O 102…”

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confidence: 99%

Processing algorithms for signals with time-varying frequencies.

Ju¹

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III However, PTFT generally requires a huge computational complexity as it involves in multi-dimensional (MD) calculation. One of the primary objectives of this research project is to propose fast algorithms to calculate PTFTs efficiently. We have proposed a general class of fast algorithms for the computation of the PTFTs of complex-valued length-aPb input sequence, where a, band p are integers. The proposed algorithms provide the flexibility to support various input sequence lengths by setting the parameters of a, band p, and have regular computational structures for an easy implementation. Analysis and comparison on the computational complexity are also reported in terms of the numbers of complex additions and complex multiplications. It is verified that the computational complexity of the PTFTs is significantly reduced by using the proposed algorithms. Next, fast algorithms for the computation of the PTFTs of real-valued sequences are presented. Similar to the fast algorithms for complex-valued input sequences, the proposed algorithms can support various input sequence lengths by setting the parameters of a, band p, and have regular computational structures. Analysis and comparison on the computational complexity are made in terms of the numbers of real additions and real multiplications. With the conjugate symmetric property, the presented fast algorithms, with a = 3, 4 and 8, effectively reduce the computational complexity to be about one-half of that needed by the reported fast algorithms for complex-valued sequences. In many practical applications, real-time processing is required. Under these conditions, the software implementation may not be fast enough. To support high processing throughput, it is necessary to implement the fast algorithms in hardware. We propose a hardware-oriented radix-2 2 algorithm for PTFT of coefficients carry desirable information. For complex-valued PPSs, the PTFT serves as the MLE of the phase coefficients, and can be efficiently computed using the generalized class of fast algorithms presented in Chapter 4, which supports arbitrary order of PTFT with the dimension size being aPb, where a, P and b are positive integers. In practice, many applications inherently deal with real-valued signals. The natural signals such as speech, marine mammal sounds, heart rate etc. are realvalued. In image and audio watermarking schemes [36, 37], the real-valued chirp signals are embedded as watermark signals, where each chirp rate corresponds

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Fast algorithms for polynomial time frequency transform

Cited by 9 publications

References 19 publications

A novel approach for DFT computation

A novel approach for DFT computation

Signal processing methods based on the local polynomial fourier transform

Processing algorithms for signals with time-varying frequencies.

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