The intrinsic structure of FFT and a synopsis of SFT

Transcription

1 Global Journal o Pure and Applied Mathematics. ISSN Volume 12, Number 2 (2016), pp Research India Publications The intrinsic structure o FFT and a synopsis o SFT Hwajoon Kim 1 Kyungdong University, School o IT Engineering, 11458, Kyungdong Univ. Rd. 27, Yangju, Gyeonggi, S. Korea. Abstract We have checked the intrinsic structure o ast Fourier transorm, centering on Danielson-Lanczos lemma, and investigated a synopsis o sparse Fourier transorm. AMS subject classiication: 34A25, 28A05. Keywords: Fast Fourier transorm. 1. Introduction I a given unction (x)has only values at inite points, we think over the discrete Fourier transorm (DFT). This DFT concerns large amounts o equal speed data, as it occur in compression, signal processing, analysis images, time series analysis, and various simulation problems. The ast Fourier transorm (FFT) is a practical method o computing DFT that needs only O(N)log 2 N operations instead o O(N 2 ). Although this algorithm is not novel, the importance is still valid because the existing commercialized system is based on FFT. Additionally, this FFT has an important application to Radon transorm which is widely applicable to tomography, and the short-term ast Fourier transorm weightily is applied to the analysis o brain waves. It is well-known act that the Fourier transorm o has the orm o (w)= ˆ (x)e iwx dw, 1 Corresponding author.

2 1672 Hwajoon Kim and we can interpret that a unction is the given signal and is the requency spectrum o. Let us deine the DFT o the given signal =[ 0,, N 1 ] T to be the vector ˆ =[ˆ 0,, N 1 ˆ ] with the components N 1 n = Nc n = k e inx k, where k = (x k ) and x k = 2πk/N [7]. This is the requency spectrum o the signal, and we can write ˆ = F N, where the N N Fourier matrix F N =[e nk ] has the entries e nk = e inx k = e 2πink/N = w nk,w= w N = e 2πi/N or n, k = 0,,N 1. Here, since N is normally large, we need quite a number o operations. To avoid this diiculty, we use FFT, a computational method or the DFT, which needs only O(N log N) instead o O(N). This FFT can get possible the Fourier matrix to break down the given problem into smaller problems by using k=0 w 2 2M = (e 2πi/N ) 2 = e 2πi/M = w M or N = 2M, sometimes called the Danielson-Lanczos lemma. With relation to this Cooley-Tukey FFT, several researches have been pursued and progressed [1-6, 9-11]. The Hartley transorm [2], an eicient real Fourier transorm algorithm, gave a urther increase in speed, Winograd transorm algorithm [10] and Sande- Tukey algorithm [9] gave big inluence as well. The sparse Fourier transorm(sft) is the most advanced algorithm recently presented by Katabi, and it is a kind o improved FFT which is increased in speed as ten times as FFT. Signals whose Fourier transorms include a relatively small number o heavily weighted requencies are called sparse. This SFT determines the weights o a signals having heavily weighted requencies; the sparser the signal, the greater the speedup which the algorithm provides[8]. The researches that identiy the most heavily weighted requency is being proceeded. For k is the sparsity o the signal spectrum and or the typical case o n is a power o 2, the SFT is known to require O(log n nk log n) only as the number o needed operations or the l /l 2 guarantee, and the importance o speed-up problem is increased in connection with big data ones. In a word, SFT is a compressed version o DFT and it is using the sparsity o the spectrum o the signal. O course, this SFT is closely related to the concept o streaming which save run time which is required to read the entire original data set. In this article, we have checked the intrinsic structure o FFT and a synopsis o the FFT and SFT. 2. The intrinsic structure o ast Fourier transorm To begin with, let us start by an example. Let us consider the case o the sample values N = 8. Then w = e 2πi/N = e πi/4 = 1 (1 i) 2

3 The intrinsic structure o FFT and a synopsis o SFT 1673 and so w nk = ( 1 2 (1 i)) nk. Since the requency spectrum o the signal is wrote as ˆ = F N,wehave ˆ = F 8 = = w w 2 w 3 w 4 w 5 w 6 w 7 1 w 2 w 4 w 6 w 8 w 10 w 12 w 14 1 w 3 w 6 w 9 w 12 w 15 w 18 w 21 1 w 4 w 8 w 12 w 16 w 20 w 24 w 28 1 w 5 w 10 w 15 w 20 w 25 w 30 w 35 1 w 6 w 12 w 18 w 24 w 30 w 36 w 42 1 w 7 w 14 w 21 w 28 w 35 w 42 w w i iw 1 w i iw 1 i 1 i 1 i 1 i 1 iw i w 1 iw i w w i iw 1 w i iw 1 i 1 i 1 i 1 i 1 iw i w 1 iw i w It is well-known act that the given vector =[ 0 N 1 ] T is split into two vectors with M components each, namely, even and odd. We determine the DFTs and ev =[ˆ ev,0 ev,2 ev,n 2 ] T = F M ev od =[ˆ od,1 od,3 od,n 1 ] T = F M od involving the same M M matrix F M [7]. From these things we have the components o the DFT o by n = n ev,n + w N od,n (2) and or n = 0,,M (1) ˆ n+m = ˆ ev,n w N n ˆ od,n (3) Example 2.1. From the case o N = 8, let us check the validity o the above ormulas (2) and (3). Solution. When N = 8, we have w = w N = 1 2 (1 i) and M = N/2 = 4. Thus

4 1674 Hwajoon Kim w = w M = e 2πi/4 = e πi/2 = i. From the above statements, ˆ ˆ ev = 2ˆ = F 4 ev = 1 i 1 i = 4ˆ 6 1 i 1 i 6 and similarly, we have 1ˆ od = 3ˆ = F 4 od = 5ˆ i 1 i i 1 i Using the ormulas (2) and (3), we have = i i i 2 4 i i i i 3 5 i 7 0 = 0 ev,0 + w N od,0 = ( ) + ( ) ˆ 1 = ˆ ev,1 + w N 1 ˆ od,1 = ( 0 i i 6 ) + w( 1 i i 7 ) = 2+4 = 2 ev,2 w N od,2 = ( ) ( i)( ) 7 = 3+4 = 3 ev,3 w N od,3 = ( 0 + i 2 4 i 6 ) ( iw)( 1 + i 3 5 i 7 ). This agrees with the equation (1).. 3. A synopsis o Sparse Fourier transorm The concept o compress sensing (CS) has aroused many researcher s interest in signal processing ield. We are using sample signals in their entire ones either to save space while storing them or to save time while sending them. In the compressing process, we sample only the qualiying parts o the signals, and we call it the signal sparsity. In a word, we can compress data by using the signal sparsity. Even i the given signal is not sparse, we can use the best k-sparse approximation o its Fourier transorm. The SFT applies almost same principle as CS to Fourier transorm, and computes the signiicant coeicients in the requency domain. Let us see the needed deinitions. The requency spectrum o the signal is written as ˆ = F N in section 2, and equivalently, we can write N 1 n = Nc n = k e inx k, k=0

5 The intrinsic structure o FFT and a synopsis o SFT 1675 where k = (x k ) and x k = 2πk/N. Thus we can write F 1 N = e inx k. Since 1 = F N ˆ,wehave k = N 1 k=0 n e inx k or x k = 2πk/N. In case that a vector contains a single requency, we can write k = n e inx k = n e in2πk/n. In this case, let us compute k+1 / k = e 2πnki/N = cos (2πk/N) + i sin (2πk/N). This requires only 2 entries, and so, aster than the existing FFT [5]. [4] insists that this SFT does not estimating large coeicients, but subtracting them and recursing on the reminder, it identiies and estimates the k largest coeicients in one shot, in a manner akin to sketching/streaming algorithms. The approximation ˆ o satisies the ollowing l 2 /l 2 guarantee or a unction whose Fourier transorm ˆ: ˆ 2 C min g 2, k sparse g where C is some approximation actor and the minimization is over k-sparse signals[11]. Recently, the approximation is perormed or l /l 2 guarantee: ˆ ˆ 2 ɛ ˆ g 2 2 /k + δ 2 1, with probability 1 1/n. It is known that this l /l 2 guarantee is better than l 2 /l 2 one in speed-up and eiciency. The detailed techniques are ound in [1], [4] and [5]. Next, let us check the computational side. Normally, the SFT hashes the Fourier coeicients o the input signal into a small number o bins, and the vector ˆ computed by the algorithm satisies l 2 l 2 or l /l 2 guarantee: The sum and the weighted sum o the Fourier coeicients stored inside i-th bin are deined as û i = n ˆ n, û i = n n ˆ n respectively, or n in i-th bin. I the given signal is sparse, we can make each bin contains just one coeicient (or requency). Otherwise, i the given signal is not sparse or sparse enough, that is, i each bin contains more than one coeicient, then we have to approximate it to sparse. I k = O(N) and the number o unidentiied coeicients is k, then we can remove k k rom the given signal and we say k -sparse. Consequently, the identiied coeicients k k is removed rom bins, and the process is repeated to each coeicients are identiied. Reerences [1] A. Akavia, Deterministic sparse Fourier approximation via ooling arithmetic progressions, COLT. (2010), [2] R. Bracewell, The ast Hartley transorm, Proc. IEEE 72 (1984),

6 1676 Hwajoon Kim [3] S. K. Q. Al-Omari and A. Kilicman, On the generalized Hartley and Hartley-Hilbert transormations, Adv. Di. Eqs., 222 (2013), [4] A. C. Gilbert, P. Indyk and M. Iwen, Recent Ddvelopments in the sparse Fourier transorm, Sig. Proce. Mag., IEEE 31 (2014), [5] H. Hassanieh, P. Indyk, D. Katabi and E. Price, Simple and practical algorithm or sparse Fourier transorm, Proceedings o the twenty-third annual ACM-SIAM symposium on Discrete Algorithms (2012). [6] Hj. Kim and S. Lekcharoen, A cooley-tukey modiied algorithm in ast Fourier transorm, Kor. J. Math. 19 (2011), [7] E. Kreyszig, Advanced Engineering Mathematics, Wiley, Singapore, [8] MIT news, The aster-than-ast Fourier transorm, Jan. 18, 2012, edu/2012/aster-ourier-transorms [9] J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, Springer Verlag, [10] S. Winograd, On computing the discrete Fourier transorm, Math. Compu. 32 (1978), [11] L. Ying, Sparse Fourier transorm via butterly algorithm, SIAM J. Sci. Compu. 31 (2009),