the discrete fourier transform part 2

... time, as the sum of all the Chapter 8- The Discrete Fourier Transform 159 Sample number 0 16 32 48 64 -2 -1 0 1 2 a. x1[ ], signal being analyzed Sample number 0 16 32 48 64 -2 -1 0 1 2 b. x2[ ], ... 16 -8 -4 0 4 8 0 2 4 6 8 10 12 14 16 -8 -4 0 4 8 0 2 4 6 8 10 12 14 16 -8 -4 0 4 8 0 2 4 6 8 10 12 14 16 -8 -4 0 4 8 0 2 4 6 8 10 12 14 16 -8 -4 0 4 8 0 2 4 6 8 10 12 14 16 -8 -4 0 4 8 0 2 4 6 8 10 12 14 ... to the frequency. This is the frequency response of a low-pass filter. Chapter 8- The Discrete Fourier Transform 143 0 2 4 6 8 10 12 14 16 -8 -4 0 4 8 0 2 4 6 8 10 12 14 16 -8 -4 0 4 8 0 2 4...

Ngày tải lên: 13/09/2012, 09:49

... is called the discrete Fourier transform of the N points h k . Let us denote it by H n , H n ≡ N−1  k=0 h k e 2 ikn/N ( 12. 1.7) The discrete Fourier transform maps N complex numbers (the h k ’s) ... operations to generate the required powers of W . So, the discrete Fourier transform appears to be an O(N 2 ) process. These appearances are deceiving! The discrete Fourier transform can, in fact, ... standard answer was this: Define W as the complex number W ≡ e 2 i/N ( 12. 2.1) Then ( 12. 1.7) can be written as H n = N−1  k=0 W nk h k ( 12. 2 .2) In other words, the vector of h k ’s is multiplied...

Ngày tải lên: 21/01/2014, 18:20

... denote the repeating part of a decimal expansion, x =0. 123 ⇒ 1000x = 123 . 123 = 123 + x ⇒ 999x = 123 ⇒ x = 123 999 Other examples of irrational numbers include π =3.1415 926 5358979 323 84 626 4338 327 95 028 84197169399375105 820 9749 ... =3.1415 926 5358979 323 84 626 4338 327 95 028 84197169399375105 820 9749 e =2. 71 828 1 828 45904 523 536 028 74713 526 624 9775 724 70936999595749669 Let ˆx n denote the n-digit decimal expansion of an arbitrary real num- ber x. Then ˆx n is ... out the roots of p(x) ans = 1.3005191730 720 6 + 1.10944 723 819596i 1.3005191730 720 6 - 1.10944 723 819596i -0.755047 925 01755 + 1 .27 501061 923 774i -0.755047 925 01755 - 1 .27 501061 923 774i -1.0909 424 9610903 Matlab...

Ngày tải lên: 31/03/2014, 15:21

... + 3 cos  π 2 n  = 3 + 3 2 e  2 8 2n + 3 2 e − 2 8 2n = 1 8  24 + 12 e  2 8 2n + 12 e  2 8 (8 2) n  . So by coefficient matching, we see that X[k] = {24 , 0, 12, 0, 0, 0, 12, 0}. Example. ... 3 3 0 3 2 2 3 0 n=N-1 0 1 322 4 1 4 2 4n=0 n=1 If the sequence is the same whether you read it out CW or CCW, then it is circularly even. By considering the points around the circle, we conclude the following. An ... = 2 δ(ω −ω 0 ) ∗R(ω) = 2 R(ω − ω 0 ), where the above δ is a Dirac impulse. Thus X[k] = X(ω)    ω= 2 N k = 2 R  2 N k −ω 0  . When ω 0 = 2 N k 0 , then the sinc is sampled only at the...

Ngày tải lên: 08/06/2014, 20:19

... fractional Fourier transform. In this work, we disclose eigenvectors of the DFT matrix inspired by the ideas behind bilinear transform. The bilinear transform maps the analog space to the discrete ... the analog s-domain is mapped to the unit circle one-to-one without aliasing in the discrete z-domain, it is appropriate to use it in the discretization of the eigenfunctions of the Fourier transform. ... The discrete fractional Fourier transform, ” IEEE Transactions on Signal Processing, vol. 48, no. 5, pp. 1 329 –1337, 20 00. [7] S C. Pei, W L. Hsue, and J J. Ding, Discrete fractional Fourier transform...

Ngày tải lên: 21/06/2014, 16:20

... 38 4 02 350 120 Cmul 22 22 22 120 120 120 Structural 25 2 25 2 25 2 124 124 124 Total 350 3 42 3 12 646 594 366 The gate count is measured as equivalent gates as used in cell- based ASIC design. The ... DFT length 4 8 16 32 64 Matrix Size — 26 640 80 640 — — Mult. ∗ [22 ] Latency 22 —— Winograd Size 5 129 14 137 36 893 — — Latency 22 2—— CSD-CZT Size 10 349 14 1 92 23 630 41 426 78 061 Latency 4444 ... becomes X[k] = W k 2 /2 N N −1  n=0  x[ n]W n 2 /2 N  W −(k−n) 2 /2 N . (6) The computation of the DFT is therefore done in three steps: (1) N multiplications of x[n]withW n 2 /2 N ; (2) linear convolution...

Ngày tải lên: 22/06/2014, 23:20

... ND2% 120 0 IF K% > J% THEN GOTO 124 0 121 0 J% = J%-K% 122 0 K% = K% /2 123 0 GOTO 120 0 124 0 J% = J%+K% 125 0 NEXT I% 126 0 ' 127 0 FOR L% = 1 TO M% 'Loop for each stage 128 0 LE% = CINT (2^ L%) 129 0 ... = N% /2 3170 N4% = N%/4-1 3180 FOR I% = 1 TO N4% 3190 IM% = ND2%-I% 320 0 IP2% = I%+ND2% 321 0 IPM% = IM%+ND2% 322 0 REX(IP2%) = (IMX(I%) + IMX(IM%)) /2 323 0 REX(IPM%) = REX(IP2%) 324 0 IMX(IP2%) = ... an Chapter 12- The Fast Fourier Transform 23 5 TABLE 12- 4 The Fast Fourier Transform in BASIC. 1000 &apos ;THE FAST FOURIER TRANSFORM 1010 'Upon entry, N% contains the number of points in the DFT,...

Ngày tải lên: 13/09/2012, 09:50

... we restrict the mathematics to be real numbers, problems arise. In other words, these problems are not solved by the complex Fourier transform, they are introduced by the real Fourier transform. In the ... ways that the real Fourier transform is awkward. When the complex Fourier transform was introduced, the problems vanished. Wonderful, we said, the complex Fourier transform has solved the difficulties. While ... Fourier Transforms A time domain signal can be either continuous or discrete, and it can be either periodic or aperiodic. This defines four types of Fourier transforms: the Discrete Fourier Transform...

Ngày tải lên: 13/09/2012, 10:20

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Ngày tải lên: 24/12/2013, 02:16