### Table 3: Radix-2r twiddle factor storage as a function of input order. 4.3 Summary of twiddle factor storage requirements The preferred combinations of data allocation and FFT type is summarized in Table 3. For multi-dimensional FFT each axis has its set of twiddle factors. The twiddle factors for an axis is a subset of the twiddle factors for the longest axis. With axes of length P1 P2 : : : Pk the minimum number of twiddle factors is max`(R ? 1)P` R . With separate storage of the twiddle factors for each axis the total storage is P`(R ? 1)P` R , which is still less than the storage required for a one-dimensional FFT of size `P`. The Inverse Discrete Fourier Transform can be computed as a Discrete Fourier Transform by using conjugate twiddle factors.

1992

### Table1. Some Properties of the Fourier Transformation Signal Fourier Transform

"... In PAGE 2: ... The discrete Fourier trans- formation (FT) F(u; v) of a 2D discrete image f[x; y] 2 IRN N is de ned as F(u; v) = 1 N N?1 X j=0 N?1 X k=0 f[j; k] exp(?2 i(uj + vk) N ) (1) with i = p?1 and u; v = 0; 1; :::; N ? 1. Using the FT properties shown in Table1 , the following characteristics of the amplitude spectrum A of F(u; v) can be derived: it is invariant with respect to translation, inverse-variant with respect to scaling and variant with respect to rotation. Thus, features based on the amplitude spectrum of an image are translation invariant.... ..."

### Table 4: Basic Properties of the Discrete Fourier Transform

1995

"... In PAGE 13: ... (We extend the de nition to non-periodic sequences of the same length N, by regarding them as periodic, as de ned earlier in this section). Table4 lists the discrete counterparts of the properties given in Table 3 for the continuous case. We assume that we are dealing only with well de ned combinations.... ..."

Cited by 9

### Table 4: Basic Properties of the Discrete Fourier Transform

1995

"... In PAGE 14: ... (We extend the de nition to non-periodic sequences of the same length N, by regarding them as periodic, as de ned earlier in this section). Table4 lists the discrete counterparts of the properties given in Table 3 for the continuous case. We assume that we are dealing only with well de ned combinations.... ..."

Cited by 9

### Table 4: Basic Properties of the Discrete Fourier Transform

1995

"... In PAGE 13: ... (We extend the de nition to non-periodic sequences of the same length N, by regarding them as periodic, as de ned earlier in this section). Table4 lists the discrete counterparts of the properties given in Table 3 for the continuous case. We assume that we are dealing only with well de ned combinations.... ..."

Cited by 9

### Table 4: Basic Properties of the Discrete Fourier Transform

1995

"... In PAGE 14: ... (We extend the de nition to non-periodic sequences of the same length N, by regarding them as periodic, as de ned earlier in this section). Table4 lists the discrete counterparts of the properties given in Table 3 for the continuous case. We assume that we are dealing only with well de ned combinations.... ..."

Cited by 9

### Table 2: Parameters for discrete Fourier Transforms. Frequency Time

"... In PAGE 17: ...3.4) and (3.5) are used as the Fourier transforms in MATLAB. Now we have to find out what the difference between the (continuous) Fourier transforms and the discrete Fourier transforms is. The parameters for the discrete Fourier transforms and the corresponding figures are shown in Table2 and Figure 4. Analysis: We start this analysis with the (continuous) Fourier transform of (3.... ..."

### TABLE 2 Upper and lower bound for discrete Fourier transform.

### Table 6: Discrete Fourier transform vs. wavelet-packet transform for spare representation example

"... In PAGE 16: ... To motivate the choice of wavelet over Fourier consider the function f(x1; x2) = b0 + b1x1 + b2x2 + b3x1x2, and the associated samples show in Table 5. If we perform a discrete (trigonomic) Fourier and a wavelet-packet transform on the data, we obtain the results presented in Table6 . The wavelet transform is seen to provide a sparser representation of the feature variables, re ecting the orthogonal basis in the feature space.... ..."