Fast Trigonometric Transforms

4.7.4 Fast Trigonometric Transforms

Two real transforms, the discrete sine transform (DST) and discrete cosine transform (DCT) , are of interest. There are several variations of these. We define the first two as follows:

• Given real data f j , j = 1 : N − 1, compute

• Given real data f j , j = 0 : N, compute

(The double prime on the sum means that the first and last term are to be halved.)

4.7. The Fast Fourier Transform 513 The real and imaginary parts of the DFT matrix F N consist of cosines and sines,

F N =C N − iS N , where 2π

(C N ) kj = cos kj

, k, j = 0 : N − 1. (4.7.26) N

(S N ) kj = sin kj

The DST and DCT transforms can be expressed in matrix form as y =S 2N ( 1 : N − 1, 1 : N − 1)f,

y =C 2N ( 0 : N, 0 : N) ˜ f, respectively, where ¯ f =( 1 2 f 0 ,f 1 ,...,f N −1 , 1 2 f N ) T .

These two transforms can be computed by applying the FFT algorithm (for real data) to an auxiliary vector ˜ f formed by extending the given data vector f either into an odd or even sequence. For DST-1 the data f j , j = 1 : N − 1, are extended to an odd sequence of length 2N by setting

f 0 =f N = 0,

f 2N−j ≡ −f j , j = 1 : N − 1. For example, the data f = {f 1 ,f 2 ,f 3 }, (N = 4) are extended to

f ˜ = {0, f 1 ,f 2 ,f 3 ,

0, −f 3 , −f 2 , −f 1 }.

The extended vector satisfies ˜ f = −T 2N f ˜ , and thus by Table 4.7.1 the DFT of ˜ f will be pure imaginary. For DCT-1 the data f j , j = 0 : N, are extended to an even sequence of length 2N by setting

f 2N−j ≡f j , j = 1 : N.

For example, the data f = {f 0 ,f 1 ,f 2 ,f 3 ,f 4 }, (N = 4) are extended to

f ˜ = {f 0 ,f 1 ,f 2 ,f 3 ,f 4 ,f 3 ,f 2 ,f 1 }

so that ˜ f =T 2N f ˜ . By Table 4.7.1 the DFT of ˜ f will then be real. Theorem 4.7.6 (Van Loan [366, Sec. 4.4]).

Let f j , j = 1 : N − 1, form a real data vector f and extend it to a vector ˜ f with

f ˜ 0 =˜ f N = 0 so that ˜ f = −T 2N ˜ f . Then y( 1 : N − 1) is the DST of ˜ f , where

y = y(0 : 2N − 1) = F 2N f. ˜

Let f j , j = 0 : N, form a real data vector f and extend it to a vector ˜ f so that

f ˜ =T 2N ˜ f . Then y( 0 : N) is the DCT of f , where

1 y = y(0 : 2N − 1) = F 2N f. ˜

There is an inefficiency factor of two in the above procedure. This can be eliminated by using a different auxiliary vector. For details we refer to [294, pp. 420–421]; [366, Sec. 4.4].

514 Chapter 4. Interpolation and Approximation In some application areas variants of the above transforms called DST-2 and DCT-2

turn out to be more useful. We define them as follows: • Given real data f j , j = 1 : N, compute

• Given real data f j , j = 0 : N − 1, compute

The DST-2 and DCT-2 transforms can also be obtained by extending the data vector f . For DST-2 the data vector f j , j = 0 : N −1, is extended to an odd sequence of length

2N. For example, the data {f 1 ,f 2 ,f 3 ,f 4 }, (N = 4) are extended to

f ˜ = {f 1 ,f 2 ,f 3 ,f 4 , −f 4 , −f 3 , −f 2 , −f 1 }. The extended vector satisfies f = −T 2N f , and thus by Table 4.7.1 the DFT of f will be

imaginary. For DCT-2 the data f j , j = 0 : N, are extended to an even sequence of length 2N.

For example, the data {f 0 ,f 1 ,f 2 ,f 3 }, (N = 4) are extended to

f ˜ = {f 0 ,f 1 ,f 2 ,f 3 ,f 3 ,f 2 ,f 1 ,f 0 }.

so that f = T 2N f . By Table 4.7.1 the DFT of f will then be real. We give without proof the following result, which allows the computation of DST-2

and DCT-2 from the FFT.

Theorem 4.7.7.

Let f j , j = 1 : N, form a real data vector f and extend it to a vector ˜ f so that

f ˜ = −T 2N ˜ f . Then y( 1 : N) is the DST-2 of f , where

y = y(0 : 2N − 1) = < 2N F 2N f, ˜

where

< 2N = diag (1, ω 4N ,...,ω 2N 4N ).

Let f j , j = 0 : N − 1, form a real data vector f and extend it to a vector ˜ f so that

f ˜ =T 2N ˜ f . Then y( 0 : N − 1) is the DCT-2 of f , where

1 y = y(0 : 2N − 1) = < 2N F 2N f. ˜

The two-dimensional DCT-2 transform has the property that, for a visual image, most of the information is concentrated in the first few coefficients of the DCT. For this reason the DCT-2 transform is often used in image compression algorithms. 165

165 This transform is used in the JPEG (Joint Photographic Experts Group) compression algorithm for image processing. Each 8×8 block in the image is transformed by a two-dimensional DCT-2 transform; see Strang [340].

4.7. The Fast Fourier Transform 515