ࡱ> ]_\r7 bjbjUU "J7|7|xlt!t!t!8!!|3P"" # # # # # ,#2222222$94 Y62E4# # #4#4#2^% # #=3^%^%^%4# # #2^%4#2^%^%`(:),N) #D" }%t!#`:) N) S303D) 6^%6N)^%X@@HYPERLINK "..\\Audio\\FFT.doc"Previous lectureHYPERLINK "JPEG.DOC"Next Lecture HYPERLINK "..\\320Syllabus.doc" Syllabus HYPERLINK "..\\..\\Homework\\Week9\\Week9.doc" HomeworkDiscrete Cosine Transform (DCT) We saw in the previous lecture that we can recreate audio signals by adding up sine or cosine waves in the right ratios. There are mathematical formulas that tell us the right way to mix these [co]sine waves. One way is the Discrete Fourier Transform. Another is the Discrete Cosine Transform. There is a good illustration on page 79 of Symes showing why the DCT gives better compression for non-periodic signals. A 1-D array V of N numbers is transformed to an array T by:  c(0) = (1/N) c(k) = (2/N) for k `" 0. T is the DCT of V. JPEG does 2-D DCT on blocks of size 8x8. Thus we are most concerned with the case N=8. We can make a table of all the values involved and write this as a matrix equation: T = C M1 V where T and V are 1x8 columns. C is an 8x8 diagonal matrix whose diagonal is 1/"8 on the upper left and 0.5 at the other 7 positions. The entries for M1 are given by the table: 1.0000001.0000001.0000001.0000001.0000001.0000001.0000001.0000000.9807850.8314700.5555700.195090-0.195090-0.555570-0.831470-0.9807850.9238800.382683-0.382683-0.923880-0.923880-0.3826830.3826830.9238800.831470-0.195090-0.980785-0.5555700.5555700.9807850.195090-0.8314700.707107-0.707107-0.7071070.7071070.707107-0.707107-0.7071070.7071070.555570-0.9807850.1950900.831470-0.831470-0.1950900.980785-0.5555700.382683-0.9238800.923880-0.382683-0.3826830.923880-0.9238800.3826830.195090-0.5555700.831470-0.9807850.980785-0.8314700.555570-0.195090 We can get back from T to V by the inverse discrete cosine transform (IDCT):  We can see that T contains coefficients that are used to build the original signal V from cosines of different frequencies. Thus the DCT can be viewed as finding coefficients. We can see that the table of cosines for the IDCT is just the transpose of M1 for the DCT. These transform matrices are orthogonal, which means that the transpose is equal to the inverse. (This is not true for non-orthogonal matrices.) We can right multiply the forward DCT matrix equation by the transpose: 1/C T = M1 V M1T 1/C T = M1 M1T V = V We can note that T(0) is simply the average value of V(0) V(N-1). It is called the DC coefficient. The others are called AC. If we plot the first 4 rows of the 8 point DCT we see DC, period, 1 period, 1.5 periods:  The last 4 lines plot as:  [Miano examples: Fig 7-2, Table 7.1, Fig 7.4, Table 7.2, Fig 7.5] The Discrete Fourier Transform (DFT) A 1-D array V of N numbers is transformed to an array T by:  where  We can write this as T = C V (Mc j Ms) Note that M1 does not equal Mc. The cosine transform is not the cosine part of the Fourier transform. In the case of N = 8, the real and imaginary transform matrices for the DFT are in the two tables below. Mc = 1.0000001.0000001.0000001.0000001.0000001.0000001.0000001.0000001.0000000.7071070.000000-0.707107-1.000000-0.7071070.0000000.7071071.0000000.000000-1.0000000.0000001.0000000.000000-1.0000000.0000001.000000-0.7071070.0000000.707107-1.0000000.7071070.000000-0.7071071.000000-1.0000001.000000-1.0000001.000000-1.0000001.000000-1.0000001.000000-0.7071070.0000000.707107-1.0000000.7071070.000000-0.7071071.0000000.000000-1.0000000.0000001.0000000.000000-1.0000000.0000001.0000000.7071070.000000-0.707107-1.000000-0.7071070.0000000.707107 Ms = 0.0000000.0000000.0000000.0000000.0000000.0000000.0000000.0000000.0000000.7071071.0000000.7071070.000000-0.707107-1.000000-0.7071070.0000001.0000000.000000-1.0000000.0000001.0000000.000000-1.0000000.0000000.707107-1.0000000.7071070.000000-0.7071071.000000-0.7071070.0000000.0000000.0000000.0000000.0000000.0000000.0000000.0000000.000000-0.7071071.000000-0.7071070.0000000.707107-1.0000000.7071070.000000-1.0000000.0000001.0000000.000000-1.0000000.0000001.0000000.000000-0.707107-1.000000-0.7071070.0000000.7071071.0000000.707107 We can plot the first four lines of Mc DC, 1 period, 2 periods, 3 periods:  The 2-D DCT It is possible to do a 2-D DCT by doing a 1-D DCT on the rows, followed by a 1-D DCT on the columns. We do the 2-D case of the DCT by using the transform matrix M: EMBED Equation.3 The transform can be expressed as a matrix: T = M V MT V = MT T M By using some clever mathematics (see chapter 10 in Miano) it is possible to perform the 8x8 matrix multiplication with just 29 addition operations and 5 multiplication operations. It turns out that the matrix M is the same as the 1-D matrix M1 except that it incorporates the coefficients C. M = C M1. The entries for M are: 0.3535530.3535530.3535530.3535530.3535530.3535530.3535530.3535530.4903930.4157350.2777850.097545-0.097545-0.277785-0.415735-0.4903930.4619400.191342-0.191342-0.461940-0.461940-0.1913420.1913420.4619400.415735-0.097545-0.490393-0.2777850.2777850.4903930.097545-0.4157350.353553-0.353553-0.3535530.3535530.353553-0.353553-0.3535530.3535530.277785-0.4903930.0975450.415735-0.415735-0.0975450.490393-0.2777850.191342-0.4619400.461940-0.191342-0.1913420.461940-0.4619400.1913420.097545-0.2777850.415735-0.4903930.490393-0.4157350.277785-0.097545 HYPERLINK "..\\Audio\\FFT.doc"Previous lectureHYPERLINK "JPEG.DOC"Next Lecture HYPERLINK "..\\320Syllabus.doc" Syllabus HYPERLINK "..\\..\\Homework\\Week9\\Week9.doc" Homework CS 320  PAGE 6 Oct 11, 00  !1234HIJVWXYz{| z | W X > ? '('("#$%023456 jMU jEHU OJQJ^JH*H* jEHUjUjmUjU0JjU jUG3X hfddddd$$Ifl\:,"064 la$If v  2 D V h z | 44Ff+Ff0 $$Ifa$   ' 1 ; E N W X a k u  44Ff! Ff& $$Ifa$    ! + 4 > ? 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