1. The Report of Monographic Study
A New Algorithm to Compute the Discrete Cosine Transform
A Fast Algorithm for Modified Discrete Cosine Transform
指導教授 : 尤信程 老師
報 告 者 : 侯侑成
報告地點 : 資工所辨公室
報告時間 : 90年12月6日(星期四) AM 1:30~3:00

2. Outline Introduction to Discrete Cosine Transform
Lee’s FDCT Algorithm and Example
Conclusions of Lee’s Algorithm
Time Domain Aliasing Cancellation (TDAC)
Derivation of Fast Inverse MDCT
Summary of the Fast MDCT
Conclusion Remarks
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3. Introduction
The discrete cosine transform (DCT) is one of the most widely used tools in the area of digital signal processing, because it has energy packing capabilities and also approaches the statistically optimum Karhunen-Loeve transform for highly correlated signals.
In the DCTs we present the definitions for the four DCTs as classified by Wang. It can be seen that DCT-Ⅱ is asymptotically equivalent to KLT for Markov-1 signals as ρ→1. Since this asymptotic behavior is independent of N, this explains the empirical results in which DCT-Ⅱ seems to have the best variance distribution compared with the other non-KLT discrete transforms.
Among the various algorithms for the DCT, the Lee fast cosine transform algorithm has received the most attention.
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4. A Fast DCT Algorithm The N-point DCT is Its inverse(IDCT)is given by
Where and otherwise , and define C such that
Then the N-point IDCT becomes
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5. FDCT Algorithm(cont’d)
Decomposing x(k) into even and odd indexes of n (assuming that N is even)
where
since
We rewrite g(k) in the form
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6. FDCT Algorithm(cont’d)
Since
we have
So,if we define
then
because so
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7. FDCT Algorithm(cont’d)
Now we define
and
Finally, we get
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8. Flowgraph of Lee’s DCT
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9. Conclusions of Lee’s Algorithm
Lee’s algorithm, based on the symmetry of the kernel to decompose an N-point IDCT into the sum of two N/2-point IDCT and decompose it further by repeating this process recursively. The DCT can be obtained by “transpose” the IDCT.
The number of real multiplications thus appears to be (N/2)log2N for an N-point FCT with N=2m, which is about half the number required by existing efficient algorithms.
REF.
Chen
Wang
Lee
Vetterll
Suehiro
Hou
Loeffler
Mult.
Add. 16 26 13 29 12
12(7)
29(18) 11
Number of operations for some 8-point fast DCT algorithm
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10. Conclusions of LFDCT (cont’d)
However, It has very regular first stage, but has irregular data flow in the last stage. For large length transforms, the indexing of last part of the Lee’s Algorithm may cause difficulties. We show that the regularity of the LFDCT can be improved by arranging the input data into the Hadamard order. We then show the close relationship between the LFDCT and the RCFDCT.
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11. Figure(1)
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12. The Karhunen-Loeve Transform
Consider N sampled points of a zero mean random vector x, then x expanded in terms of Фi is We seek the best representation of a given random function in the MSE sense. ( )
The KLT is said to be an optimal transform because it has the following properties
It completely decorrelates the signal in the transform domain.
It minimizes the MSE in bandwidth reduction or data compression.
It contains the most variance (energy) in the fewest number of transform
coefficients.
It minimizes the total representation entropy of the sequence.
It is immediately clear that the basis functions are dependent on the auto-covariance and therefore cannot be predetermined.
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13. The Family of Forward DCTs
The family of forward DCTs can be defined as follows. They are very similar, and type Ⅱ, called DCT-Ⅱ, is the most widely used.
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14. Forward Discrete Cosine Transform
In one definition the cosine transform of x(n) is the real part of X(ω). The real part of X(ω) is also equal to the Fourier transform of the even part of x(n) defined by xe(n)=[x(n)+x(-n)]/2.
Therefore, the cosine transform of x(n) is equal to the Fourier transform of xe(n). Now, if x(n) is causal, i.e., x(n)=0 for n <0, then xe(n) and , therefore the cosine transform uniquely specifies x(n).
In that case, xe(n) can be thought of as an even extension of x(n). Therefore the cosine transform of a causal x(n) can be obtained as the Fourier transform of an even extension of x(n). This view point forms the basis of the derivation of the DCT following.
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15. Let be a 2N-point even extension of define by:
Forward DCT (cont’d)
The DFT of is given by
Let be a 2N-point even extension of define by:
Therefore,
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16. Forward DCT (cont’d)
x(n)
y(n)
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17. Time Domain Aliasing Cancellation (TDAC)
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18. Derivation of Fast Inverse MDCT
The inverse MDCT kernels are as following
and
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19. Derivation of FIMDCT(cont’d)
It can be seen that if k0and k0+1 disappear, equations will have the form similar to standard inverse DCT
Let i=k+k0, equation is decomposed to
Noting that
Then,
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20. Derivation of FIMDCT(cont’d)
Keeping index k instead of i, we have the
where
Repeating the same procedure to h(n) yields
where
Substitution of g(n) and h(n) into gives
where
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21. Derivation of FIMDCT(cont’d)
We decompose into two parts
where
Finally, we obtain
where
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22. Summary of the Fast MDCT
By simple time shift and reverse, MDCT is computed using Lee’s Fast Discrete Cosine Transform (LFDCT). The number of real multiplications is reduced for 2N2 to (N/2)log2N+N.
We summarize the fast algorithm of IMDCT as following. Denote by X(k), k=0,1,…,2N-1, the input data sequence,
a) Circular data shift
b) Sum of adjacent terms
c) Constructing the half sequence
d) Computing N-point IDCT using FCT
e) Post processing, multiplied by coefficients
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23. Conclusion Remarks
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