修改型離散餘弦轉換之新快速演算法

Abstract

本論文對修改型離散餘弦轉換(MDCT)與其反轉換(IMDCT)提出兩個快速計算方法。第一個IMDCT的快速計算方法可分成三個步驟步驟，首先將信號乘以N/2點的餘弦數列，再做N/2點DCT的快速計算，接著再將計算出來的信號做簡單的遞迴加減運算，就可以求得N點IMDCT的值。第二個新的快速計算方法的步驟剛好跟第一個新的快速計算方法相反，先將信號做遞迴加減法的運算，再對運算出來的信號做N/2點的IDCT的計算，接著再將計算出來的信號乘以N/2點的餘弦數列，就可以求出N點IMDCT的值。將上述兩個新的快速IMDCT的信號流程圖做轉置就可以得到兩個新的快速MDCT的計算方式。 本論文對文獻中和新的快速MDCT的計算方法做計算量的比較和定點誤差的模擬，新的方法計算量和文獻中的方法方法一樣好，但是定點誤差卻比已經提出的方法來的小，所以，本論文提出的新的快速MDCT計算方法在數值的精確度上有較好的表現。

In this thesis, we present two new algorithms for computing the inverse modified discrete cosine transform(IMDCT) such that the computation complexity is equal to but the numerical performance is better than existing fast algorithms. To compute N-point IMDCT, the first algorithm is realized in the following sequence: multiplication of the N/2 input data by an N/2-value cosine sequence, N/2-point fast DCT, and a simple recursive addition. The second algorithm, closely reversing the computation procedures of the first algorithm, is realized in order by simple recursive addition, N/2-point IDCT, and multiplication of the IDCT output by an N/2-value cosine sequence. MDCT realization can be simply obtained by transposing the signal flow graph for evaluating the IMDCT. For the proposed two IMDCT algorithms, we have analyzed the realization complexity and simulated the fixed-point error. Comparing with the existing fast IMDCT algorithms in literature, we observe from the analysis and simulation results that the new algorithms have better numerical accuracy and thus can be realized with short word length, resulting in more efficient realization.