PROBLEMS

7.1. Given the expressions for the DFT, the DCT, and the MDCT,

image

image

Figure 7.5. FFT analysis/synthesis within the two bands of QMF bank.

Write the three transforms in matrix form as follows

XT = Hx,

where H is the transform matrix, and x and XT denote the input and transformed vector, respectively. Note the structure in the transform matrices.

7.2. Give the signal flowgraph of the FFT butterfly structure for an 8-point DFT, an 8-point DCT, and an 8-point MDCT. Specify clearly the values on the nodes and the branches. [Hint: See Problem 6.16 and Figure 6.18 in Chapter 6.]

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