Explain why the factorization steps in Eqs. (8.24) and (8.25) save CPU cycles. About how much faster is the factorized approach to the DCT than the straightforward definition in (8.17)? (In both approaches, assume that of course the cosine matrix values are pre-stored outside any loop.)

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The Answer is The factorization steps in Eqs. (8.24) and (8.25) save CPU cycles because they reduce the number of multiplications and additions required to compute the DCT. In the straightforward definition in Eq. (8.17), each element of the DCT matrix is computed using a series of multiplications and additions, which can be computationally expensive.

On the other hand, the factorized approach in Eqs. (8.24) and (8.25) breaks down the DCT computation into smaller, more efficient steps. This reduces the overall number of arithmetic operations required, leading to a faster computation of the DCT.

The exact speedup achieved by the factorized approach will depend on the specific implementation and hardware, but in general, it can be significantly faster than the straightforward definition in Eq. (8.17). In some cases, the factorized approach can be several times faster, especially for larger DCT matrices. This makes it a more efficient and practical choice for real-time or high-performance applications.

Explain Why the Factorization Steps in Eqs