Find an efficient procedure to compute the discrete cosine & sin transforms using fft

Can't figure this out to save my life. Please help if possible

Consider the discrete cosine and sine transforms of a real sequence of numbers

$\displaystyle {x_j, j=0,....,2N-1}$

$\displaystyle C_k = \sum_{j=0}^{2N-1} x_j cos(\frac{jk\pi}{N}),$

$\displaystyle S_k = \sum_{j=0}^{2N-1} x_j sin(\frac{jk\pi}{N})$

Find an efficient procedure to compute $\displaystyle C_k, S_k$ using a Fast Fourier Transform of $\displaystyle \{x_j, j=0,...2N-1\}$.

Can someone give me a hint/explain to me the way to go about doing this problem? I am completely lost. Thanks!

Re: Find an efficient procedure to compute the discrete cosine & sin transforms using

Quote:

Originally Posted by

**drewprogden** Can't figure this out to save my life. Please help if possible

Consider the discrete cosine and sine transforms of a real sequence of numbers

$\displaystyle {x_j, j=0,....,2N-1}$

$\displaystyle C_k = \sum_{j=0}^{2N-1} x_j cos(\frac{jk\pi}{N}),$

$\displaystyle S_k = \sum_{j=0}^{2N-1} x_j sin(\frac{jk\pi}{N})$

Find an efficient procedure to compute $\displaystyle C_k, S_k$ using a Fast Fourier Transform of $\displaystyle \{x_j, j=0,...2N-1\}$.

Can someone give me a hint/explain to me the way to go about doing this problem? I am completely lost. Thanks!

What is the real part of the DFT of a real sequence? What is the imaginary part?

CB