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Say a discrete signal $x(n)$ and its Z-transform $X(z)$, discrete time Fourier transform $X(e^{j\omega})$, and discrete Fourier transform $X(k)$. The relationships between DFT and ZT / DTFT are respectively: $$ X(k) = X(z)|_{z=e^{j2\pi k/N}} \tag{1.1} $$ $$ X(k) = X(e^{j\omega})|_{\omega=2\pi k/N} \tag{1.2} $$ Since the original signal $x(n)$ can be ...


What you have to show is that $$W_3^2+W_3+1=0\tag{1}$$ with $W_3=e^{-j2\pi/3}$. It's a straightforward exercise to prove $(1)$. In general we have $$\sum_{k=0}^{N-1}W_N^k=0,\qquad N>1\tag{2}$$ with $W_N=e^{-j2\pi/N}$. Eq. $(2)$ can be shown by using the formula for the geometric sum.


Behind an FFT or a DCT operator, which can be implemented as a matrix product. Depending on the shape of your discrete vectors, you may have $b=D\times a$ and $c=F\times a$. So you can find your answer with a suitable matrix product. You can also find fast algorithms by decomposing each of the above matrices in simpler matrix product, or using lifting-like ...

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