Show that the discrete Fourier transform (DFT) is linear. Solution The continuou

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The continuous Fourier transform is defined as f(nu) = F_t[f(t)](nu) (1) = int_(-infty)^inftyf(t)e^(-2piinut)dt. (2) Now consider generalization to the case of a discrete function, f(t)->f(t_k) by letting f_k=f(t_k), where t_k=kDelta, with k=0, ..., N-1. Writing this out gives the discrete Fourier transform F_n=F_k[{f_k}_(k=0)^(N-1)](n) as F_n=sum_(k=0)^(N-1)f_ke^(-2piink/N). (3) The inverse transform f_k=F_n^(-1)[{F_n}_(n=0)^(N-1)](k) is then f_k=1/Nsum_(n=0)^(N-1)F_ne^(2piikn/N). (4) Discrete Fourier transforms are extremely useful because they reveal periodicities in input data as well as the relative strengths of any periodic components. There are a few subtleties in the interpretation of discrete Fourier transforms, however. In general, the discrete Fourier transform of a real sequence of numbers will be a sequence of complex numbers of the same length. In particular, if f_k are real, then F_(N-n) and F_n are related by F_(N-n)=F^__n, (5) for n=0, 1, ..., N-1, where z^_ denotes the complex conjugate. This means that the component F_0 is always real for real data.

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