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DCSP-13 Jianfeng Feng Department of Computer Science Warwick Univ., UK

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DFT for spectral estimation One of the uses of the DFT is to estimate the frequency spectrum of a signal. The Fourier transform of a function produces a spectrum from which the original function can be reconstructed by an inverse transform. So it is reversible. (signal processing: transforms, demo)

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In order to do that, it preserves not only the magnitude of each frequency component, but also its phase. This information can be represented as a 2- dimensional vector or a complex number, or as magnitude and phase (polar coordinates). In graphical representations, often only the magnitude (or squared magnitude) component is shown. This is also referred to as a power spectrum.

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In particular, given a continuous time signal x(t), the goal is to determine its frequency components by taking a finite set of samples as a vector, (x(0), x(1),…, x(N-1)), and computing its DFT as in the previous section.

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The fact that we window the signal x(n) (i.e. we take a finite set of data) will have an effect on the frequency spectrum we compute, and we have to be aware of that when we interpret the results of the DFT. This can be seen by defining a window sequence

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and the DTFT of the windowed signal Now notice that we can relate the right-hand side of the preceding expression to the DFT of the finite sequence x(0), …,x(N-1)

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To see the effect of the windowing operation on the frequency spectrum, let us examine the case when the signal x(n) is a complex exponential, such as x(n) = exp( j 0 n)

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N=10, = 0.5

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Two effects of DFT 1.Lose of resolution 1.Leakage of energy from mainlobe to sidelobes

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As we have seen, the DFT is the basis of a number of applications, and is one of the most important tools in digital signal processing. The problem with this algorithm is that a brute-force implementation would hardly be feasible in real time, even for a data set of modest length.

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Fortunately, the DFT can be computed in a very efficient way, exploiting the very structure of the algorithm we have presented by using the fast Fourier transform (FFT), which is an efficient way of computing he DFT. It is know that for a data set of length N, the complexity of the FFT grows as N log N, the complexity of the DFT computed by brute force grow as N 2. FFT is widely used in many applications.

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Sampling and reconstruction The question we consider here is under what conditions we can completely reconstruct the original signal x(t) from its discretely sampled signal x(n).

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