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$\displaystyle {\cal D} = \Bigg\{ \cos\Big(\frac{\pi k(t + \frac12)}{N}\Big)\Bigg\}  

\bigcup \Bigg\{\delta_k(t)\Bigg\} ,

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$\displaystyle X(t) = 3 \cos\Big(\frac{\pi 8(t + \frac12)}{100}\Big) - 5 \delta_{10}(t) + 15

\delta_{20}(t) 

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\begin{figure}

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\psfig{figure=/afs/cs/user/vyass/Fall05/IMC_CAMERA_READY_Aug03/IMC_RESULTS/example.eps,width=200pt,height=200pt}

 & \psfig{figure=/afs/cs/user/vyass/Fall05/IMC_CAMERA_READY_Aug03/IMC_RESULTS/exampledct.eps,width=200pt,height=200pt}\\


a. An example signal $X$\  which has a short representation & b. The discrete cosine transform (DCT) of the example signal $X$.  \\

over the redundant dictionary ${\cal D}$. & \\

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$\displaystyle X(t) = \frac{1}{100} \sum_{k=0}^{99} \hat X(k) 

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$ X(t)$%
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$\displaystyle X(t) = 3 \cos\Big(\frac{\pi 8(t + \frac12)}{100}\Big) - 5 \delta_{10}(t) + 15

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$\displaystyle \lambda_t \  = \  {\rm argmax}_{j} \

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$ \bm{P}_t$%
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$\displaystyle \bm{r}_t = \bm{v} - \bm{P}_t \, \bm{v}.

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$ t$%
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$ t < m$%
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\begin{figure}

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\psfig{figure=/afs/cs/user/vyass/Fall05/IMC_CAMERA_READY_Aug03/IMC_RESULTS/SPECTRUM_ANALYSIS/spectrum_25.eps,width=200pt,height=200pt}

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Port 25/TCP & Port 4662/TCP \\

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$ -f$%
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\begin{figure}

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$ D_{F}$%
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$ D_{W}$%
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$ D_H$%
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$ D_{H+S}$%
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High Volume Ports & Scan Target Ports & P2P Ports & Random Ports \\

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Real part & Imaginary part \\

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Real Part & Imaginary Part \\

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 High Volume Ports& P2P Ports \\

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Port 80 & Port 6346 & Port 9898 & Port 43726 \\
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 Port 80 & Port 1214 \\
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$ 168 \times 12 = 2016$%
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\end{document}
