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				<p style="Section Title" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Discrete Cosine Transforms</p>
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				<p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">The discrete cosine transform (DCT) is an important and widely used analysis tool in signal and image processing. The DCT is particularly valuable for coding and data compression applications, and it forms the core of the JPEG and MPEG algorithms for image compression.</p>
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				<p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Like the discrete Fourier transform (DFT), the DCT defines an orthogonal transformation which maps a signal from the time domain to the frequency domain. However, the DCT is a real-valued transform, and does not involve complex numbers. The DFT is a complex-valued transform.</p>
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				<p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">This section gives a brief introduction to the discrete cosine transforms. Algorithms for the DCT are discussed in the next section. For a comprehensive discussion of the DCT and applications involving the DCT, refer to the book by <f family="Times New Roman" charset="0">
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				<p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">The Fourier cosine transform (FCT) g(<f family="Symbol" charset="2" size="12">
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				<p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">The discrete cosine transform is a discretized version of this equation. There are four commonly used orthogonal discrete cosine transforms: DCT-I, DCT-II, DCT-III, and DCT-IV. </p>
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					</region> if k = 0 or n, and 1 otherwise.</p>
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				<p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">The DCT-IV is defined as</p>
			</text>
		</region>
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				<p style="Heading 1" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Functions for Computing DCTs</p>
			</text>
			<indexes>
				<phrase>DCTs, functions for computing</phrase>
			</indexes>
		</region>
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				<p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Functions for computing the various DCTs are easily defined. For example, the DCT-II is given by:</p>
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																			<ml:real>2</ml:real>
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																		</ml:apply>
																		<ml:real>1</ml:real>
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																<ml:parens>
																	<ml:apply>
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														<ml:apply>
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														</ml:apply>
													</ml:apply>
												</ml:apply>
											</ml:apply>
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										<ml:real>0</ml:real>
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											<ml:real>1</ml:real>
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								</ml:apply>
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				</ml:define>
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			<pageBreak/>
		</region>
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				<p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Similarly, the inverse DCT-II is:</p>
			</text>
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															</ml:apply>
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				<p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">The DCT-IV is:</p>
			</text>
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				<p style="Heading 1" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Computing DCTs</p>
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				<p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Now look at an example of using these transforms. First, use a single sinusoid for the test signal.</p>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Here are the DCT-II and DCT-IV transforms of the signal. </p>
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				<p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Next, look at a constant valued signal.</p>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">And the DCT-II and DCT-IV transforms of this signal. </p>
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				<p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Both cosine transforms show the frequency content of the signals. However, there is substantial <f family="Times New Roman" charset="0">
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					</f> away from the frequencies of the signals, which appears as "ringing." This effect is known as the "Gibbs phenomenon." This ringing may be reduced using special tapers on the data. More more information on the Gibbs phenomenon is given in the section on <b>
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					</b>.</p>
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				<phrase>ringing signal</phrase>
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