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                <c val="#008080">SECTION 3 - SPECTRAL ANALYSIS</c>
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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">Compute the cepstrum and complex cepstrum of a real-valued, multichannel signal. They take only one argument:</p>
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          <b>x</b>, a real-valued vector or matrix. <b>cepstrum</b> accepts only vectors. <b>ccepstrum</b> accepts vectors and matrices, and acts upon the individual columns if x is a matrix.</p>
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            <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">This is the definition of the Inverse transform after applying a natural logarithm to the Fourier transform of x(n). The logarithm maps convolution in the time domain to addition in the cepstral domain, making the cepstrum is a convenient tool for signal deconvolution, which is mapped to subtraction. </inlineAttr>
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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">Calculate the cepstrum and recenter the result:</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">The original signal cannot be recovered from the cepstrum, because the phase information is not retained. The complex cepstrum, shown below, is invertible.</p>
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        <p style="Heading 1" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">Example - Complex cepstrum </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">The complex cepstrum is similar to the cepstrum, but it uses the complex logarithm.</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">The following example simulates a signal going through a </p>
        <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">communication channel which introduces echoes (following section 12.8.1 of Oppenheim and Schaefer, Prentice-Hall, 1989). Comparing the complex cepstrum of the output signal to those of the input and channel sequences demonstrates that output complex cepstrum is the sum of the complex cepstra of the input and channel. This is similar to adding the spectra of two signals in the frequency domain, but it is done in the time-domain.</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">First, define the channel response as an FIR filter that introduces attenuated copies of another sequence (echoes).</p>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">Here is the sequence that will be convolved with the channel response:</p>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">Compare the complex cepstrum of the output signal to the cepstra of the original sequences. Results are recentered for comparison.</p>
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