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        <p style="Chapter Head" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">WAVELET EXTENSION PACK</p>
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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">Time-Frequency Analysis of a Two-Tone 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">Time-frequency analysis is concerned with how the frequency content of a signal changes in time. Since wavelets, wavelet packets, and local cosine packets are localized in time and frequency, they can be used to describe the time-frequency structure of a signal. Using any of these transforms, you can construct a time-frequency plot for a 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">In the plots below, the x-axis (horizontal) represents time and the y-axis (vertical) represents frequency. The time-frequency plane is divided into rectangles which correspond to the location and frequency of each basis vector. The length of each rectangle along the x-axis is given by the length of the associated basis vector. The width along the y-axis corresponds to the bandwidth of the vector. Each rectangle has constant area since the bandwidth of a signal is inversely proportional to its length. The modulus of the corresponding coefficient determines the gray level of each rectangle; darker rectangles indicate higher energy at that time-frequency location.</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 first test signal shown below is composed of two sinusoids of equal duration, with one that is twice the frequency of the other. The two tones are separated in time and frequency so the resulting time-frequency plots should have two distinct peaks. </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">You can analyze the two-tone signal with the discrete wavelet transform (DWT), the wavelet packet transform (WPT), and the local cosine transform (LCT). Remember that the DWT uses a fixed basis, which is generated here using the <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">wavebs</inlineAttr></b></f> function. The WPT and LCT can have optimized bases. Begin with the DWT:</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 time-frequency plot shows two distinct peaks separated in time (the horizontal, or x-axis) and in frequency (the vertical, or y-axis). The localization in frequency is rather poor as a result of leakage in the wavelet filters. Later you will graphically see the advantage of the optimization that the WPT allows.</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, use wavelet packets to analyze the two-tone signal. A time-frequency plot can be constructed for any basis extracted from a wavelet packet table. For example, use the best basis algorithm to find an optimal basis.</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 resolution of the two peaks is better in the WPT than the DWT because of the optimal selection of basis functions by the best basis algorithm. Note that there is still some leakage in the wavelet packet filters, appearing at higher frequencies than the true tones. Nevertheless, the best basis produces an excellent representation of our 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">Finally, repeat the analysis using the local cosine transform in conjunction with the best basis algorithm.</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 transform gives an excellent representation of the two tones, because the basis functions for the local cosine transform are well-matched to the signal. This time-frequency plot shows that some basis vectors are better than others for representing certain signals. No single basis will be perfect for all data.</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 example works with a linear chirp signal. A linear chirp is a sine wave whose frequency increases or decreases linearly over time. The frequency at any time is given by:</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">You can work with a short segment of the ice signal for efficiency:</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">For another example of a time-frequency plot on a different signal, see the example on <f family="Times New Roman" charset="0"><b><u><inlineAttr line-through="false"><c val="#000"><link href="./compress.xmcd" title="Links to section" popup="false">Signal and Image Compression</link></c></inlineAttr></u></b></f><inlineAttr font-weight="normal" underline="false"><c val="#000">.</c></inlineAttr></p>
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