Given a sequence x(n), find the 1-level DWT coefficients...

Given a sequence $x(n)$, find the 1-level DWT coefficients using the $9 / 7$ filter. Assume whole-point symmetry at the borders. Verify that the reconstructed signal is the same as the input. (i) $$ \{1,-4,1,3,3,1,3,0,2,2,3,1,-5,2,0,3\} $$ (ii) $$ \{1,2,1,1,3,4,0,3,1,3,2,-1,0,1,4,-3\} $$ (iii) $$ \{-2,0,2,-2,1,0,3,1,-2,1,2,-1,2,0,-2,1\} $$

Given a sequence $x(n)$, find the 1-level DWT coefficients using the $9 / 7$ filter. Assume whole-point symmetry at the borders. Verify that the reconstructed signal is the same as the input.
(i)
$$
\{1,-4,1,3,3,1,3,0,2,2,3,1,-5,2,0,3\}
$$
(ii)
$$
\{1,2,1,1,3,4,0,3,1,3,2,-1,0,1,4,-3\}
$$
(iii)
$$
\{-2,0,2,-2,1,0,3,1,-2,1,2,-1,2,0,-2,1\}
$$

Key Concepts

Discrete Wavelet Transform (DWT)

Discrete Wavelet Transform is a mathematical technique used to decompose a signal into different frequency bands and temporal resolutions. It allows for a multi-resolution analysis of signals by applying wavelet filters that capture both the smooth parts (approximation coefficients) and the details (detail coefficients) of the signal. DWT is widely used in signal processing and image compression due to its efficiency in capturing localized frequency information.

9/7 Filter

The 9/7 filter is a specific biorthogonal wavelet filter used in the DWT, commonly utilized in image compression standards like JPEG2000. This filter is designed to provide a balance between effectively capturing signal details and achieving smooth signal approximations, with 9 coefficients for the analysis low-pass filter and 7 coefficients for the analysis high-pass filter, ensuring good performance in reconstruction accuracy.

Level-1 Decomposition

Level-1 decomposition in the context of DWT refers to the process of breaking down the original signal into two sets of coefficients: one representing the low-frequency (approximation) components and the other representing the high-frequency (detail) components. This single-level analysis is the first step in multi-resolution analysis, allowing subsequent levels to capture increasingly finer details of the signal.

Whole-Point Symmetry at Borders

Whole-point symmetry is a boundary-handling strategy used in signal processing to extend the signal at its borders by mirroring around a whole point. This symmetric extension minimizes edge effects during convolution with wavelet filters, ensuring that the transform operates consistently at the boundaries and aids in achieving accurate analysis and reconstruction.

Signal Reconstruction and Perfect Reconstruction

Signal reconstruction in the context of the DWT involves using the inverse transform to recombine the approximation and detail coefficients in order to recover the original signal. The perfect reconstruction property ensures that if the analysis and synthesis processes are correctly implemented, the reconstructed signal will be identical to the input signal. This property is critical in applications like compression and denoising, where data integrity is essential.

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