2.4 Linear transformations and basis functions

Linear transformations are the backbone of signal processing, allowing us to manipulate and analyze complex data. They preserve key relationships between vectors, making them invaluable for understanding how signals change and interact in various systems.

Basis functions provide a framework for representing signals in different domains. By changing the basis, we can simplify complex signals, extract important features, and apply transformations that reveal hidden patterns in the data.

Linear Transformations

Properties of linear transformations

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• Preserve vector addition enables combining transformed vectors in the same way as the original vectors
• Preserve scalar multiplication allows scaling transformed vectors by the same factor as the original vectors
• Map the zero vector to the zero vector maintains the origin of the vector space
• Preserve linear combinations ensures that any linear combination of vectors is transformed into the corresponding linear combination of their transformed counterparts

Matrix representation of transformations

• Transformation matrix $A$ contains the images of the standard basis vectors $e_1, e_2, \ldots, e_n$ as its columns
• Matrix multiplication $Ax$ computes the linear transformation of vector $x$ by mapping each component to its corresponding transformed value
• Composition of linear transformations corresponds to matrix multiplication of their respective transformation matrices ($T_2 \circ T_1 \leftrightarrow A_2A_1$)
• Inverse of a linear transformation, if it exists, is represented by the inverse of its transformation matrix ($T^{-1} \leftrightarrow A^{-1}$)

Basis Functions

Role of basis functions

• Provide a coordinate system for representing vectors in a vector space or signals in a function space
• Enable unique representation of any vector or signal as a linear combination of the basis functions
• Allow for compact and efficient representation of signals by capturing their essential features
• Facilitate analysis, processing, and transformation of signals in different domains ($\text{time} \leftrightarrow \text{frequency}$)

Basis changes for signals

• Change of basis matrix $A$ contains the new basis vectors expressed in terms of the original basis
• Coefficients in the new basis $c_{\text{new}}$ are obtained by multiplying the original coefficients $c_{\text{old}}$ by the change of basis matrix: $c_{\text{new}} = Ac_{\text{old}}$
• Inverse change of basis matrix $A^{-1}$ maps the coefficients back to the original basis: $c_{\text{old}} = A^{-1}c_{\text{new}}$
• Basis changes can be used to simplify signal representation, extract features, or apply transformations

Common basis functions in processing

• Fourier basis represents signals as a sum of complex exponentials with different frequencies
  • Enables frequency-domain analysis and filtering ($\text{low-pass}$, $\text{high-pass}$, $\text{band-pass}$)
  • Used in applications such as audio processing, telecommunications, and radar
• Wavelet basis represents signals at different scales and positions using scaled and shifted versions of a mother wavelet
  • Captures both frequency and time information, providing a multi-resolution analysis
  • Used in applications such as image compression ($\text{JPEG 2000}$), denoising, and feature extraction
• Polynomial basis represents signals as a sum of powers of a variable ($1$, $x$, $x^2$, $\ldots$)
  • Used in curve fitting, interpolation, and approximation problems
• Legendre and Hermite bases are orthogonal polynomial bases used in solving differential equations and quantum mechanics, respectively