contents


discrete signal vectors


  • a generic discrete signal:
    • \( x[n] = …,1.23, -0.73, 0.89, 0.17, -1.15, -0.26,… \)
    • set of ordered number sequence
  • four classes of signals
    • finite length
    • infinite length
    • periodic
    • finite support
  • digital signal processing:
    • signal analysis
    • signal synthesis
  • number sets:
    • \( \mathbb{N} \): natural numbers \( [1,\infty) \)
      • whole numbers \( [0,\infty) \)
    • \( \mathbb{Z} \): integers
    • \( \mathbb{Q} \): rational numbers
      • inclusive of recurring mantissa
    • \( \mathbb{P} \): irrational numbers
      • non-repeating and non-recurring mantissa
      • \( \pi \) value, \( \sqrt{2} \)
    • \( \mathbb{R} \): real numbers (everything on the number line)
      • includes rational and irrational numbers
    • \( \mathbb{C} \): complex numbers
      • includes real and imaginary numbers

number-sets

fig: number sets

vector space framework:


  • justification for dsp application:
    • common framework for various signal classes
      • inclusive of continuous-time signals
    • easy explanation of Fourier Transform
    • easy explanation of sampling and interpolation
    • useful in approximation and compression
    • fundamental to communication system design
  • paradigms for vector space application to discrete signals
    • object oriented programming
      • the instantiated object can have unique property values
      • but for a given object class, the properties and methods are the same
    • lego
      • various unit blocks, same and different
      • units assembled in different ways to build different complex structures
      • similarly, a complex signal is broken down into a combination of basis vectors for analysis
  • key takeaways
    • vector spaces are general objects
    • vector spaces are defined by their properties
    • once signal properties satisfy vector space conditions
      • vector space tools can be applied to signals

vector spaces


  • some vector spaces
    • \(\mathbb{R}^2\): 2D space
    • \(\mathbb{R}^3\): 3D space
    • \(\mathbb{R}^N\): N real numbers
    • \(\mathbb{C}^N\): N complex numbers
    • \(\ell_2(\mathbb{Z}) \):
      • square-summable infinite sequences
    • \(L_2([a,b]) \):
      • square-integrable functions over interval \([a,b] \)
  • vector spaces can be diverse
  • some vector spaces can be represented graphically
    • helps visualize the signal for analysis insights
  • \(\mathbb{R}^2\): \(\textbf{x} = [x_0, x_1]^T \)
  • \(\mathbb{R}^3\): \(\textbf{x} = [x_0, x_1, x_2]^T \)
    • both can be visualized in a cartesian system fig: vector in 2D space
  • \( L_2([a,b]) \): \( \textbf{x} = x(t), t \in [-1,1] \)
    • function vector space
    • can be represented as sine wave along time fig: L2 function vector
  • others cannot be represented graphically:
    • \(\mathbb{R}^N\) for \( N > 3 \)
    • \(\mathbb{C}^N\) for \( N > 1 \)

vector space axioms


  • informally, a vector space:
    • has vectors in it
    • has scalars in it, like \( \mathbb{C} \)
    • scalers must be able to scale vectors
    • vector summation must work
  • formally, \( \forall \text{ } \textbf{x,y,z} \in V, \text{ } and \text{ } \alpha, \beta \in \mathbb{C} \) (\(V\): vector space)
    • \(\textbf{x} + \textbf{y} = \textbf{y} + \textbf{x} \)
      • commutative addition
    • \( (\textbf{x} + \textbf{y}) + \textbf{z} = \textbf{x} + (\textbf{y} + \textbf{z}) \)
      • distributive addition
    • \( \alpha(\textbf{x} + \textbf{y}) = \alpha \textbf{y} + \alpha \textbf{x} \)
      • distributive scalar multiplication over vectors
    • \( (\alpha + \beta)\textbf{x} = \alpha \textbf{x} + \beta \textbf{x} \)
      • distributive vector multiplication over scalers
    • \( \alpha(\beta \textbf{x}) = \alpha(\beta \textbf{x}) \)
      • associative scalar multiplication
    • \( \exists \text{ } 0 \in V \text{ } | \text{ } \textbf{x} + 0 = 0 + \textbf{x} = \textbf{x} \)
      • null vector, \( 0 \), exists
      • addition of \( 0 \) and another vector \(\textbf{x}\) returns \(\textbf{x}\)
      • summation with null vector is commutative
    • \( \forall \text{ } \textbf{x} \in V \text{ } \exists \text{ } (-\textbf{x}) \text{ } | \text{ } \textbf{x} + (-x) = 0 \)
      • an inverse vector exists in vector space such that adding the vector with it’s inverse yields the null vector
  • examples:
    • \( \mathbb{R}^N \): vector of N real numbers
      • two vectors from this space look like:
        • \(\textbf{x} = [x_0, x_1, x_2, … x_N]^T \)
        • \(\textbf{y} = [x_0, x_1, x_2, … x_N]^T \)
      • the above mentioned rules apply to these vectors and can be verified
    • \( L_2[-1,1] \)

inner product


  • aka dot product: measure of similarity of two vectors
    • \( \langle \cdot, \cdot \rangle: V \times V \rightarrow \mathbb{C} \)
  • takes two vectors and outputs a scaler which indicates how similar the two vectors are
  • inner product axioms
    • \( \langle x+y, z \rangle = \langle x, z \rangle + \langle y, z \rangle \)
      • distributive over vector addition
    • \( \langle x,y \rangle = \langle y,x \rangle^* \)
      • commutative with conjugation
    • \( \langle x, \alpha y \rangle = \alpha \langle x,y \rangle \)
      • distributive with scalar multiplication
      • when scalar scales the second operand
    • \( \langle \alpha x,y \rangle = \alpha^* \langle x,y \rangle \)
      • distributive with scalar multiplication
      • conjugate scalar if it scaling the first operand
    • \( \langle x,x \rangle \geq 0 \)
      • self inner product ( \in \mathbb{R})
    • \( \langle x,x \rangle = 0 \Leftrightarrow x = 0 \)
      • if self inner product is 0, then the vector is the null vector
    • if \( \langle x,y \rangle = 0 \text{ } and \text{ } x,y \neq 0 \),
      • then \( x \) and \( y \) are orthogonal
  • inner product is computed differently for different vector spaces
  • in \( \mathbb{R}^2 \) vector space:
    • \( \langle x,y \rangle = x_0y_0 + x_1y_1 = \Vert x \Vert \Vert y \Vert \cos \alpha \)
      • where \( \alpha \): angle between \(x\) and \(y\)
    • when two vectors are orthogonal to each other
      • \( \alpha = 90^{\circ} \), so \( \cos 90^{\circ} = 0 \), so \( \langle x,y\rangle = 0 \)
  • in \( L_2[-1,1]\) vector space:
    • \( \langle x,y \rangle = \int_{-1}^1 x(t) y(t) dt \)
      • norm: \( \langle x,x \rangle = \Vert x \Vert^2 = \int_{-1}^1 \sin^2(\pi t)dt \)
    • the inner product of a symmetric and an anti-symmetric function is 0
      • i.e. they are orthogonal to each other and cannot be expressed as a factor of the other in any way
      • example 1:
        • \( x = \sin(\pi t) \) - anti-symmetric
        • \( y = 1 - \vert t \vert \) - symmetric
        • \( \langle x,y \rangle = \int_{-1}^1 (\sin(\pi t))(1 - \vert t \vert) dt = 0 \)

        inner-product-sym-antisym

        fig: inner product of a symmetric and an anti-symmetric function

      • example 2:
        • \( x = \sin( 4 \pi t) \)
        • \( y = \sin( 5 \pi t) \)
        • \( \langle x,y \rangle = \int_{-1}^1 (\sin(4 \pi t))(\sin(5 \pi t)) dt = 0 \)

norm and distance


  • norm of a vector: - inner product of a vector with itself - square of the norm (length) of a vector - \( \langle x,x \rangle = x_0^2 + x_1^2 = \Vert x \Vert ^ 2 \)
  • distance between two vectors:
    • the norm of the difference of the two vectors
  • the distance between orthogonal vectors is not zero
  • in \( \mathbb{R}^2 \), norm is the distance between the vector end points
    • \( \Vert x - y \Vert \) is the difference vector
    • \( \Vert x - y \Vert = \sqrt{(x_0 - y_0)^2 + (x_1 - y_1)^2} \)
      • connects the end points of the vectors \(x \) and \(y \)
    • see triangle rule of vector addition, and pythagorean theorem
  • in \( L_2[-1,1] \), the norm is the mean-squared error:
    • \( \int_{-1}^1 \vert x(t) - y(t) \vert^2 dt \)

signal spaces


completeness


  • consider an infinite sequence of vectors in a vector space
  • if it converges to a limit within the vector space
    • then said vector space is “complete”
    • also called Hilbert Space
  • limiting operation is ambiguous, definition varies from one space to the other
  • so some limiting operation may fail and point outside the vector space
    • such vector spaces are not said to be complete

common signal spaces


  • while vectors spaces can be applied to signal processing
    • not all vector spaces can be used for all signals
  • different signal classes are managed in different spaces
  • \( \mathbb{C}^N \): vector space of N complex tuples
    • valid signal space for finite length signals
      • vector notation: \( \textbf{x} = [x_0, x_1, … x_N]^T \)
      • where \( x_0, x_1 … x_N \) are complex tuples
    • also valid for periodic signals
      • vector notation: \( \tilde{\textbf{x}} \)
    • all operations are well defined and intuitive
    • inner product: \( \langle \textbf{x,y} \rangle = \sum_{n=0}^{N-1} x^*[n]y[n] \)
      • well defined for all finite-length vectors in ( \mathbb{C}^N)
  • the inner product for infinite length signals explode in \( \mathbb{C}^N \) - inappropriate for infinite length signal analysis
  • \( \ell_2(\mathbb{Z}) \): vector space of square-summable sequences - requirement for sequences to be square-summable: - \( \sum \vert x[n] \vert^2 < \infty \) - sum of squares of elements of the sequence is less than infinity - all sequences that live in this space must have finite energy - “well-behaved” infinite-length signals live in \( \ell_2(\mathbb{Z}) \) - vector notation: \( \textbf{x} = […, x_{-2}, x_{-1}, x_0, x_1, … ]^T \)
  • lot of other interesting infinite length signals do not live in \( \ell_2 \)
    • examples:
      • \( x[n] = 1 \)
      • \( x[n] = \cos(\omega n) \)
    • these have to be dealt with case-by-case

basis


  • a basis is a building block of a vector space
    • a vector space usually has a few basis vectors called bases
    • like the lego unit blocks
  • any element in a vector space can be
    • built with a combination of these bases
    • decomposed into a linear combination of these bases
  • the basis of a space is a family of vectors which are least like each other
    • but they all belong to the same space
    • as a linear combination, the basis vectors capture all the information within their vector space
  • fourier transform is simply a change of basis

vector families


  • \( { \textbf{w}^{(k)} } \): family of vectors - \( k \): index of the basis in the family
  • canonical \( \mathbb{R}^2 \) basis: \( \textbf{e}^k \)
    • \( \textbf{e}^{(0)} = \begin{bmatrix} 1 \ 0 \end{bmatrix} \text{; } \textbf{e}^{(1)} = \begin{bmatrix} 0 \ 1 \end{bmatrix} \)
    • this family of basis vectors is denoted by \( \textbf{e}^k \)
  • any vector can be expressed as a linear combination of ( \textbf{e}^k) in ( \mathbb{R}^2 )
    • \( \begin{bmatrix} x_0 \ x_1 \end{bmatrix} = x_0\begin{bmatrix} 1 \ 0 \end{bmatrix} + x_1\begin{bmatrix} 0 \ 1 \end{bmatrix} \)
    • \( \textbf{x} = x_0 \textbf{e}^{(0)} + x_1 \textbf{e}^{(1)} \)
  • graphical example:
    • \( \begin{bmatrix} 2 \ 1 \end{bmatrix} = 2\begin{bmatrix} 1 \ 0 \end{bmatrix} + 1\begin{bmatrix} 0 \ 1 \end{bmatrix} \)
    • \( \textbf{x} = 2 \textbf{e}^{(0)} + 1 \textbf{e}^{(1)} \)

R2-basis

fig: linear combination of canonical ( \textbf{e}^k) in (\mathbb{R}^2)

  • non-canonical \(\mathbb{R}^2 \) basis example: \( \textbf{v}^k \)
    • \( \textbf{v}^{(0)} = \begin{bmatrix} 1\ 0 \end{bmatrix} \text{; } \textbf{v}^{(1)} = \begin{bmatrix} 1\ 1 \end{bmatrix} \)
  • any vector can be expressed as a linear combination of these vectors in \( \mathbb{R}^2 \)
    • the coefficients of the bases will be different compared to the canonical bases
  • graphical example:
    • \( \begin{bmatrix} 2 \ 1 \end{bmatrix} = \alpha \textbf{v}^{(0)} + \beta \textbf{v}^{(1)} \)
    • \( \begin{bmatrix} 2 \ 1 \end{bmatrix} = \alpha \begin{bmatrix} 1\ 0 \end{bmatrix} + \beta \begin{bmatrix} 1\ 1 \end{bmatrix} \)
      • by rule of parallelogram vector addition
    • \( \alpha = 1 \text{; } \beta = 1 \)

R2-basis

fig: linear combination of non-canonical \( \textbf{v}^k \) in \(\mathbb{R}^2\)

  • only vectors which are linearly independent can be the basis vectors of a space
  • infinite dimensional spaces bases:
    • some limitations have to be applied to obtain basis vectors of infinite dimension
    • \( \textbf{x} = \sum_{k=0}^{\infty} \alpha_k \textbf{w}^{(k)} \)
  • a canonical basis of \(\ell_2(\mathbb{Z})\) - \( \textbf{e}^{k} = \begin{bmatrix} .\\.\\.\\ 0\\ 0\\ 1\\ 0\\ 0\\ 0\\ .\\.\\.\\ \end{bmatrix} \), \(k\) -th position, \( k \in \mathbb{Z} \)
  • function vector spaces:
    • basis vector for functions: \( f(t) = \sum_{k}\alpha_{k}\textbf{h}^{(k)}(t) \)
  • the fourier basis for functions over an interval \( [-1,1] \): - \( \frac{1}{\sqrt{2}}, \cos\pi t, \sin\pi t, \cos2\pi t, \sin2\pi t,\cos3\pi t, \sin3\pi t, \ldots \) - any square-integrable function in \([-1,1]\) can be represented as a linear combination of fourier bases - a square wave can be expressed as a sum of sines
  • formally, in a vector space \( H \),
  • a set of \( K \) vectors from \(H\), \(W = { \textbf{w}^{(k)}}_{k=0,1,\ldots,K-1} \) is a basis for \( H \) if: ```
    1. \(\forall \in H \): \( \textbf{x} = \sum_{k=0}^{K-1}\alpha_k\textbf{w}^{(k)} \), \( \alpha_k \in \mathbb{C} \)
    2. the coefficients \( \alpha_k\) are unique
      • this implies linear independence in the vector basis
      • \( \sum_{k=0}^{K-1} \alpha_k\textbf{w}^{(k)} = 0 \Leftrightarrow \alpha_k = 0, k=0,1,\ldots,K-1 \) ```

orthonormal basis


  • the orthogonal bases are the most important
    • of all possible bases for a vector space
  • orthogonal basis: \( \langle \textbf{w}^{(k)},\textbf{w}^{(n)} \rangle = 0 \) for \( k \neq n\)
    • vectors of an orthogonal basis are mutually orthogonal
    • their inner product with each other is zero
  • in some spaces, the orthogonal bases are also orthonormal
    • i.e. they are unit norm
    • their length \( \Vert \textbf{w}^{(k)}\Vert = 1 \)
  • the inner product of any two vectors in the orthonormal bases is the difference between their indices
    • \( \langle \textbf{w}^{(k)}, \textbf{w}^{(n)} \rangle = \delta[n-k]\)
  • gran-schmidt algorithm can be used to orthonormalize any orthogonal bases
  • obtaining the bases coefficients \( \alpha_k \) for bases can be involved and challenging
    • \( \textbf{x} = \sum_{k=0}^{K-1} \alpha_k\textbf{w}^{(k)} \)
      • \( \textbf{x} \): a vector as the linear combination of \(K\) basis vectors \( \textbf{w}^{(k)} \),
      • with corresponding coefficients \( \alpha_k \)
    • however, they are easy to obtain with an orthonormal basis
      • \( \alpha_k = \langle \textbf{w}^{(k)},\textbf{x} \rangle\)

change of basis


  • \( \textbf{x} = \sum_{k=0}^{K-1} \alpha_k\textbf{w}^{(k)} = \sum_{k=0}^{K-1} \beta_k\textbf{v}^{(k)}\)
    • \( \textbf{v}^{(k )}\) is the target basis, \(\textbf{w}^{(k )}\) is the original basis
  • if \( { \textbf{v}^{(k )} } \) is orthonormal:
    • \( \beta_h = \langle v^{(h)}, \textbf{x} \rangle \)
    • \( = \langle \textbf{v}^{(h)}, \sum_{k=0}^{K-1} \alpha_k\textbf{w}^{(k)} \rangle \)
    • \( = \sum_{k=0}^{K-1} \alpha_k \langle \textbf{v}^{(h)}, \textbf{w}^{(k)} \rangle \)
    • \( = \sum_{k=0}^{K-1} \alpha_k c_{hk} \)
    • \( = \begin{bmatrix} c_{00} & c_{01} & \ldots & c_{0(K-1)}\\ & & \vdots & \\ c_{(K-1)0} & c_{(K-1)1} & \ldots & c_{(K-1)(K-1)} \end{bmatrix} \begin{bmatrix} \alpha_0\ \vdots\\ \alpha_{K-1} \end{bmatrix} \)
  • this forms the core of the discrete fourier transform algorithm for finite length signals
  • can be applied to elementary rotations of basis vectors in the euclidean plane
    • the same vector has different coefficients in the original and the rotates bases
    • the rotation matrix is obtained by the matrix multiplication of the original and the target bases
    • the rotation matrix applied to a vector in the original bases yields the coefficients of the same vector in the rotated bases
    • the matrix multiplication of the rotation matrix with its inverse yields the identity matrix

subspaces


  • subspaces can be applied to signal approximation and compression
  • with vector \( \textbf{x} \in V \) and subspace \(S \subseteq V \)
    • approximate \( \textbf{x} \) with \( \hat{\textbf{x}} \in S \) by
    • take projection of the vector \( \textbf{x} \) in \( V \) on \( S \)
  • due to the adaptation of vector space paradigm for signal processing
    • this geometric intuition for approximation can be extended to arbitrarily complex vector spaces

vector subspace


  • a subspace is a subset of vectors of a vector space closed under addition and scalar multiplication
  • classic example:
    • \( \mathbb{R}^2 \subset \mathbb{R}^3 \)
    • in-plane vector addition and scalar multiplication operations do not result in vectors outside the plane
    • \( \mathbb{R}^2 \) uses only 2 of the 3 orthonormal basis of \( \mathbb{R}^3 \)
  • the subspace concept can be extended to other vector spaces
    • \( L_2[-1,1]\): function vector space
      • subspace: set of symmetric functions in \(L_2[-1,1]\)
      • when two symmetric functions are added, they yield symmetric functions
  • subspaces have their own bases
    • a subset of their parent space’s bases

least square approximations


  • \( { \textbf{s}^{(k)} }_{k=0,1,\ldots,K-1} \) orthonormal basis for \( S \)
  • orthogonal projection:
    • \( \hat{\textbf{x}} = \sum_{k=0}^{K-1} \langle \textbf{s}^{(k)},\textbf{x} \rangle \textbf{s}^{(k)} \)
  • the orthogonal projection: the “best” approximation of \( \textbf{x} \) over \(S\) - because of two of its properties - it has minimum-norm error: - \( arg \text{ } min_{y\in S} \Vert x - y \Vert = \hat{\textbf{x}}\) - orthogonal projection minimizes the error between the original vector and the approximated vector - this error is orthogonal to the approximation: - \( \langle \textbf{x} - \hat{\textbf{x}}, \hat{\textbf{x}} \rangle = 0\) - the error and the basis vectors of the subspace are maximally different - they are uncorrelated - the basis vectors cannot capture any more information in the error
  • example: polynomial approximation
    • approximating from vector space \(L_2[-1,1] \) to \( P_N[-1,1] \)
    • i.e. vector space of square-integrable functions to a subspace of polynomials of degree \(N-1\)
    • generic element of subspace \( P_{N}[-1,1] \) has form
      • \( \textbf{p} = a_0 + a_1t + \ldots + a_{N-1}t^{N-1} \)
    • a naive, self-evident basis for this subspace:
      • \( \textbf{s}^{(k)} = t^k, k = 0,1,\dots,N-1 \)
      • not orthonormal, however

approximation with Legendre polynomials


  • example goal:
    • approximate \( \textbf{x} = \sin t \in L_2[-1,1]\) to \( P_3[-1,1] \)
      • \( P_3[-1,1] \): polynomials of the degree 2
  • build orthonormal basis from naive basis
    • use Gram-Schmidt orthonormalization procedure for naive bases:
      • \( { \textbf{s}^{(k)}} \rightarrow { \textbf{u}^{(k)}} \)
      • \( { \textbf{s}^{(k)}} \): original naive bases
      • \( { \textbf{u}^{(k)}} \): orthonormalized naive bases
    • this algorithm takes one vector at a time from the original step and incrementally produces an orthonormal set
      1. \( \textbf{p}^{(k)} = \textbf{s}^{(k)} - \sum_{n=0}^{k-1} \langle \textbf{u}^{(n)},\textbf{s}^{(n)} \rangle \textbf{u}^{(n)} \)
        • for the first naive basis vector, normalize it with 1
        • project the second naive basis vector on to the normalized first basis
        • then subtract this projection from the second basis vector to get the second normalized basis
        • this removes the the first normalized basis’s component from the second naive basis
      2. \( \textbf{u}^{(k)} = \frac{\textbf{p}^{(k)}}{\Vert\textbf{p}\Vert^{(k)}} \)
        • normalize the extracted vector
    • this process yields:
      • \( \textbf{u}^{(1)} = \sqrt{\frac{1}{2}} \)
      • \( \textbf{u}^{(2)} = \sqrt{\frac{3}{2}}t \)
      • \( \textbf{u}^{(3)} = \sqrt{\frac{5}{8}}(3t^2-1) \)
      • and so on
    • these are known as Legendre polynomials
    • they can be computed to the arbitrary degree,
      • for this example, up to degree 2
  • project \( \textbf{x} \) over the orthonormal basis
    • simply dot product the original vector \(x\) over all the legendre polynomials i.e. the orthogonal basis of the \(P_3[-1,1]\) subspace
    • \( \alpha_k = \langle \textbf{u}^{(k)}, \textbf{x} \rangle = \int_{-1}^{1} u_k(t) \sin t dt \)
      • \( \alpha_0 = \langle \sqrt{\frac{1}{2}}, \sin t \rangle = 0 \)
      • \( \alpha_1 = \langle \sqrt{\frac{3}{2}}t, \sin t \rangle \approx 0.7377 \)
      • \( \alpha_2 = \langle \sqrt{\frac{5}{8}}(3t^2 -1), \sin t \rangle = 0 \)
  • compute approximation error
    • so using the orthogonal projection
      • \( \sin t \rightarrow \alpha_1\textbf{u}^{(1)} \approx 0.9035t \)
      • this subspace has only one non-zero basis:
        • \( \sqrt{\frac{3}{2}}t \)
  • compare error to taylor’s expansion approximation
    • well known expansion, easy to compute but not optimal over interval
    • taylor’s approximation: \( \sin t \approx t \)
    • in both cases, the approximation is a straight line, but the slopes are slightly different (\(\approx\) 10% off)
      • the taylor’s expansion is a local approximation around 0,
      • the legendre polynomials method minimizes the global mean-squared-error between the approximation and the original vector
      • the error of the legendre method has a higher error around 0
      • however, the energy of the error compared to the error of the taylor’s expansion is lower in the interval
    • error norm:
      • legendre polynomial based approximation:
        • \( \Vert \sin t - \alpha_1\textbf{u}^{(1)} \Vert \approx 0.0337 \)
      • taylor series based approximation:
        • \( \Vert \sin t - t \Vert \approx 0.0857 \)

haar spaces


  • haar spaces are matrix spaces
    • note: matrices can be reshaped for vector operations
  • encodes matrix information in a hierarchical way
    • finds application in image compression and transmission
  • it has two kinds of basis matrices
    • the first one encodes the broad information
    • the rest encode the details, which get finer by the basis index
  • each basis matrix has positive and negative values in some symmetric pattern
  • the basis matrix will implicitly compute the difference between image areas
    • low-index basis matrices take differences between large areas
    • high-index matrices take differences in smaller, localized areas
  • this is a more robust way of encoding images for transmission methods prone to losses on the way
  • if images are transmitted as simple matrices, they are prone to being chopped is loss in communication occurs during transmission

  • haar encoding transmits coefficients not pixel by pixel but hierarchically in the level of detail
    • so if communication loss occurs, the broad idea of the image is still conveyed
    • while continued transmission will push up the detail level
  • approximation of matrices to harr space is an example of progressive encoding

references