Notes of Digital Audio Signal Processing, Lecture 2.

Dot product

  • \(D(\mathbf{u},\mathbf{v})=\sum_{i=1}^3u_iv_i\)
  • The correlation is related to dot product, see here.
  • norm (\(\left\lVert\mathbf{u}\right\rVert = D(\mathbf{u},\mathbf{u})=\sum_{i=1}^3u_i^2\)) \(\rightarrow\) dot product (\(D(\mathbf{u},\mathbf{v})=\sum_{i=1}^3u_iv_i\)) \(\rightarrow\) Energy \(\left\lVert\mathbf{u}\right\rVert ^2\).
  • \(D(\mathbf{u},\mathbf{u}) = \mathbf{u}^\intercal \mathbf{u}\)
  • inner product
  • Signal in discrete time of lenght \(N\) has a dimension of \(N\);
  • Orthogonal \(\mathbf{s_1}^\intercal \mathbf{s_2} = 0\), meaning, nonzeros in \(\mathbf{s_1}\) correspond to zeros in \(\mathbf{s_2}\) \(\rightarrow\) Frequency does not overlap ??

Matrix

  • A matrix is a system.
  • Hadamard matrix \(\rightarrow\) Hadamard Transform is an example of a generalized class of Fourier transform.
  • Rotation matrix
  • In Matlab, u.*v equals diag(u)*v (element-wise multiplication), where diag(u) is the temporal envelope.
  • In Matlab, \(B^{-1}C=\)B\C and \(B/C^{-1}=\)B/C.
  • The relationship between the deconvolution and the inverse of a matrix
  • Toeplitz matrix and its "upside down" version - Hankel matrix \(\rightarrow\) filter correlation \(\rightarrow\) Transmission line matrix (Waveguide)
  • VanderMonde matrix \(\rightarrow\) damped sine wave (inversion) \(\rightarrow\) noise cancellation
  • Matrices might not be inversable just like one might not recover the original signal from its projection onto one axis.
  • \((\mathbf{ABC})^\intercal = \mathbf{C}^\intercal\mathbf{B}^\intercal\mathbf{A}^\intercal\)
  • Eigenvector \(\mathbf{v}\) and eigenvalues \(\lambda\). "Eigen" origins from German for "proper".

Functions and Polynomials

  • Linear (gain or interpolation), exponential (the feedback loop) and polynomial functions (spline interpolation, harmonic distortion or representing any functions)
  • Chebyshev Polynomial and distortion
  • Roots of polynomials \(p_n(x_i) = 0\)
    • roots (order limits)
    • \(p_n(x) = (x-x_1)p_{n-1}(x) = a_n\prod_{i=1}^n(x-x_i)\) ??

Rational

  • \(f(x) = \frac{Q}{P}\)
  • Filter frequency response is a rational function (for most of cases), e.g., an exception, viscosity loss of pipe \(\rightarrow\) \(\sqrt{f}\) \(\rightarrow\) irrational function

Complex numbers

Imaginary

  • Matlab considers a number to be complex (\(\mathbb{C}\))
  • Complex number is defined because it does not exist in \(\mathbb{R}\) or is just not defined before?
  • \(j, -1, -j, 1\) for \(j^n\), where \(n=1,2,3,4\).
  • Imaginary, a good word, but imaginary is not real imaginary

Phase and angle

  • atan2, "2" because it accepts two arguments and angle in Matlab uses atan2
  • phase in(de)crease infinitely but how?

Conjugate

  • conjugate meaning the opposite angle
  • real coefficients of polynomial \(\rightarrow\) roots must be grouped by pairs
  • Euler's formula\(\rightarrow\)
  • \(e^{j\theta} = \cos\theta+j\sin\theta\) where \(e^{j\theta}\) is the analytic signal, the analytic representation of the real-value function (analytic continuation)
  • Transfer complex to real after passing a linear system is true but it is not true for a nonliear processing. WHY?

Unity Circle

  • For root of unity \(z^N=1\), there are \(N\) Nth root because it is an Nth-order polynomial \(1-z^N=0\).
  • Reciprocal of \(z\) \(\rightarrow\) unit circle (in(out)side) \(\rightarrow\) stability ((un)stable)
  • polynomial
    • Complex conjugate root theorem: real coefficients \(\rightarrow\) roots are conjugate pairs.
    • symmetrical coefficients \(\rightarrow\) roots are conjugate inverse pairs
    • when, \(|z|=1\), \(z^*=\dfrac{1}{z}\), roots are pairs of both inverse and conjugate, and are on unit circle.
  • For complex vectors \(\mathbf{u}\) and \(\mathbf{v}\),
    • \(D(u,v) = \sum_{i=1}^{N}\bar{u}_iv_i=u^{*T}v = u^Hv\), \(H\) means transposed and conjugated \(\rightarrow\) u' in Matlab
    • u.' only does the transpose
    • \(D(u,v)=\overline{D(v,u)}\)