Digital Audio Signal Processing-Lecture 2 (Notes)
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
equalsdiag(u)*v
(element-wise multiplication), wherediag(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".
- \(T(\mathbf{v})=\lambda\mathbf{v}\): \(T\) is a linear transform and \(\mathbf{v}\) and \(\lambda\) are its eigenvector and eigenvalue.
- The spectrum of a matrix is the set of its eigenvalues and each eigenvector represents one frequency or one dimension/direction. Check Eigendecomposition of a matrix.
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 andangle
in Matlab usesatan2
- 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- \(D(u,v)=\overline{D(v,u)}\)
- \(D(u,v) = \sum_{i=1}^{N}\bar{u}_iv_i=u^{*T}v = u^Hv\), \(H\) means transposed and conjugated \(\rightarrow\)