Digital Audio Signal Processing Lecture 6 (Notes)

Notes of Digital Audio Signal Processing, Lecture 6.

Inverse System

• $y=\mathcal{T}\{x\}$ and $x = \mathcal{T_i}\{y\}$, where $\mathcal{TT_i} = \mathcal{T_iT} = \mathbf{I}$
• $h*h_i = h_i*h = \delta$
• E.g., michrophone with a flare that is a high-pass system, needs an inverse system to get rid of the HP effect.
• E.g., $h[n]=u[n]$ (accumulator) $\leftrightarrow$ $h_i[n]=\delta[n]-\delta[n-1]$, where $h*h_i = \delta$
• Infinite impulse response $\longleftrightarrow$ finite impulse response, where the stability needs to be checked when inversion is from finite IR to infinite IR.
• $power \propto amplitude^2$
  • Power dB = $10\log_{10}(\dfrac{p}{p_0}) = 20\log_{10}(\dfrac{a}{a_0})$
  • Amplitude dB = $5\log_{10}(\dfrac{p}{p_0}) = 10\log_{10}(\dfrac{a}{a_0})$

Frequency Response of LTI System

• $p_{f_0}[n] = e^{2\pi jf_0 n}$ is the eigenvector of a filter
• Eigenvalue & eigenvector of LTI system
  • $$\begin{align} y[n] &= h*p_{f_0} = \sum\limits_{k\in \mathbb{Z}}h[k]p_{f_0}[n-k] \\ &= \sum\limits_{k\in \mathbb{Z}}h[k]e^{2\pi jf_0 (n-k)} \\ &= e^{2\pi jf_0 n}\sum\limits_{k\in \mathbb{Z}}h[k]e^{-2\pi jf_0k} \\ &= p_{f_0}[n]H(f_0) \end{align}$$
  • where $p_{f_0}$ is the eigenvector and $H(f_0)$ is the eigenvalue.

Notable notes