Digital Audio Signal Processing Lecture 6 (Notes)

Notes of Digital Audio Signal Processing, Lecture 6.

\(y=\mathcal{T}\{x\}\) 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 = \(5\log_{10}(\dfrac{p}{p_0}) = 10\log_{10}(\dfrac{a}{a_0})\)

\(p_{f_0}[n] = e^{2\pi jf_0 n}\) 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.