Digital Audio Signal Processing Lecture 1 (Notes)
Notes of Digital Audio Signal Processing, Lecture 1.
Introduction
- Tools and concepts used in the course: quantitative, math, symbolic
- Symbolic representation
Symbolic Representation
- numbers: \(i,j,k,l,m,n\) (interger), \(x,y,z,t\) (coordinate).
- Interger: count object: \(\mathbb{Z,V}\)
- Real: \(\mathbb{R}\) + rational \(x=\dfrac{p}{q}\)
Properties Links
- \(\subset\) (includes), \(\in\) (belong to)
- \(s_i\), where \(i\in[0,1\cdots]\)
- \(m=\dfrac{1}{N}\sum\limits_{i=0}^{N-1}s_i\)
Sequence and Series
Sequence
- Definition: ordered set of values (mathmatical objects).
- Arithmetic sequence
- \[ u_n= \begin{cases} a,\quad n=0, \\ u_{n-1}+b,\quad n>0. \end{cases} \]
- or \(u_n=a+nb\).
- Geometric sequence
- \[ u_n= \begin{cases} a,\quad n=0, \\ u_{n-1}\cdot b,\quad n>0. \end{cases} \]
- or \(u_n=a\cdot b^n\).
- Harmonic sequence
- \(u_k[n]=a_k\cos(2\pi f_k n +\phi_k)\)
Series
- Definition: \(S_n=\sum\limits_{i=0}^n u_i\)
- Arithmetic series \[ \begin{align} S_n&= \sum\limits_{i=0}^n (b+ia) \\ &= \sum\limits_{i=0}^n b \sum\limits_{i=0}^n ia \\ &= (n+1)b + a\dfrac{n(n+1)}{2} \\ &= (n+1)\cdot(b+\dfrac{an}{2}). \end{align} \]
- Geometric series \[S_n=\sum\limits_{i=0}^n ba^i=b\cdot\sum\limits_{i=0}^n a^i = b\left(\dfrac{1-a^{n+1}}{1-a}\right)\], when \(n\rightarrow \infty\)
- \(|a|<1 \rightarrow S_n=b\dfrac{1}{1-a}\)
- \(|a|>1 \rightarrow S_n=\pm\infty\)
- Fourier series \(S_K[n]=\sum\limits_{k=0}^Ku_k[n]=\sum\limits_{k=0}^K a_k\cos(2\pi f_k n+\phi_k)\)