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 $S_K[n]=\sum\limits_{k=0}^Ku_k[n]=\sum\limits_{k=0}^K a_k\cos(2\pi f_k n+\phi_k)$

Vectors & Matrices