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: Z,V
- Real: R + rational x=pq
Properties Links
- ⊂ (includes), ∈ (belong to)
- si, where i∈[0,1⋯]
- m=1NN−1∑i=0si
Sequence and Series
Sequence
- Definition: ordered set of values (mathmatical objects).
- Arithmetic sequence
- un={a,n=0,un−1+b,n>0.
- or un=a+nb.
- Geometric sequence
- un={a,n=0,un−1⋅b,n>0.
- or un=a⋅bn.
- Harmonic sequence
- uk[n]=akcos(2πfkn+ϕk)
Series
- Definition: Sn=n∑i=0ui
- Arithmetic series Sn=n∑i=0(b+ia)=n∑i=0bn∑i=0ia=(n+1)b+an(n+1)2=(n+1)⋅(b+an2).
- Geometric series Sn=n∑i=0bai=b⋅n∑i=0ai=b(1−an+11−a), when n→∞
- |a|<1→Sn=b11−a
- |a|>1→Sn=±∞
- Fourier series SK[n]=K∑k=0uk[n]=K∑k=0akcos(2πfkn+ϕk)