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
  • (includes), (belong to)
  • si, where i[0,1]
  • m=1NN1i=0si

Sequence and Series

Sequence

  • Definition: ordered set of values (mathmatical objects).
  • Arithmetic sequence
    • un={a,n=0,un1+b,n>0.
    • or un=a+nb.
  • Geometric sequence
    • un={a,n=0,un1b,n>0.
    • or un=abn.
  • Harmonic sequence
    • uk[n]=akcos(2πfkn+ϕk)

Series

  • Definition: Sn=ni=0ui
  • Arithmetic series Sn=ni=0(b+ia)=ni=0bni=0ia=(n+1)b+an(n+1)2=(n+1)(b+an2).
  • Geometric series Sn=ni=0bai=bni=0ai=b(1an+11a), when n
    • |a|<1Sn=b11a
    • |a|>1Sn=±
  • Fourier series SK[n]=Kk=0uk[n]=Kk=0akcos(2πfkn+ϕk)

Vectors & Matrices