DSP

1.Introduction

Pythagoras

Most Reality can be described in terms of numbers and measurement
Parmenides

plant the seed of fndemental dichotomy, the fundemental difference between the reality and the ideal reality
Plato

metaphysics
Zeno's Paradox
Galileo's Calculus and Descartes' Coordinate
Fourier Analysis
Nyquist and Shanon: The Sampling Theorem

the amplitude discretization are good for storage,processing and transmission

because of Noise, the digital signal is better

x[n](n < 0) is 0 because it will unbounded

scaling
- $y[n]=lpha x[n]$
sum
- y[n] = x[n] + z[n]
product
- y[n] = x[n] * z[n]
shift by k(decay)
- y[n] = x[n - k]
- shift of a finite-length:finite support
- shift of a finite-length:periodic extension
Energy
- $E_x=\sum_{n=-\infty}^{+\infty}{\left|x[n] ight|}^2$
Power
- $$P_x=\lim_{N o \infty} rac{1}{2N+1} \sum_{n=-N}^N{\left|{x[n]} ight|}^2$
- The window's size is 2N+1
Energy&Power:Periodic Signal -
- Energy
  - $E_{ ilde{x}}= \infty$
- Power
  - $P_{ ilde{x}}= rac{1}{N} \sum_{n=0}^{N-1}{\left| { ilde{x}[n]} ight|}^2$

The oscillatory heartbeat is measured by:
- a frequency $\omega$ (unit:radians)
- an initial phase $\phi$ (unit:radians)
- an amplitude A
- a trigonometric function $x[n]=Acos(\omega n+\phi)$
The Euler's formulas
- $x[n]=Acos(\omega n+\phi)+Ajsin(\omega n+\phi)=e^{j(\omega n+\phi)}$
The advantages of using complex exponentials rather than trigonometric
- sine & cosine "live" together
- phase shift is simple multiplication

when the $\omega$ is larger and larger, when it is near $2$\pi$$ we can see the wagonwheel seems stop

Discrete
- n: no physical dimension
- periodicty: samples
Physical
- measured in Hz
- where system clock take part in interpolation
- The real frequency is $f=MT_s$