快速fourier变换

Fourier theory made easy(?)

Fourier theory made easy (?)

a sine wave 5*sin(2π4t) Amplitude =5 Frequency =4 Hz seconds

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 - 8 - 6 - 4 - 2 0 2 4 6 8 5*sin (24t) Amplitude = 5 Frequency = 4 Hz seconds A sine wave

A sine wave signal 5*sin(274t) Amplitude =5 F1 =4H Sampling rate=256 samples/ secon Sampling duration

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 - 8 - 6 - 4 - 2 0 2 4 6 8 5*sin(24t) Amplitude = 5 Frequency = 4 Hz Sampling rate = 256 samples/second seconds Sampling duration = 1 second A sine wave signal

An undersampled signal sin(28t), SR=8.5 Hz 0.20.4 141.61.82

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 - 2 -1.5 - 1 -0.5 0 0.5 1 1.5 2 sin(28t), SR = 8.5 Hz An undersampled signal

The Nyquist Frequency The nyquist frequency is equal to one-half of the sampling frequency. The Nyquist frequency is the highest frequency that can be measured in a signal

The Nyquist Frequency • The Nyquist frequency is equal to one-half of the sampling frequency. • The Nyquist frequency is the highest frequency that can be measured in a signal

Fourier series Periodic functions and signals may be expanded into a series of sine and cosine functions http://www.falstad.com/fourier/12/

http://www.falstad.com/fourier/j2/ Fourier series • Periodic functions and signals may be expanded into a series of sine and cosine functions

The Fourier transform a transform takes one function(or signal) and turns it into another function(or signal)

The Fourier Transform • A transform takes one function (or signal) and turns it into another function (or signal)

The Fourier transform a transform takes one function(or signal) and turns it into another function(or signal) Continuous fourier transform close your eyes if you don t like integrals

The Fourier Transform • A transform takes one function (or signal) and turns it into another function (or signal) • Continuous Fourier Transform: close your eyes if you don’t like integrals

The Fourier transform a transform takes one function(or signal) and turns it into another function(or signal) Continuous fourier transform H(= h()e2mdt h()=「H()=2mdf

The Fourier Transform • A transform takes one function (or signal) and turns it into another function (or signal) • Continuous Fourier Transform: ( ) ( ) ( ) ( )    − −  − = = h t H f e df H f h t e dt ift ift   2 2

The Fourier transform a transform takes one function(or signal) and turns it into another function(or signal) The discrete Fourier transform Hn=∑he 2mikn/N H 2m//N

• A transform takes one function (or signal) and turns it into another function (or signal) • The Discrete Fourier Transform: The Fourier Transform   − = − − = = = 1 0 2 1 0 2 1 N n ikn N k n N k ikn N n k H e N h H h e  