sine wave 5sin2 4 t AmplitudeA 5 Frequencyf

正弦波(sine wave) 5*sin(2 4 t) Amplitude(振幅A) = 5 Frequency(頻率f) = 4 Hz Phase 相位 秒(seconds)

正弦波信號 5*sin(2 4 t) Amplitude = 5 Frequency = 4 Hz Sampling rate (SR 採樣頻率) = 256 samples/second Sampling duration (採樣總時間) = 1 second 秒(seconds)

Euler's formula

傅立葉轉換 Fourier Transform • A transform takes one function (or signal) and turns it into another function (or signal)

Continuous Fourier Transform 頻率域(Frequency Domain) 時間域(Time Domain)

離散傅立葉轉換 Discrete Fourier Transform

快速傅立葉轉換(FFT) Fast Fourier Transform • The Fast Fourier Transform (FFT) is a very efficient algorithm for performing a discrete Fourier transform • FFT algorithm published by Cooley & Tukey in 1965 • 怎麼做!! • 一般計算要求，採樣點是 2的次方. 8, 16, 32, …

Data flow diagram for N=8: a decimation-in-time radix-2 FFT breaks a length-N DFT into two length-N/2 DFTs followed by a combining stage consisting of many size-2 DFTs called "butterfly" operations (so-called because of the shape of the data-flow diagrams).

還有 • 卷積 Convolution

Fourier Transform and Convolution Let Then Convolution 時間域 Multiplication 頻率域

頻率域(f) 時間域(t) Find g(t) by Fourier transform !! IFT FT FT

Famous Fourier Transforms 傅立葉變換對 Sine wave Delta function

Famous Fourier Transforms Gaussian

Famous Fourier Transforms Sinc function Square wave

Famous Fourier Transforms Exponential Lorentzian

FFT of FID (Free induction decay, 自由感應衰減)

• • • T：總採樣時間 , Sampling time , Duration N：總採樣點 , Sample points Ts：採樣週期 , Sampling period Fs：採樣頻率, Sampling rate df：頻率間隔, 頻率解析度 , Frequency interval , Frequency space , Frequency spacing T/N = Ts = 1/Fs , Fs=N/T , df = 1/T = 1/(N*Ts) = Fs/N 頻率範圍 –TN/2 ~ TN/2 • • ps. 採樣點 64點為例, 設頻率由小而大為 -32, -31, …, -1, 0, 1, 2, …, 31 fft(x) 0, 1, …, 31, -32, -31, …-1 fftshift(fft(x)) -32, -31, …, -1, 0, 1, 2, …, 31

Measuring multiple frequencies

Measuring multiple frequencies

簡單的 Signal Filtering(低通) T = 100 間隔 0. 01 秒

band-pass (low-pass)

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