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Sampling theory Fourier theory made easy Sampling, FFT and Nyquist Frequency A sine wave 8 5*sin (24t) 6 Amplitude = 5 4 Frequency = 4 Hz 2 0 -2 -4 -6 -8 0 0.1 0.2 0.3 0.4 0.5 seconds 0.6 0.7 0.8 0.9 1 We take an ideal sine wave to discuss effects of sampling A sine wave signal and correct sampling 8 5*sin(24t) 6 Amplitude = 5 4 Frequency = 4 Hz 2 Sampling rate = 256 samples/second 0 -2 Sampling duration = 1 second -4 -6 -8 0 0.1 0.2 0.3 0.4 0.5 seconds 0.6 0.7 0.8 0.9 1 We do sampling of 4Hz with 256 Hz so sampling is much higher rate than the base frequency, good Thus after sampling we can reconstruct the original signal An undersampled signal Here sampling rate is 8.5 Hz and the frequency is 8 Hz Sampling rate Red dots represent the sampled data sin(28t), SR = 8.5 Hz 2 1.5 1 0.5 0 -0.5 -1 -1.5 -2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Undersampling can be confusing Here it suggests a different frequency of sampled signal Undersampled signal can confuse you about its frequency when reconstructed. Because we used to small frequency of sampling. Nyquist teaches us what should be a good frequency The Nyquist Frequency 1. The Nyquist frequency is equal to one-half of the sampling frequency. 2. The Nyquist frequency is the highest frequency that can be measured in a signal. Nyquist invented method to have a good sampling frequency We will give more motivation to Nyquist and next we will prove it Fourier series is for periodic signals • As you remember, periodic functions and signals may be expanded into a series of sine and cosine functions http://www.falstad.com/fourier/j2/ 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: H f h t e h t H f e 2ift dt 2ift df The Fourier Transform • A transform takes one function (or signal) and turns it into another function (or signal) • The Discrete Fourier Transform: N 1 H n hk e 2ikn N k 0 N 1 1 2ikn N hk H n e N n 0 Fast Fourier Transform 1. The Fast Fourier Transform (FFT) is a very efficient algorithm for performing a discrete Fourier transform 2. FFT principle first used by Gauss in 18?? 3. FFT algorithm published by Cooley & Tukey in 1965 4. In 1969, the 2048 point analysis of a seismic trace took 13 ½ hours. 5. • Using the FFT, the same task on the same machine took 2.4 seconds! • We will present how to calculate FFT in one of next lectures. Now you can appreciate applications that would be very difficult without FFT. Examples of FFT Famous Fourier Transforms 2 1 Sine wave 0 -1 -2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 In time 300 250 200 Delta function 150 100 50 0 0 20 40 60 80 100 120 In frequency Calculated in real time by software that you can download from Internet or Matlab Famous Fourier Transforms 0.5 0.4 0.3 Gaussian 0.2 0.1 0 0 5 10 15 20 25 30 35 40 45 50 In time 6 5 4 Gaussian 3 2 1 0 0 50 100 150 200 250 In frequency Famous Fourier Transforms 1.5 1 Sinc function 0.5 0 -0.5 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 In time 6 5 4 Square wave 3 2 1 0 -100 -50 0 50 100 In frequency Famous Fourier Transforms 1.5 1 Sinc function 0.5 0 -0.5 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 In time 6 5 4 Square wave 3 2 1 0 -100 -50 0 50 100 In frequency Famous Fourier Transforms 1 0.8 0.6 Exponential 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 In time 30 25 20 Lorentzian 15 10 5 0 0 50 100 150 200 250 In frequency FFT of FID 1. If you can see your NMR spectra on a computer it’s because they are in a digital format. 2. From a computer's point of view, a spectrum is a sequence of numbers. 3. Initially, before you start manipulating them, the points correspond to the nuclear magnetization of your sample collected at regular intervals of time. 4. This sequence of points is known, in NMR jargon, as the FID (free induction decay). 5. Most of the tools that enrich iNMR are meant to work in the frequency domain; they are disabled when the spectrum is in the time domain. 6. Indeed, the main processing task is to transform the time-domain FID into a frequency-domain spectrum. t F t sin 2ft exp FFT of FID T 2 2 T2=0.5s 1 SR=sampling rate 0 f = 8 Hz SR = 256 Hz T2 = 0.5 s -1 -2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 In time 70 60 50 40 30 20 10 0 0 20 40 60 80 100 120 In frequency FFT of FID t F t sin 2ft exp T 2 2 f = 8 Hz SR = 256 Hz T2 = 0.1 s 1 T2=0.1s Effect of change of T2 from previous slide 0 -1 -2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 14 In time 12 10 8 6 4 2 0 0 20 40 60 80 100 120 In frequency FFT of FID T2 = 2s t F t sin 2ft exp T 2 Effect of change of T2 from previous slide 2 1 0 -1 -2 f = 8 Hz SR = 256 Hz T2 = 2 s 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 In time 200 150 100 50 0 0 20 40 60 80 100 120 In frequency Effect of changing sample rate Change of sampling rate, we see pulses 2 1 0 In time -1 -2 f = 8 Hz T2 = 0.5 s 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 70 35 60 30 50 25 40 20 30 15 20 10 10 5 0 0 10 20 30 40 50 60 0 In frequency • Lowering the sample rate: Effect of changing sample rate SR = 256 kHz – Reduces the Nyquist frequency, which SR = 128 kHz • Reduces the maximum measurable frequency • Does not affect the frequency resolution 2 SR = 256 Hz SR = 128 Hz 1 0 -1 -2 Circles appear more often 0 0.2 0.4 0.6 0.8 1 1.2 f = 8 Hz T2 = 0.5 s 1.4 1.6 1.8 2 70 35 60 30 Peak for circles and crosses in the same frequency 50 40 25 20 30 15 20 10 10 5 0 0 10 20 30 40 In time 50 60 0 In frequency Effect of changing sample rate • Lowering the sample rate: – Reduces the Nyquist frequency, which • Reduces the maximum measurable frequency • Does not affect the frequency resolution To remember Effect of changing sampling duration 2 1 0 -1 -2 f = 8 Hz T2 = .5 s 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 2 4 6 8 10 12 14 16 18 20 In time 70 60 50 40 30 20 10 0 In frequency Effect of reducing the sampling duration from ST = 2s to ST = 1s ST = Sampling Time duration 2 1 ST = 2.0 s ST = 1.0 s 0 -1 -2 f = 8 Hz T2 = .5 s 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 2 4 6 8 10 12 14 16 18 20 In time 70 60 50 40 30 20 10 0 In frequency • Reducing the sampling duration: – – Lowers the frequency resolution Does not affect the range of frequencies you can measure Effect of changing sampling duration • Reducing the sampling duration: – Lowers the frequency resolution – Does not affect the range of frequencies you can measure To remember Effect of changing sampling duration T2 = 20 s 2 1 0 -1 -2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 In time 200 150 100 50 0 f = 8 Hz T2 = 2.0 s 0 2 4 6 8 10 12 14 16 18 20 In frequency Effect of changing sampling duration T2 = 0.1s 2 ST = 2.0 s ST = 1.0 s 1 0 f = 8 Hz T2 = 0.1 s -1 -2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 2 4 6 8 10 12 14 16 18 20 In time 14 12 10 8 6 4 2 0 In frequency Measuring multiple frequencies 3 2 f1 = 80 Hz, T21 = 1 s f = 90 Hz, T2 = .5 s 1 f3 = 100 Hz, T23 = 0.25 s 2 2 0 -1 -2 -3 SR = 256 Hz 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 In time 120 100 80 60 40 20 0 0 20 40 60 80 100 120 In frequency conclusion: you can read the main frequencies which give you the value of your NMR signal, for instance logic values 0 and 1 in NMR –based quantum computing Good sampling is important for accuracy Measuring multiple frequencies 3 2 f1 = 80 Hz, T21 = 1 s f = 90 Hz, T2 = .5 s 1 f3 = 200 Hz, T23 = 0.25 s 2 2 0 -1 -2 -3 SR = 256 Hz 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 In time 120 100 80 60 40 20 0 0 20 40 60 80 100 120 In frequency Sampling Theorem of Nyquist Nyquist Sampling Theorem f x Continuous signal: x Shah function (Impulse train): sx s x x nx n 0 projected x x0 Sampled function: Sampled and discretized f s x f x sx f x x nx0 n Multiplication in image domain Sampling Theorem: multiplication in image domain is convolution in spectral Sampled function: image Sampling frequency Shah function (Impulse train): f s x f x sx f x x nx0 1 x0 n 1 n FS u F u S u F u u x0 n x0 FS u F u A A umax x0 umax u 1 Only if u max 1 2 x0 u x0 We do not want trapezoids to overlap Nyquist Theorem If u max FS u 1 2 x0 A x0 Aliasing u umax 1 x0 When can we recover F u from FS u ? Only if u max We can use Then 1 (Nyquist Frequency) 2 x0 x0 C u 0 u 1 2 x0 otherwise F u FS u C u and f x IFTF u Sampling frequency must be greater than 2umax Nyquist Theorem; We can recover F(u) from Fs(u) when the sampling frequency is greater than 2 u max Aliasing in 2D image High frequencies Low frequencies Some useful links • • • • • • http://www.falstad.com/fourier/ – Fourier series java applet http://www.jhu.edu/~signals/ – Collection of demonstrations about digital signal processing http://www.ni.com/events/tutorials/campus.htm – FFT tutorial from National Instruments http://www.cf.ac.uk/psych/CullingJ/dictionary.html – Dictionary of DSP terms http://jchemed.chem.wisc.edu/JCEWWW/Features/McadInChem/mcad008/FT 4FreeIndDecay.pdf – Mathcad tutorial for exploring Fourier transforms of free-induction decay http://lcni.uoregon.edu/fft/fft.ppt – This presentation Conclusions 1. Signal (image) must be sampled with high enough frequency 2. Use Nyquist theorem to decide 3. Using two small sampling frequency leads to distortions and inability to reconstruct a correct signal. 4. Spectrum itself has high importance, for instance in reading NMR signal or speech signal.