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Sampling theory Fourier theory made easy

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Sampling, FFT and Nyquist Frequency

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5*sin (2 4t) Amplitude = 5 Frequency = 4 Hz seconds A sine wave We take an ideal sine wave to discuss effects of sampling

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5*sin(2 4t) Amplitude = 5 Frequency = 4 Hz Sampling rate = 256 samples/second seconds Sampling duration = 1 second A sine wave signal and correct sampling 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

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An undersampled 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 Sampling rate Undersampling can be confusing Here it suggests a different frequency of sampled signal Red dots represent the sampled data Here sampling rate is 8.5 Hz and the frequency is 8 Hz

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The Nyquist Frequency one-half of the sampling frequency. 1.The Nyquist frequency is equal to one-half of the sampling frequency. that can be measured 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

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Fourier series is for periodic signals As you remember, periodic functions and signals may be expanded into a series of sine and cosine functions

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

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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

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

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

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Fast Fast Fourier Transform very efficient algorithm 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 In 1969, the 2048 point analysis of a seismic trace took 13 ½ hours. 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. 5.Now you can appreciate applications that would be very difficult without FFT.

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Examples of FFT

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Famous Fourier Transforms Sine wave Delta function In time In frequency Calculated in real time by software that you can download from Internet or Matlab

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Famous Fourier Transforms Gaussian In time In frequency

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Famous Fourier Transforms Sinc function Square wave In time In frequency

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Famous Fourier Transforms Sinc function Square wave In time In frequency

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Famous Fourier Transforms Exponential Lorentzian In time In frequency

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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.

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FFT of FID SR=sampling rate In time In frequency T2=0.5s

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FFT of FID In time In frequency Effect of change of T2 from previous slide T2=0.1s

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In time In frequency Effect of change of T2 from previous slide FFT of FID T2 = 2s

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Effect of changing sample rate In time In frequency Change of sampling rate, we see pulses

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changing sample rate Effect of changing sample rate In time In frequency SR = 256 kHz SR = 128 kHz Lowering the sample rate: –Reduces the Nyquist frequency, which Reduces the maximum measurable frequency Does not affect the frequency resolution Circles appear more often Peak for circles and crosses in the same frequency

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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

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Effect of changing sampling duration In time In frequency

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Effect of reducing the sampling duration from ST = 2s to ST = 1s In time In frequency ST = Sampling Time duration Reducing the sampling duration: –Lowers the frequency resolution –Does not affect the range of frequencies you can measure

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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

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Effect of changing sampling duration In time In frequency T2 = 20 s

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Effect of changing sampling duration In time In frequency T2 = 0.1s

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Measuring multiple frequencies In time 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

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Measuring multiple frequencies In time In frequency

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Sampling Theorem of Nyquist

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Nyquist Sampling Theorem Continuous signal: Shah function (Impulse train): Sampled function: projected Sampled and discretized Multiplication in image domain

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Sampling Theorem: multiplying image by Impulse train in image domain Continuous signal: Shah function (Impulse train): Sampled function:

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Sampling Theorem: multiplication in image domain is convolution in spectral Sampled function: Only if Sampling frequency Shah function (Impulse train): image We do not want trapezoids to overlap

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Nyquist Theorem If Aliasing When can we recover from ? Only if (Nyquist Frequency) We can use Thenand Sampling frequency must be greater than Nyquist Theorem; greater We can recover F(u) from Fs(u) when the sampling frequency is greater than 2 u max

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Aliasing in 2D image Low frequencies High frequencies

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Some useful links –Fourier series java applet –Collection of demonstrations about digital signal processing –FFT tutorial from National Instruments –Dictionary of DSP terms 4FreeIndDecay.pdfhttp://jchemed.chem.wisc.edu/JCEWWW/Features/McadInChem/mcad008/FT 4FreeIndDecay.pdf –Mathcad tutorial for exploring Fourier transforms of free-induction decay –This presentation

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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.

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