1. Sampling theory Fourier theory made easy

2. Sampling, FFT and Nyquist Frequency

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

4. A sine wave signal and correct sampling
5*sin(24t)
Amplitude = 5
Frequency = 4 Hz
Sampling rate = 256 samples/second
Sampling duration =
1 second
We do sampling of 4Hz with 256 Hz so sampling is much higher rate than the base frequency, good
seconds
Thus after sampling we can reconstruct the original signal

5. An undersampled signal
Here sampling rate is 8.5 Hz and the frequency is 8 Hz
Sampling rate
Red dots represent the sampled data
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

6. 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.
We will give more motivation to Nyquist and next we will prove it
Nyquist invented method to have a good sampling frequency

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

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

9. 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

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

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

12. Fast Fourier Transform
The Fast Fourier Transform (FFT) is a very efficient algorithm for performing a discrete Fourier transform
FFT principle first used by Gauss in 18??
FFT algorithm published by Cooley & Tukey in 1965
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.
Now you can appreciate applications that would be very difficult without FFT.

13. Examples of FFT

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

15. Famous Fourier Transforms
Gaussian
In time
Gaussian
In frequency

16. Famous Fourier Transforms
Sinc function
In time
Square wave
In frequency

17. Famous Fourier Transforms
Sinc function
In time
Square wave
In frequency

18. Famous Fourier Transforms
Exponential
In time
Lorentzian
In frequency

19. FFT of FID
If you can see your NMR spectra on a computer it’s because they are in a digital format.
From a computer's point of view, a spectrum is a sequence of numbers.
Initially, before you start manipulating them, the points correspond to the nuclear magnetization of your sample collected at regular intervals of time.
This sequence of points is known, in NMR jargon, as the FID (free induction decay).
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.
Indeed, the main processing task is to transform the time-domain FID into a frequency-domain spectrum.

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

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

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

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

24. 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
SR = 256 kHz
SR = 128 kHz
Circles appear more often
In time
Peak for circles and crosses in the same frequency
In frequency

25. 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

26. Effect of changing sampling duration
In time
In frequency

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

28. 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

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

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

31. 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

32. Measuring multiple frequencies
In time
In frequency

33. Sampling Theorem of Nyquist

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

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

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

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

38. Aliasing in 2D image
High frequencies
Low frequencies

39. Some useful links http://www.falstad.com/fourier/
Fourier series java applet
Collection of demonstrations about digital signal processing
FFT tutorial from National Instruments
Dictionary of DSP terms
Mathcad tutorial for exploring Fourier transforms of free-induction decay
This presentation

40. Conclusions Signal (image) must be sampled with high enough frequency
Use Nyquist theorem to decide
Using two small sampling frequency leads to distortions and inability to reconstruct a correct signal.
Spectrum itself has high importance, for instance in reading NMR signal or speech signal.