2011. 1002. b. sampling fourier

Sampling theory

Fourier theory made easy

Sampling, FFT Sampling, FFT and Nyquist and Nyquist FrequencyFrequency

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 (24t)

Amplitude = 5

Frequency = 4 Hz

seconds

A sine wave

We take an ideal sine wave to discuss effects of sampling

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(24t)

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

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

2sin(28t), SR = 8.5 Hz

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 confusingHere 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

The Nyquist Frequency

1. The Nyquist frequency is equal to one-half one-half of the sampling frequency.of the sampling frequency.

2. The Nyquist frequency is the highest frequency that can be measured 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

http://www.falstad.com/fourier/j2/

Fourier series is for periodic signals

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

The Fourier TransformFourier 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



• Continuous Fourier Transform:

dfefHth

dtethfH

ift

ift

2

2


• The Discrete Fourier Transform:


1

0

2

1

0

2

1 N

n

Niknnk

N

k

Niknkn

eHN

h

ehH

FastFast Fourier Transform1. The Fast Fourier Transform (FFT) is a very efficient algorithm 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. • 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.

Examples of Examples of FFTFFT

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-2

-1

0

1

2

0 20 40 60 80 100 1200

50

100

150

200

250

300

Famous Fourier Transforms

Sine wave

Delta function

In timeIn time

In frequencyIn frequencyCalculated in real time by software that you can download from Internet or Matlab


0 5 10 15 20 25 30 35 40 45 500

0.1

0.2

0.3

0.4

0.5

0 50 100 150 200 2500

1

2

3

4

5

6

Gaussian

Gaussian

In timeIn time

In frequencyIn frequency


-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.5

0

0.5

1

1.5

-100 -50 0 50 1000

1

2

3

4

5

6

Sinc function

Square wave

In timeIn time



Sinc function

Square wave

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.5

0

0.5

1

1.5

-100 -50 0 50 1000

1

2

3

4

5

6

In timeIn time



Exponential

Lorentzian

0 50 100 150 200 2500

5

10

15

20

25

30

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

In timeIn time


FFT of FID1. 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.

FFT of FID

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-2

-1

0

1

2

0 20 40 60 80 100 1200

10

20

30

40

50

60

70

f = 8 Hz SR = 256 HzT2 = 0.5 s

SR=sampling rate

In timeIn time


2exp2sin

TtfttF

T2=0.5s

FFT of FID

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-2

-1

0

1

2

0 20 40 60 80 100 1200

2

4

6

8

10

12

14

f = 8 HzSR = 256 HzT2 = 0.1 s

In timeIn time


Effect of change of T2 from previous slide

2exp2sin

TtfttF

T2=0.1s

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-2

-1

0

1

2

0 20 40 60 80 100 1200

50

100

150

200

f = 8 Hz SR = 256 HzT2 = 2 s

In timeIn time


Effect of change of T2 from previous slide

FFT of FID

2exp2sin

TtfttF

T2 = 2s

Effect of Effect of changing sample changing sample raterate

0 10 20 30 40 50 600

10

20

30

40

50

60

70

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-2

-1

0

1

2

0 10 20 30 40 50 600

5

10

15

20

25

30

35

f = 8 Hz T2 = 0.5 s

In timeIn time


Change of sampling rate, we see pulses

Effect of changing sample ratechanging sample rate

0 10 20 30 40 50 600

10

20

30

40

50

60

70

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-2

-1

0

1

2

0 10 20 30 40 50 600

5

10

15

20

25

30

35

SR = 256 HzSR = 128 Hz

f = 8 HzT2 = 0.5 s

In timeIn time


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

Effect of changing Effect of changing sample ratesample 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

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-2

-1

0

1

2

0 2 4 6 8 10 12 14 16 18 200

10

20

30

40

50

60

70

f = 8 Hz T2 = .5 s

In timeIn time


Effect of reducing the sampling duration from ST = 2s to ST = 1s

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-2

-1

0

1

2

0 2 4 6 8 10 12 14 16 18 200

10

20

30

40

50

60

70

ST = 2.0 sST = 1.0 s

f = 8 HzT2 = .5 s

In timeIn time


ST = Sampling Time duration

• Reducing the sampling duration:– Lowers the frequency resolution– Does not affect the range of frequencies you can measure


• Reducing the sampling duration:– Lowers the frequency resolution– Does not affect the range of frequencies you

can measure

To remember


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-2

-1

0

1

2

0 2 4 6 8 10 12 14 16 18 200

50

100

150

200

f = 8 Hz T2 = 2.0 s

In timeIn time


T2 = 20 s


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-2

-1

0

1

2

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

ST = 2.0 sST = 1.0 s

f = 8 Hz T2 = 0.1 s

In timeIn time


T2 = 0.1s

Measuring multiple frequenciesMeasuring multiple frequencies

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-3

-2

-1

0

1

2

3

0 20 40 60 80 100 1200

20

40

60

80

100

120

f1 = 80 Hz, T21 = 1 s f2 = 90 Hz, T22 = .5 sf3 = 100 Hz, T23 = 0.25 s

SR = 256 Hz

In timeIn time

In frequencyIn frequencyconclusion: 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

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-3

-2

-1

0

1

2

3

0 20 40 60 80 100 1200

20

40

60

80

100

120

f1 = 80 Hz, T21 = 1 s f2 = 90 Hz, T22 = .5 sf3 = 200 Hz, T23 = 0.25 s

SR = 256 Hz

In timeIn time


Sampling Sampling Theorem of Theorem of

NyquistNyquist

Nyquist Sampling TheoremNyquist Sampling TheoremContinuous signal:

Shah function (Impulse train):

xf

x

Sampled function:

n

s nxxxfxsxfxf 0

xs

x0x

n

nxxxs 0

projected

Sampled and discretized Multiplication in image domain

Sampling Theorem: multiplication in image domain is Sampling Theorem: multiplication in image domain is convolution in spectralconvolution in spectral

Sampled function:

n

s nxxxfxsxfxf 0

FS u F u S u F u 1x0

u nx0

n

uF

maxu

A

u

uFS

maxu

0xA

0

1x

u

Only if0

max 21x

u

Sampling frequency 0

1xShah function

(Impulse train):

image

We do not want trapezoids to overlap

Nyquist TheoremNyquist TheoremIf

0max 2

1x

u uFS

maxu

0xA

0

1x

u

Aliasing

When can we recover from ? uF uFS

Only if0

max 21x

u (Nyquist Frequency)

We can use

otherwise0

21

00 xux

uC

Then uCuFuF S uFxf IFTand

Sampling frequency must be greater than max2u

Nyquist Theorem;Nyquist Theorem;We can recover F(u) from Fs(u) when the sampling frequency is greatergreater than 2 u max

Aliasing in 2D image

Low frequencies

High 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/FT4FreeIndDecay.pdf– Mathcad tutorial for exploring Fourier transforms of free-induction decay

• http://lcni.uoregon.edu/fft/fft.ppt– This presentation

ConclusionsConclusions

1. Signal (image) must be sampled with high enough frequency

2. Use Nyquist theorem to decide3. 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.

2011. 1002. b. sampling fourier

Documents