Sampling Theory and Discrete Time Fourier Transform
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Transcript of Sampling Theory and Discrete Time Fourier Transform
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BiS351 S iBiS351 S i 20092009BiS351: SpringBiS351: Spring 20092009BioBio--Signal ProcessingSignal Processingg gg g(Part D: Discrete Time Fourier Transform (DTFT)(Part D: Discrete Time Fourier Transform (DTFT)
Discrete Fourier Transform (DFT) )Discrete Fourier Transform (DFT) )Discrete Fourier Transform (DFT) )Discrete Fourier Transform (DFT) )
Jong Chul Ye, Ph.D ([email protected])Jong Chul Ye, Ph.D ([email protected])
Dept. of Bio & Brain EngineeringKorea Advanced Institute of Science and Technology (KAIST)
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Week Topics Contents
1 Introduction Course Overview
2 Signal Basics Type of Signals, Elementary Signals
3 Linear Time Invariant Systems Continuous- and Discrete LTI Systems, Properties
4
C S
Fourier Transform of Continuous-Time Signals5
Continuous-Time Signal Analysis6
Fourier Transform Properties7
8 Fourier Series
9 Midterm Exam. Period (No Class)
10 Di t Ti F i T f10
Discrete Time Signal Analysis
Discrete-Time Fourier Transform(DTFT)11
12 Discrete Fourier Transform (DFT)
1313Z-Transform
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15 Sampling Rate Conversion
16 Final Exam
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16 Final Exam
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R R d M F K dReview: Road Map to Fourier Kingdom
Periodic Discrete
h(u) Non-Periodic Periodich(u) H(ω)
Non Periodic Periodic
Non-Periodic (Continuous Time) Fourier Series( )Fourier Transform
Periodic Discrete Time Fourier Discrete Fourier Transform (DTFT)
Transform(DFT/FFT)
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P F lPoisson Formula
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S l ThSampling Theory
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S l Th ( )Sampling Theory (cont)
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P f R C dPerfect Reconstruction Condition
• Nyquist Sampling Theory
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Al C /V dAliasing in Camera/Video
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MRI SMRI System• The static field (B0 field)
– Strong magnetic field (1.5~7T) to align the nuclei.
• Radio waves (RF)– Transforms the magnetisation so that
we can measure
• Gradient fields (Gx, Gy, Gz)S h k h h l – So that we know where the signal comes from
– So that we can create an image
• Image characteristics:• Image characteristics:– Density of nuclei– Structure surrounding nuclei; T1 & T2
– Chemical characteristics
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MRI F i T f F lMRI: Fourier Transform Formula
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Al MRIAliasing in MRI
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D ff f S l M l l T l R lDiffusion of Single Molecule: Temporal Resolution
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A k i S i N k f C l i S f f NAnkyrin-Spectrin Network of Cytoplasmic Surface of Neuron
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D T F T f (DTFT)Discrete Time Fourier Transform (DTFT)
• Discrete in time Periodic in Frequency domain
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E l 5 1Example 5.1
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E l 5 3Example 5.3
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D f DTFTDerivation of DTFT
• We assume that the Nyquist sampling period of is equal to T=1.
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D f DTFT ( )Derivation of DTFT (cont.)
Nyquist sampling
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DTFT F SDTFT vs Fourier Series
Periodic in frequencyDiscrete in time
Periodic in timeDiscrete in frequencyDiscrete in time Discrete in frequency
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T ETime Expansion
• Time expansion is very important issue.• This will be covered in more detail during “sampling rate
i ” lconversion” class.
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D ff E SDifference Equation Systems
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E l 5 18Example 5.18
• A causal LTI system described by a difference equation
F • Frequency response
• Impulse response
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E l 5 19Example 5.19
• Compute the Frequency domain transfer function of the following difference equation.
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SSummary
• Derivation of DTFT• Properties of DTFT• Linear Difference Equation
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Week Topics Contents
1 Introduction Course Overview
2 Signal Basics Type of Signals, Elementary Signals
3 Linear Time Invariant Systems Continuous- and Discrete LTI Systems, Properties
4
C S
Fourier Transform of Continuous-Time Signals5
Continuous-Time Signal Analysis6
Fourier Transform Properties7
8 Fourier Series
9 Midterm Exam. Period (No Class)
10 Di t Ti F i T f10
Discrete Time Signal Analysis
Discrete-Time Fourier Transform(DTFT)11
12 Discrete Fourier Transform (DFT)
1313Z-Transform
14
15 Sampling Rate Conversion
16 Final Exam
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16 Final Exam
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R d F K dRoadmap to Fourier Kingdom
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P d D S l ( 211 226 367 372)Periodic Discrete Signal (pp. 211-226,p. 367-372)
• Signal with period N
E l• Example:
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D P d S l ( )Discrete Periodic Signal (cont.)
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D P d S l ( )Discrete Periodic Signal (cont.)
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D f D F S Derivation of Discrete Fourier Series
DTFT Fourier SeriesDTFT Fourier Series
Periodic in freq Periodic in time
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D f I F lDerivation of Inversion Formula
• Suppose the discrete signal is periodic is time with T=N, we have Fourier series formula
Th f h f l b• Therefore, the inverse formula becomes
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D f F d F lDerivation of Forward Formula
• The continuous representation of the discrete signal x[n] is
• Furthermore, the period of x(t) is N. Therefore, we have
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D F T fDiscrete Fourier Transform
• Note that x[n] and a[k] are periodic with N.• Hence, we can just use one period of the signal for computation.• The Discrete Fourier Transform (DFT) isThe Discrete Fourier Transform (DFT) is
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D F T f (T F )Discrete Fourier Transform (Two Forms)
• Discrete Fourier Series a.k.a. Discrete Fourier Transform (DFT) in Oppenheim, Wilskly and Nawab is given by
DFT i MATLAB d O h i S h f i d fi d b• DFT in MATLAB and Oppenheim, Schafer is defined by
• Normalized Notation
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E l DFT Example: DFT computation
• Q1: Compute Q2 C X[k]• Q2: Compute X[k]
• Q3: Relationship in between ? Plot it.
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P f DFTProperties of DFT
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C l C lCircular Convolution
• Multiplication of DFT coefficients circular convolution
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E l C l C lExample: Circular Convolution
S [ ] h[ ] i i f ll i Suppose x[n], h[n] is given as following
Sh hShow that1. using DFT2 using circular convolution 2. using circular convolution
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C f DTFT l DFTComputation of DTFT convolution using DFT
• How to compute the convolution of two discrete NON-periodic signal using DFT ?
Solution: we need zero padding – Solution: we need zero padding
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F F T fFast Fourier Transform
• Note that the complexity of the original DFT is N2
• Is there any way to reduce the complexity ?Y FFT Nl (N) l– Yes…. FFT == Nlog2(N) complexity
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D i i i Ti FFT Al i h (C l & T k Al i h )Decimation-in-Time FFT Algorithm (Cooley & Tukey Algorithm)
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N/2 point DFT, Period= N/2
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D T FFT Al hDecimation-in-Time FFT Algorithms
• Note the following complexity reduction– Total computation = 2 x N/2-
DFT + N multiplication = N+N2/2
• The complexity is reduced about to halfto half
• What if we use recursively ?
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8 P FFT8-Point FFT
T l l i i O(Nl N)
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Total complexity is O(Nlog2 N)
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H kHomework
• Write MATLAB code for 64-point FFT implementation
Using recursive application of the Cooley& Tukey algorithm– Using recursive application of the Cooley& Tukey algorithm– Compare the results with the built in MATLAB function “fft”.
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SSummary
• Discrete Fourier Series• Discrete Fourier Transform• Fast Fourier Transform
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