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 (jong.ye@kaist.ac.kr)Jong Chul Ye, Ph.D (jong.ye@kaist.ac.kr)

Dept. of Bio & Brain EngineeringKorea Advanced Institute of Science and Technology (KAIST)

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

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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

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

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

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)

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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