Shearing NAME : YI-WEI CHEN STUDENT NUMBER : R

Shearing Sheep shearing? To remove (fleece or hair) by cutting or clipping. Motions in the time-frequency distribution Multiply chirp function Generalized shearing  phase is a polynomial HOW to do?

HIGHER ORDER MODULATION AND THE EFFICIENT SAMPLING ALGORITHM FOR TIME VARIANT SIGNAL JIAN-JIUN DING, SOO CHANG PEI, AND TING YU KO DEPARTMENT OF ELECTRICAL ENGINEERING, NATIONAL TAIWAN UNIVERSITY 20TH EUROPEAN SIGNAL PROCESSING CONFERENCE(EUSIPCO 2012)

Abstract

Higher order modulation scheme High order modulation with the fractional Fourier transform Minimize the area of a signal in the time-frequency domain Much reduce the number of sampling points Efficient for sampling a time variant signal (ex : voice of an animal and the speech signal)

Introduction

Shannon’s sampling theory fs : the sampling frequency △ = 1/fs : the sampling interval F : the total bandwidth The sampling frequency should be larger than the Nyquist rate suppose that the support of a signal is T(x(t )~= 0 for t t0+T), its bandwidth is F: TF value determines the lower bound of sampling points

Fundamental harmonic part of a whale voice signal STFT : TF VALUE = 2.1*1000 = 2100 CONVENTIONAL MODULATION : TF VALUE = 2.1* 100 = 210 (F1 = 440HZ)

Conventional modulation analytic signal form : Xa(f) = X(f) for f > 0 and Xa(f) = 0 for f < 0 the conventional modulation operation: f1 is chosen as 440 Hz

Sampling algorithm The innovation is…… the higher order exponential function is adopted for modulation reducing the aliasing effect before sampling  the fractional Fourier transform  the signal segmentation technique  the pre-filter

Higher order modulation

Generalized modulation operation m(t) is an nth order polynomial, and the instantaneous frequency: The STFT relations between x(t) & y(t)

Central frequency THE CENTRAL FREQUENCY (VARIES WITH TIME) OF THE WHALE VOICE SIGNAL USING A 5TH ORDER POLYNOMIAL TO APPROXIMATE THE CENTRAL FREQUENCY OF THE WHALE VOICE(P 5 (T))

Approximation Legendre polynomial expansion: central frequency of the signal is h(t) [t0, t0+T] is the support of h(t) {Lk(t) | k = 0, 1, 2, …} is the Legendre polynomial set For this example :

X 2 (t) The STFT of x2(t) where x2(t) is the result of proposed high order modulation of the analytic signal of the whale voice TF value = 35*2.1 = 73.5

Combining higher order modulation with the fractional fourier transform

Fundamental harmonic part of a whale voice signal CONVENTIONAL MODULATION (F1 = 400HZ) AFTER PERFORMING THE FRFT AND THE SCALING OPERATION, THE STFT IS ROTATED(X3)

FRFT “signal segmentation” and “bandwidth reduction” Definition : performing the Fourier transform 2α/π times placing a separating line : where H(u) = 1 for u u0 Scaling + rotating

Fundamental harmonic part of a whale voice signal THE 5TH ORDER POLYNOMIAL (BLACK LINE) TO APPROXIMATE THE CENTRAL FREQUENCY AFTER THE SCALE FRFT + PROPOSED HIGH ORDER MODULATION (TF VALUE = 21*2.4 = 50.4) (X4)

X4(t) according to the 5th order polynomial that can approximate the central frequency of x3(t)

TF value (a) The original sampling algorithm. (b) Analytic signal conversion + modulation. (c) Analytic signal conversion + FRFT + modulation. (d) Analytic signal conversion + FRFT + proposed higher order modulation

Results

Reconstruction sinc function interpolation is the inverse of the sampling operation removing the imaginary part is the inverse of the analytic function generation operation

Other simulations

Another whale voice signal

Speech signal : “for” STFT OF THE FIRST HARMONIC PART OF THE SPEECH SIGNAL THE STFT OF THE ANALYTIC SIGNAL CONVERSION + CONVENTIONAL MODULATION + SCALED FRFT OPERATIONS

Speech signal : “for” A 5TH ORDER POLYNOMIAL (BLACK LINE) TO APPROXIMATE THE CENTRAL FREQUENCY THE STFT OF THE SIGNAL AFTER HIGH ORDER MODULATION

Speech signal : “for”

Conclusion

A new signal sampling algorithm :  the higher order modulation operation  the STFT  the FRFT filter The number of sampling points is very near to the area of the nonzero region much fewer number of sampling points to represent a signal Other applications :  data transmission  communication

Thank you