ShearingNAME : YI-WEI CHEN

STUDENT NUMBER : R02942096

ShearingSheep 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 FORTIME VARIANT SIGNALJ IAN- J IUN DING, SOO CHANG PEI , AND TING YU KO

DEPARTMENT OF ELECTRICAL ENGINEERING, NATIONAL TAIWAN UNIVERSITY

20TH EUROPEAN SIGNAL PROCESSING CONFERENCE(EUSIPCO 2012)

Abstract

AbstractHigher 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 and t > t0+T), its bandwidth is F:

TF value determines the lower bound of sampling points

Fundamental harmonic part of a whale voice signalSTFT : TF VALUE = 2.1*1000 = 2100 CONVENTIONAL MODULATION : TF VALUE =

2.1* 100 = 210 (F1 = 440HZ)

Conventional modulationanalytic 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 algorithmThe innovation is……

the higher order exponential function is adopted for modulation

reducing the aliasing effect before samplingthe fractional Fourier transformthe signal segmentation techniquethe 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 frequencyTHE CENTRAL FREQUENCY (VARIES WITH TIME) OF THE WHALE VOICE SIGNAL

USING A 5TH ORDER POLYNOMIAL TO APPROXIMATE THE CENTRAL FREQUENCY OF THE WHALE VOICE(P5(T))

ApproximationLegendre 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 :

X2(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 signalCONVENTIONAL 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 and H(u) = 0 for u > u0

Scaling + rotating

Fundamental harmonic part of a whale voice signalTHE 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

Reconstructionsinc 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 FREQUENCYTHE STFT OF THE SIGNAL AFTER HIGH ORDER MODULATION

Speech signal : “for”

Conclusion

ConclusionA new signal sampling algorithm :

the higher order modulation operationthe STFTthe 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