Group Delay Spectrum Estimation Method of Improvement of SNR for MST Radar Data

(1)

International Journal of Electronics Communication and Computer Engineering

Volume 6, Issue 2, ISSN (Online): 2249–071X, ISSN (Print): 2278–4209

252

Group Delay Spectrum Estimation Method of

Improvement of SNR for MST Radar Data

Dr. K. Nagi Reddy1 , K.Haneesha2 , K. Krishna Bharathi3 , Md.Rasool4, M.Surendra5 , M .Adarsh 6

NBKR Institute of Science & Technology, Vidyanagar, Nallore(dt) email: [email protected]

Abstract: One of the primary objectives of this work is to study the use of phase based features, specifically, group delay function based features. The group delay function has been used in clean and noisy environments. The group delay function can be ill-behaved when the signal is not minimum phase (in particular, when the zeros of the transfer function are on the unit circle). To solve this, it was shown in [1] that modified group delay function (MODGDF) is a practical estimate of the minimum phase group delay function. In this paper, we propose to use the product of the power spectrum and the complex group delay function (CGDF), from the product spectrum. This spectrum combines the information from the magnitude spectrum as well as the phase spectrum. Finally computation and comparison of SNR‘s are being done on these three methods.

Keywords: Power Spectrum, Group Delay, Complex Group Delay, Signal to Noise Ratio(SNR) Modified Product Group Delay

I. I

NTRODUCTION

Traditionally, the phase spectrum of the signal has been ignored, primarily because only the principal values of the phase can be estimated from the Fourier transform. For the phase to be used, the phase function will have to be unwrapped to produce a continuous estimate [2]. On the other hand, the group delay function [3] (defined as the negative derivative of the phase function), which has properties similar to the phase, can be computed directly from the signal.

A new function called complex I-Order and II-Order Group delay spectrum estimation functions based on the derivatives of the FT phase has been proposed for spectrum estimation. The proposed complex I-Order and II-Order Group delay estimations are applied for both sinusoids in noise as well as for real data(MST Radar data) collected from the NARL Gadanki to extract useful spectral information and to compare the same with the results obtained using the group delay functions proposed [4,5]. It is observed that, the proposed complex II-Order Group delay and modified product group delay estimation methods reduces the noise levels to large extent and also significantly reduces side lobe leakage due to additive noise.

Apart from these methods, modified and product group delay spectrum estimation also been utilized for the estimation of Signal to Noise Ratio (SNR) and the comparative analysis has been done between these methods, the paper is organized in the following manner. In section 2 group delay method and modified group delay method of spectrum estimation is explained, section 3 deals with the I,II order complex group delay functions

and modified product group delay functions, section 4 illustrates the results and section 5 concludes the paper

II. G

ROUP

D

ELAY

The group delay function is defined as the negative derivative of the Fourier transform phase of a signal [2,4,6]. For a minimum phase signal, the group delay computed from the magnitude spectrum of the Fourier transform is equal to that computed from the phase spectrum [7,8]. The group delay function

τ

(

ω

)

directly from the signal

x

(

n

)

as follows [6, 9].

∑

−

=

n

n j

e

n

x

X

(

ω

)

(

)

ω

(1)

The X (

ω

) as a function of magnitude and phase can be expressed as

( )

) ) (

) ( ( )

(

) ( ) ( )

(

1

2 2

) (

ω

θ

ω

θω

R I

I R

j

X X Tan

X X

X here

e X X

−

=

+ =

=

(2)

Then the group delay

τ

(

ω

)

is defined as [3,8,10]

ω

θ

ω

τ

d

(

)

(

=

−

(3)

To avoid unwrapping, another method [4, 9-11] is used to calculate the group delay directly as:

)

(

)

(

log(

)

(

log

X

ω

=

X

ω

+

j

θ

ω

(4)

the equation (3) can be simplified as [6]

[

]

_











−

=

ω

τ

d

X

d

log

(

)

Im

)

(

(5)

(2)

253

















₊

=

₂

)

(

)

(

)

(

)

(

)

(

)

(

ω

τ

X

w

Y

X

Y

X

_R _R _I _I

(6)

where XR is a Real part of the Fourier Transform of

x(n) XI is an Imaginary part of the Fourier Transform of

x(n) , YR is a Real part of the Fourier Transform of

nx(n) and YI is an Imaginary part of the Fourier

Transform of nx(n)

2.1 Modified Group delay function

There are only meaningless peaks and valleys in the GDF. It occurs due to the power spectrum in the denominator in Equation (6).In order to make the GDF meaningful, a modification to the GDF has been proposed

by replacing the power spectrum

X

(

ω

)

2 with the

cepstrally smoothed power [8] spectrum

S

(

ω

)

2in Equation (6). This gives

















₊

=

₂

)

(

)

(

)

(

)

(

)

(

)

(

ω

τ

S

w

Y

X

Y

X

_R _R _I _I

(7)

We denote X (k) and Y (k) as Fourier Transforms of x(n) and y(n) respectively. Then the samples of group delay function can be written as follows

III. C

OMPLEX

G

ROUP

D

ELAY

F

UNCTIONS

Complex group delay function can be formulated from the definition of the Fourier Transform(1) as follows

) (

)

(

)

(

ω

=

X

ω

e

jΦω

X

₍₈₎

In equation (8),

X

(

ω

)

is the frequency magnitude

response of the filter,

Φ

(

ω

)

is the filter response and

ω

is continuous frequency measured in radians/seconds. Taking derivative of

X

(

ω

)

with respective to

ω

on both sides of equation (8) and simplify we get I-order complex group delay

τ

(

ω

)

(

)

(

)

(

1 1

ω

τ

ω

τ

ω

τ

=

R

+

j

I (9)

Where 2 ) ( ) ( ) ( Re 1 ω τ ω ω ω X Y X Y X X d dX j

al R R I I

R + = =                   2 ) ( ) ( ) ( 1 ω τ ω ω ω X Y X Y X X d dX j

imag R I I R

I − = =                  

Here the group delay appears as a complex quantity of which the real part

τ

R

(

ω

)

is the group delay obtained

from the traditional definition for the group delay. The second term in equation (9) appeared as the imaginary quantity of the complex group delay

τ

(

ω

)

.It is obtained that, the dimensions of the

(

)

1

ω

τ

I are also same with dimensions of

(

)

1

ω

τ

R .

The formulation of the proposed method is being derived by performing the derivative of (9) with respect to

ω

and by doing some mathematical manipulations it can be shown[5]













+













=

ω

τ

ω

τ

ω

τ

d

j

d

(

)

R

(

)

I

(

)

(10) where

ω

τ

d

(

)

is the Complex II- order Group Delay and

(

)

2 1 2 2 2

2 )

(

)

(

I I I I R R I

X

Z

X

Z

X

Y

d

τ

ω

τ

=

−

















₋

₊

=

(11) 2 1

)

(

)

2 (

)

2 (

2 2 R I R R R I I I R R

X

Y

Z

X

Y

Z

X

d

τ

ω

τ

=













+

−

=

(12)

where ZR = Real part of the Fourier Transform of n 2

x(n) ZI = Imaginary part of the Fourier Transform of n

2

x(n)

3.1 Modified Product Group delay

In this paper, we define the product spectrum Q(ω_{) as} the product of the power spectrum and the GDF is The

product of (11) and

S

2

(

ω

)

as fellows.

2

)

(

)

(

)

(

2

ω

τ

ω

X

Q

=

_R

(13)

The product spectrum is influenced by both the magnitude spectrum and the phase [11]. Another method to prevent the spikes on the group delay of signal, we will use a modified group delay as follows [12].

2 1

)

(

)

2 (

)

2 (

2

2 _A R

I R R R I I I R R

X

Y

Z

X

Y

Z

X

d

τ

ω

τ

=













+

−

=

1 2 2

)

(

=

_R _R Β−

P

w

τ

(14)

(3)

254

ained in section 4

.

IV. R

ESULTS AND

D

ISCUSSION

The Group delay Functions have been successfully implemented for spectrum estimation by simulating the atmospheric signal spectrum model using MATLAB and tested for sinusoidal varying Doppler trace. The results are compared with the various methods like conventional group delay I,II order complex group delay and modified group delay methods .This comparison is being done by varying parameters A and B of equation (14).SNR is calculated for the MST Radar data by varying the parameters A and B and the SNR values are tabulated for the optimum values of A=0.9 and B=0.2. SNR is calculations are carried out foe MST Radar data in 6 directions namely East, west, north, south, Zenith-X and Zenith-y for the different methods of spectrum estimation. From these observations presented in the Table 1 using the five methods like Normal FFT Method Conventional group delay method, I-order Group Delay, II-order Group Delay Modified Product group delay it is inferred that the II-order Group delay method is superior when compared to the other three methods in general and with the existing FFT method in particular. The II-order Group delay method yields nearly 7dB SNR improvement over the FFT method where the improvement in SNR is highly desirable to have a better Doppler trace visibility. The modified group delay method is also yields an improvement of 6db which is also a considerable improvement when compared with the traditional FFT method.

V. C

ONCLUSIONS

A new spectral estimation method based modified complex group delay has been proposed. This newly proposed method has been compared successfully with the Group Delay methods proposed by [2] and the FFT method. The proposed method provides clear PSD and also suppresses the spikes generated due to noise in the spectrum compared with the above methods to a greater extent at higher altitudes. Nearly 7dB SNR improvement is observed using the II-Order complex group delay spectral estimation method compared to the conventional FFT method. And The modified group delay method is also yields an improvement of 6dB which is also a considerable improvement when compared with the traditional FFT method. Hence it is concluded that both the II order and Modified complex group delay methods for the parameter extraction of the real time radar data and can be suggested an alternative for the FFT method.

Table 1:Comparative of SNR values for different methods of spectrum estimation

S.No Method Average SNR in dB

1 Group Delay -22.85 2 I-order Group Delay -21.7855 3 II-order Group Delay -18.3075

4 FFT Method -25.00

5 Modified Product group delay

19.00

R

EFERENCES

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[11] Donglai Zhu and Kuldip K. Paliwal “product of power spectrum and group delay function for speech recognition”pp125-128,

ICASSP 2004

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A

UTHOR

'

S

P

ROFILE

K. Nagi Reddy