978-1-61284-798-6/11/$26.00 ©2011 IEEE 8C3-1

AN ACCURATE NUMERICAL METHOD FOR ESTIMATING THE DELAY BETWEEN TWO OMNI-DIRECTIONAL RECEIVING ELEMENTS

Kaluri V. Rangaraoand and Shridhar Venkatanarasimhan Department of Electronics and Communication Engineering, Jawaharlal Nehru Technological University, Hyderabad, India

Abstract In any two elements of a sensor array with

omni-directional receiving elements, the effect of a narrow-band (NB) signal incident on the elements is the same for both of them, except for a phase lag or lead between them. In this paper a numerical algorithm for estimating the delay between the time series acquired by these two spatially separated elements is presented. The method makes use of a first-order all pass filter (APF) for creating a delay with respect to one of the two elements and compares it with the second element. An objective function is created which varies with the delay. This objective function is minimized using gold-section univariate minimization algorithm without gradients to yield an accurate estimate of delay. It is well known that there are limitations of using first-order APF for different sampling rates while minimizing the objective function. An innovative logic and methodology to overcome these issues is presented. This algorithm is applied to a 3-element Direction Finding (DF) system covering the entire azimuth plane. Results for various signal-to-noise (SNR) conditions are presented along with the intricacies of the algorithm. The DF system shows an accuracy, i.e. standard deviation (�), of 1� or better for SNR less than 4 dB.

Keywords: Delay Estimation, All Pass Filter, Smart Antennas, Direction Finding, Gold-Section Univariate Minimization

I. Introduction Consider a simple 3-element Direction Finding

(DF) system as depicted in Figure 1. The three omni-directional elements are A-B-C forming an equilateral triangle in the azimuth plane. The incident wave makes an angle � with normal of AB. It can be seen from Figure 1 the normal of AC has rotated by an angle of �

� clock-wise with respect to the normal of AB. Thus the same incident wave makes an angle of � � �

� with respect to the normal of AC. Similarly, the

wave makes an angle of � � ��� with respect to the

normal of BC.

Figure 1. A Simple 3-Element Direction Finding

System

A. Narrow Band (NB) Signal Consider a signal (�) received at A (see

Figure 1), whose spectral energy is mostly centered around � , where � is the center frequency of the NB signal. The velocity of propagation � and � are related as � = � � where � is the wave-length. It is assumed that the signal is sampled at a sampling frequency of ��.

The normalised frequency is defined as �� = ����

. The signal (�) on sampling can be written as �

= (���) where �� = ���

. Consider a requirement of sampling a signal ��(�) = (� � �) as an estimate of signal �

� received at B. This implies that we need to construct a signal such that ��

� = (� � ���) = (� � �) . Difficulty arises

because � = ��� is not necessarily an integer. In fact,

= ����

= ��!

, is defined as samples/cycle as if the signal is a pure sine wave.

The non-integer shift � is achieved using an all pass filter pole positioning. In the trivial case of �

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being an integer, phase shift is achieved by shifting the sequence by a sample index. The samples in the original sequence are shifted by an integral multiple of ��" , where the integer N is the number of samples per

cycle. i.e. the phase shift can be quantified as ��" �,

where � is an integer. When ��� = �#�

, � = 1 and if � = 2 then ��

� = �#� and so on, limiting the

accuracy.

Ideally narrow-band (NB) signal is modelled using a band-pass filter [1] as: �

= $%�#� � $��#�

+ &('� � '�#�) (1)

where the input '� is a White Gaussian Noise(WGN) and & = �#*,

� with 0.95 < $ < 0.98.

However the NB signal can be written as pure tone for ease of presentation and understanding without any loss of generality. Signal received at the 3-elements A-B-C as depicted in Figure 1 can be written as:

� = /34 (26���) (2)

�� = /34 (26��� + 7�) (3)

�: = /34 (26��� + 7:) (4)

where 7� = 26 ;> /34 � and

7: = 26 ;> /34 (� � �

�) . The incident wave has a wave-length of � and the spacing between elements is ?. The center frequency of the narrow-band signal is assumed as � . The signal is sampled at a sampling rate of ��. The normalised frequency is �� = ��

�� and

normalised angular frequency is @� = 26��.

II. Delay Estimation (DE) A considerable amount of research has gone into

time-delay estimation [2-6] for the past three decades for a variety of applications unveiling new dimensions.

This problem is addressed from the point of view of a Direction Finding system [7] application. This algorithm for delay estimation has been tested for a wide variety of narrow-band signals that are encountered in real life. It also gives accurate results under various acceptable signal-to-noise (SNR) conditions. These results are presented and tabulated towards the end of this paper. The estimated delay is

applied for finding the angle of the direction of arrival (DOA) of the incoming signal. The DE algorithm is depicted in Figure 2 and applied for estimating delay between the time series �

and �� . DE has three

important components:

1. A simple All Pass Filter (APF) and the associated logic to make it work for all {��, 7} combinations, where �� is the normalised frequency and 7 is the delay

2. An objective function 3. A stable minimization algorithm

Figure 2. Delay Estimation Algorithm

DE is attempted in two passes. In the first pass, various angles between 0 and 360 degrees are tried out in equal steps evaluating an objective function. The angle at which minima occurs is a coarse estimate. Using this coarse minima an accurate estimate of the delay is obtained using gold section univariate minimization.

A All Pass Filter Consider the difference equation representing a

first-order All Pass Filter (APF)

��� = $��#�

� � $� + �#�

(5)

Here, � is the NB input and ��

� is the output sequence, where � is the index of the sample in the sequence. The radial pole position $ is to be chosen such that 0.1 < |$| < 0.98 due to numerical stability considerations. By executing (5), ��

� gets phase shifted by 7 with respect to �

. This phase shift is determined by pole position $ . A shift of ��

" � is obtained, such that � is a real number. It is assumed that 7 is a delay lying between 0 and 26.

Consider the following operator relationship between ��

and ��#� in equation (5).

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��#� = B#���

(6)

Same relationship also exists between ��� and

��#�� . Using these relationships, equation (5) can be

re-written as

��� = $B#���

� � $�� + B#���

(7)

= C DEF#*�#*DEFG ��

(8)

= C (DEF#*)(�#*D)(�#*DEF)(�#*D)G ��

(9)

= C (DEF#*)(�#*D)(�#*[DEFHD]H*,)G ��

(10)

= B#� C (�#*D)(�#*D)(�#*[DEFHD]H*,)G ��

(11)

we can easily see that by using the substitution B = IJK and B = cos@ + Lsin@

(1 � $B)� = (1 + $� � 2$ MN/ @)IJO

substituting the above in equation (11), we get

��� = I#JK C(�H*,#�* �� K)PQR

(�H*,#�* �� K) G ��

= SI#JKIJOT'�

= SIJ(#KHO)T'� (12)

where

� = 2 �U4 #� #* �V� K�#* �� K (13)

As can be seen from equation (12), the desired phase-shift delay angle 7 and the angle � are related as:

7 = @ � � (14)

� = @ � 7 (15)

Using the equation (13), it may be possible to choose $ such that the phase shift is given by (14)

$ = �W�(R,)

�W�(R,) ��K#�V�K

(16)

Figure 3 shows the plot of �� and ��

� . We chose to shift by 69. 6� , which corresponds to 5.8 samples, which is a non integral number of samples. The number of samples per cycle was chosen as

= 30 , i.e., normalized frequency was �� = ��Z .

Using equation (16), $ was obtained as 0.7373 . Phase-shift was done as shown in Figure 3: blue circles with dot inside represent input signal and red pluses output signal. Green circles represent the sequence ��#^

, i.e. integral shift by 5 samples. As it can be seen from Figure 3, a new sequence ��

� =��#^._

is generated by passing �� through difference

equation (5). So, from the explanation given above and from Figure 3, it can be concluded that we can choose any arbitrary real phase-shift. Before using the value of $, we must check that 0 < |$| < 1.

Figure 3. Non Integral Shifting of `a

b

Figure 4 shows the similar plot for a narrow-band signal ��

shifted by 5.8 samples resulting ��� =

��#^._ . The narrow-band signal is generated using

equation (1).

Figure 4. Non Integral Shifting of NB Signal `a

b

B. Limitations of Simple APF For stability reasons, the $ values are

normally considered within some thresholds such as 0.1 < |$| < 0.98, rather than using values close to 0 and 1.

Figure 5 shows a plot of �� and 7 depicting unsuitable $ values (filled space) for APF for a range of 0� to 360�, where as the unfilled space is suitable. It is clear that, given a normalized frequency ��, it may not be always possible to find a suitable $ for creating delay using APF, for some ranges of phase-shift angles 7.

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Figure 5. Simple All Pass Filter Performance

C. Modified All Pass Filter In order to overcome the problem of getting the

delayed response for such �� and 7, where $ is out of range for a single filter, it is proposed to use cascade of two filters, each giving a shift of d

� . This substantially improves the usable {��, 7} space. For example, let �� = �

�Z and the desired phase delay be 225� . The $ value obtained using equation (16) is 1.0910, which is unstable. Hence, we use the angle ��^

��

, i.e., 112. 5� , for which $ turns out to be 0.8688. Even after splitting the shift angle into two parts for a given normalized frequency, ��, we may not be able to find a suitable $ for some ranges of phase-shift angles 7. In order to further narrow down the set of values of �� and 7 for which $ gets out of bounds (even with two filters), it is proposed to use cascade of three filters, each giving a shift of d

� .

Figure 6 shows how most of the {��, 7} space is usable with this logic (see Figure 7). In the rare case when $ is unsuitable even with three filters, the process reaches a hard stop.

Figure 6. Modified All Pass Filter Performance

Figure 7. Logic State Diagram

C.1. Logic The function efe($) checks if the value of $ is

within the limits i.e., 0.1 < |$| < 0.98. efe($) =1 if 0.1 < |$| < 0.98 else efe($) =0. Figure 7 summarizes the logic in the form of a state diagram.

D. Minimization Algorithm The minimization algorithm is gold section

univariate method without derivatives [8] with some pre-processing. The desired objective function is defined over g samples as:

hj(7) = k lmpF (q�m

r#qmr),

j (17)

where �� is the output of the modified all pass filter

with input �� as depicted in Figure 2. A sample

minimization process is depicted in Figure 8 and described below. In this method, two end-points 7W and 7t are used as initial values. These values are obtained by performing a coarse sweep over one cycle (0 to 360 degrees).

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Figure 8. Minimization by Gold Section

The intermediate points 7 and 7u are chosen such that 7 = 7W + &(7t � 7W) and 7u = 7 +&(7t � 7 ) where & = �#v^

� . The objective function, say hj , is evaluated at 7W, 7t, 7 and 7u . If hj(7 ) w hj(7u), 7t takes the value of 7W and 7W takes the value of 7u in the next iteration. Otherwise, 7W takes the value of 7 and 7 takes the value of 7u in the next iteration. Typically a fixed number of iterations is used and the minimum value is quickly obtained quite accurately.

The top part of Figure 8 shows the initial values of 7W and 7t and the objective function as a function of 7 to be minimized. The middle part of Figure 8 shows how 7W converges rapidly in about 20 iterations. The bottom part shows the narrow band signal.

E. Performance of DE The above algorithm is tested with two different

types of signals (pure tone and an amplitude modulated) under various Signal to Noise Ratio (SNR) conditions and sampling rates by varying (samples/cycle).

�� = /34 (26��� + 7�) + x� (18)

�� = [y + z /34 (26�~�)][ /34 (26��� + 7�)] +

x� (19)

where x� is a zero mean White Gaussian Noise (WGN). A reference signal is generated with 7� = 0 . The phase-delay 7� is varied from �170� to +170� . Phase delay 7�� is estimated

using the DE algorithm. An error �d = (7� � 7��) over this range is generated.

It was found that the distribution of �d is almost normal with zero mean. Standard deviation �d of �d is computed for SNR -2dB to 15dB for both types of signals. The entire process is repeated for N = 3 to 15. Results are depicted in Figure 9. It can be seen that the performance is satisfactory with �d is less than 2� for f � > 6 while at the maximum signal power and noise power is same it is around 4�.

Figure 9. Performance of DE Algorithm

III. Direction Finding with 3 Elements The triangular array of sensors ABC, shown in

Figure 1 is used for estimating the DOA. Using DE, the phase delay between B and A, 7�� and similarly, 7�: and 7��: are given by:

�26 ;> sin�� � , �26 ;

> sin��t� , �26 ;> sin��W�

respectively. Where ��t = �� � �� and ��W = �� � ��

� in which �� is the two-quadrant angle as seen by pair AB. Using the above relation a triplet � ={�� , ��t, ��W} is generated, for a given DOA � . The vector � is used to find the four-quadrant estimates ��W, ��t and �� as seen by each antenna pair BC, CA and AB respectively.

Due to geometric reasons each ��W , ��t and �� are accurate only in some segment of {0� �N 360�}, and this is known apriori. Once this estimate is found along with its quadrant, the best possible pair

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among the three segments of the triangle ABC are used to arrive at an accurate angle �� the DOA.

A. Performance of Direction Finding In evaluating the performance (see Figure 10) we

have chosen ;> = 0.2 . An error �O = (� � ��) is

computed over wide SNR and sampling rates. The value �O is depicted in Figure 10. A superior performance of the DF system is achieved due to the choice of the best segment pair and also due to the excellent well behaved DE algorithm. An accuracy (�O) of 1� or better for SNR less than 4 dB. It is worth noting that this DF system exhibits an error less than 2. 5� at 0 dB SNR conditions and around 3� at -2dB. Impact of sampling rate is not significant below 0dB and N � 4.

IV. Conclusions The DE algorithm described in this paper gives

accurate results. This can be used in a DF system such as the triangular array described in this paper for accurate results. An accuracy (�) of 1� or better for SNR less than 4 dB for the three-element DF system is reported. The results can be compared to eigenanalysis methods such as [9-11] and to Time-Difference of Arrival (TDOA) methods such as [12-15]. This method gives superior performance as seen in Figure 10.

Figure 10. Performance of DF System

References [1] Kaluri V Rangarao and Ranjan K Mallik, Digital Signal Processing A Practitioners Approach. England: John Wiley International, 2005.

[2] Y.T.Chan et al., A Parameter Estimation Approach to Time-Delay Estimation and Signal Detection, IEEE Trans on Acoustic, Speech and Signal Processing vol ASSP 28 No 1 pp 8-15 Feb 1980

[3] Allan G Piersol, Time Delay Estimation Using Phase Data, IEEE Trans on Acoustic, Speech and Signal Processing vol ASSP 29 No 3 pp 471-477 Jun 1981

[4] Ehud Weinstein and Anthony J Weiss, Fundamental Limitations in Passive Time-Delay Estimation - Part I: Narrow Band Systems, IEEE Trans on Acoustic, Speech and Signal Processing vol ASSP 31 No 2 pp 472-486 Apr 1983

[5] P.C.Ching and H.C.So, Two Adaptive Algorithms for Multipath Time Delay Estimation, IEEE Journal of Oceanic Engineering vol 19 No 3 pp 458-463 July 1994

[6] B Yegnanarayana et al ., Processing of Reverberant Speech for Time-Delay Estimation, IEEE Trans on Speech and Audio Processing vol 13 No 6 pp 1110-1118 Nov 2005

[7] R.K. Mallik, K.V. Rangarao, and U.M.S. Murty; DOA Estimation by Least Squares Approach Electronics Letters, vol. 34, no. 12, pp. 1187-1189, June 11 1998.

[8] Richard Brent, Algorithms for Minimization without Derivatives. Prentice - Hall, Inc. (1973).

[9] Ralph O. Schmidt; Multiple Emitter Location and Signal Parameter Estimation, IEEE Transactions on Antennas and Propagation, Vol. AP-34, No.3, March 1986.

[10] Anthony J. Weiss and Benjamin Friedlander; Direction Finding for Diversely Polarized Signals using Polynomial Rooting, ©1991 IEEE.

[11] Richard Roy and Thomas Kailath; ESPRIT-Estimation of Signal Parameters via Rotational Invariance Techniques, IEEE Transactions on Acoustics, Speech and Signal Processing, Vol. 37, No. 7, July 1989.

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[12] Rangarao, K. V. and Satish, K.; an Elegant Method for Delay Estimation in Mobile Applications TENCON 2008 - 2008, TENCON 2008. IEEE Region 10 Conference 19-21 November 2008 Page(s):1 - 3

[13] Hassan Elkamchouchi and Mohamed Abd Elsalam Mofeed; Direction-Of-Arrival Methods (CIA) and Time Difference of Arrival (TDOA) Position Location Technique, Twenty Second National Radio Science Conference (NWC 2005), March 15-17, 2005, Cairo - Egypt.

[14] Gaoming Huang, Luxi Yang and Zhenya He; Time-Delay Direction Finding Based on Canonical Correlation Analysis, 0-7803-8834-8/05 © 2005 IEEE.

[15] Johan Falk, Peter Händel and Magnus Jansson; Estimation of Receiver Frequency Error in a TDOA-Based Direction-Finding System, 0-7803-8622-1/04 ©2004 IEEE.

Acknowledgements I greatly acknowledge the faculty of ECE,

JNTUH, particularly Prof. Pratap L. Reddy and Prof. N. V. Ramana Rao for providing environment. I will be failing in my duties if I don't thank my colleagues Dr. G. Bhoopathy, Director & Distinguished Scientist and Dr. Lakshiminarayan of DLRL(DRDO) for funding this work with a vision.

This work is funded by Defence Electronics Research Laboratory (DRDO), Ministry of Defence, Government of India.

Email Addresses Kaluri V. Rangaraoand, kaluri@ieee.org

Shridhar Venkatanarasimhan, svnara12@yahoo.com

Biographies Dr. Ranga Rao earned his BSEE from Andhra

University in 1974 and MSCS from the Indian Institute of Science, Bangalore in 1977. He received MSSE in 1991 from the US Naval Post Graduate School and Ph.D. from the Indian Institute of Technology (Madras) in 1994. He is a Senior member IEEE. He is author of a book from John Wiley(UK) on DSP applications. Currently he is a Professor at Dept of ECE, JNTU-H, Hyderabad, India. Before that, he was Chief Scientist of Tanla Solutions, VP Engineering at Sarayu Softech, Chief Engineer at General Electric (Hyderabad), Vice President Satyam Computer Services, CEO of GMR Vasavi Infotech and a Senior Scientist in DRDO.

He has been a Visiting Professor at IITD, Visiting Scholar at Oklahoma State University, Visiting Professor at IIIT-H, and Visiting Scholar at CMU. Dr. Ranga Rao holds US and Indian patents.

Shridhar Venkatanarasimhan received BSCS from the Indian Institute of Technology (Mumbai) in 1991 and MSCS from Rutgers University in 1993. Currently he is a Research Scholar at JNTU-H.

30th Digital Avionics Systems Conference October 16-20, 2011