Open Access

Optimized Punctured ZCZ Sequence-Pair Set: Design, Analysis, and Application to Radar System

EURASIP Journal on Wireless Communications and Networking20102010:254837

DOI: 10.1155/2010/254837

Received: 23 November 2009

Accepted: 26 April 2010

Published: 28 June 2010

Abstract

Based on the zero correlation zone (ZCZ) concept, we present the definitions and properties of a set of new ternary codes, ZCZ sequence-Pair Set (ZCZPS), and propose a method to use the optimized punctured sequence-pair along with Hadamard matrix to construct an optimized punctured ZCZ sequence-pair set (OPZCZPS) which has ideal autocorrelation and cross-correlation properties in the zero correlation zone. Considering the moving target radar system, the correlation properties of the codes will not be severely affected when Doppler shift is not large. We apply the proposed codes as pulse compression codes to radar system and the simulation results show that optimized punctured ZCZ sequence-pairs outperform other conventional pulse compression codes, such as the well-known polyphase code—P4 code.

1. Introduction

Pulse compression is known as a technique to raise the signal to maximum sidelobe (signal-to-sidelobe) ratio to improve the target detection and range resolution abilities of the radar system. This technique allows a radar to simultaneously achieve the energy of a long pulse and the resolution of a short pulse without the high peak power which is required by a high energy short duration pulse [1]. One of the waveform designs suitable for pulse compression is phase-coded waveform design. The phase-coded waveform design is that a long pulse of duration https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq1_HTML.gif is divided into https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq2_HTML.gif subpulses each of width https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq3_HTML.gif . Each subpulse has a particular phase, which is selected in accordance with a given code sequence. The pulse compression ratio equals the number of subpulses https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq4_HTML.gif , where the bandwidth is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq5_HTML.gif . In general, a phase-coded waveform with longer code word, in other words, higher pulse compression ratio, can have lower sidelobe of autocorrelation, relative to the mainlobe peak, so its main peak can be better distinguished. The relative lower sidelobe of autocorrelation is very important since range sidelobes are so harmful that they can mask main peaks caused by small targets situated near large targets. In addition, the cross-correlation property of the pulse compression codes should be considered in order to reduce the interference among radars when we choose a set of pulse compression codes to work in a Radar Sensor Network (RSN).

Much time and effort was put for designing sequences with impulsive autocorrelation functions (ACFs) and cross-correlation functions (CCFs) for radar target ranging and target detection. On one hand, for aperiodic sequences, it is known that for most binary sequences of length https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq6_HTML.gif the attainable sidelobe levels are approximately https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq7_HTML.gif [2, 3] and the mutual peak cross-correlations of the same-length sequences are much larger and are usually in the order of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq8_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq9_HTML.gif . Later, set of binary sequences of length https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq10_HTML.gif with autocorrelation sidelobes and cross-correlation peak values of approximately https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq11_HTML.gif are studied in paper [4]. Besides, the small set of Kasami sequences and the Bent sequences could achieve maximum correlation values of approximately https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq12_HTML.gif . In addition to binary sequences, polyphase codes, with better Doppler tolerance and lower range sidelobes such as the Frank and P1 codes, the Butler-matrix derived P2 code, the linear-frequency-derived P3 and P4 codes were provided and intensively analyzed in [57]. Quadiphase [8] code could also reduce poor fall-off of the radiated spectrum and mismatch loss in the receiver pulse compression filter of biphase codes. Nevertheless, the range sidelobe of the polyphase codes can not be low enough to avoid masking returns from targets. Hence, considerable work has been done to reduce range sidelobes for the radar system. By suffering a small https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq13_HTML.gif loss, the authors in [9] present several binary pulse compression codes to greatly reduce sidelobes. In the previous paper [10], pulse compression using a digital-analog hybrid technique is studied to achieve very low range sidelobes for potential application to spaceborne rain radar. In the paper [11], time-domain weighting of the transmitted pulse is used and is able to achieve a range sidelobe level of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq14_HTML.gif 55 dB or better in flight tests. These sidelobe suppression methods, however, degrade the receiving resolution because of wider mainlobe.

On the other hand, for periodic sequences, the lowest periodic ACF that could be achieved for binary sequences, as in the case of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq15_HTML.gif -sequences [12, 13] or Legendre sequences, is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq16_HTML.gif . GMW [14] has the same periodic ACF properties, but posses larger linear complexity. Considering the nonbinary case, it is possible to find perfect sequences, such as two valued Golomb sequences, Ipatov ternary sequences, Frank sequences, Chu sequences, and modulatable sequences. However, it should be noted that for both binary and non-binary cases, it is impossible for the sequences to have perfect ACF and CCF simultaneously although ideal CCFs could be achieved alone. One can synthesize a set of non-binary sequences with impulsive ACF and the lower bound of CCF: https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq17_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq18_HTML.gif [15, 16], which is governed by Welch bound and Sidelnikov bound.

So far in the previous work, range sidelobes could hardly reach as low as zero. In addition, it has also been well proven that it is impossible to design a set of codes with ideal impulsive autocorrelation function and ideal zero cross-correlation functions, since the corresponding parameters have to be limited by certain bounds, such as Welch bound [15], Sidelnikov bound [16], Sarwate bound [17], and Levenshtein bound [18]. To overcome these difficulties, the new concepts, generalized orthogonality (GO), also called Zero Correlation Zone (ZCZ) is introduced. Based on ZCZ [1921] concept, we propose a set of ternary codes, ZCZ sequence-pair set, which can reach zero autocorrelation sidelobe zero mutual cross-correlation peaks during Zero Correlation Zone. We also present and analyze a method to construct such ternary codes and subsequently apply them to a radar detection system. The method is that optimized punctured sequence-pair joins together with Hadamard matrix to construct optimized punctured ZCZ sequence-pairs set. An example is presented, investigated, and studied in the radar targets detection simulation system for the performance evaluation of the proposed ternary codes. Because of the outstanding property performance and well target detection performance in simulation system, the newly proposed codes can be useful candidates for pulse compression application in radar system.

The rest of the paper is organized as follows. Section 2 introduces the definitions and properties of ZCZPS. In Section 3, the optimized punctured ZCZPS is introduced, and a method using optimized punctured sequence-pair and Hadamard matrix to construct such codes is given and proved. In Section 4, the properties and ambiguity function of optimized punctured ZCZPS are simulated and analyzed. The performance of optimized punctured ZCZPS is investigated in radar targets detection system by comparing with P4 code in Section 5. In Section 6, conclusions are drawn on optimized punctured ZCZPS.

2. Definitions and Properties of ZCZ Sequence-Pair Set

Zero Correlation Zone (ZCZ) is a new concept provided by Fan et al. [21, 22] in which the autocorrelation sidelobes and cross-correlation values are zero while the time delay is kept within ZCZ instead of the whole period of time domain. There has been considerable interest in constructing [2327] new classes of ZCZ sequences in ZCZ and studying their properties [28].

Here, we introduce sequence-pair into the ZCZ concept to construct ZCZ sequence-pair set. We consider ZCZPS https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq19_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq20_HTML.gif is a set of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq21_HTML.gif sequences of length https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq22_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq23_HTML.gif is a set of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq24_HTML.gif sequences of the same length https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq25_HTML.gif :
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_Equ1_HTML.gif
(1)
The autocorrelation function (ACF) (here we use autocorrelation to stand for the cross-correlation between two different sequences of a sequence-pair to distinguish the cross-correlation between two different sequence-pairs) of sequence-pair https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq26_HTML.gif is defined by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_Equ2_HTML.gif
(2)
The cross-correlation function of two sequence-pairs https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq27_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq28_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq29_HTML.gif is defined by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_Equ3_HTML.gif
(3)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq30_HTML.gif is the time delay and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq31_HTML.gif is the bit duration.

For pulse compression sequences, some properties are of particular concern in the optimization for any design in engineering field. They are the peak sidelobe level, the energy of autocorrelation sidelobes, and the energy of their mutual cross-correlation [4]. Therefore, the peak sidelobe level which represents a source of mutual interference and obscures weaker targets can be presented as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq32_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq33_HTML.gif is among the zero correlation zone for ZCZPS. Another optimization criterion for the set of sequence-pairs is the energy of autocorrelation sidelobes joined together with the energy of cross-correlation. By minimizing the energy, it can be distributed evenly, and the peak autocorrelation sidelobe and the cross-correlation level can be minimized as well [4]. Here, the energy of ZCZPS can be employed as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_Equ4_HTML.gif
(4)

According to (4), it is obvious to see that the energy can be kept low while minimizing the autocorrelation sidelobes and cross-correlation values of any two sequence-pairs within Zero Correlation Zone.

Hence, the ZCZPS can be constructed by minimizing the autocorrelation sidelobe of a sequence-pair and cross-correlation value of any two sequence-pairs in ZCZPS.

Definition 1.

Assume https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq34_HTML.gif to be a sequence-pair set of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq35_HTML.gif sequence-pairs and each sequence-pair is of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq36_HTML.gif bit length. If all the sequence-pairs in the set satisfy the following equation:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_Equ5_HTML.gif
(5)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq37_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq38_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq39_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq40_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq41_HTML.gif is called a ZCZ sequence-pair, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq42_HTML.gif is an abbreviation, and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq43_HTML.gif is called a ZCZ sequence-pair set, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq44_HTML.gif is an abbreviation.

3. Optimized Punctured ZCZ Sequence-Pair Set

3.1. Definition of Optimized Punctured ZCZ Sequence-Pair Set

Matsufuji and Torii have provided some methods of constructing ZCZ sequences in [29, 30]. In this section, a set of novel ternary codes, namely, the optimized punctured ZCZ sequence-pair set, is constructed by applying the optimized punctured sequence-pair [31] to the Zero Correlation Zone. Here, optimized punctured ZCZPS is a specific kind of ZCZPS.

Definition 2 (see [31]).

Sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq45_HTML.gif is the punctured sequence for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq46_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_Equ6_HTML.gif
(6)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq47_HTML.gif is the number of punctured bits in sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq48_HTML.gif . Suppose https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq49_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq50_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq51_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq52_HTML.gif -punctured binary sequence, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq53_HTML.gif is called a punctured binary sequence-pair.

Definition 3 (see [31]).

The autocorrelation of punctured sequence-pair https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq54_HTML.gif is defined as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_Equ7_HTML.gif
(7)
If the punctured sequence-pair has the following autocorrelation property:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_Equ8_HTML.gif
(8)

the punctured sequence-pair is called an optimized punctured sequence-pair [31]. Where, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq55_HTML.gif , is the energy of punctured sequence-pair.

Definition 4.

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq56_HTML.gif in Definition 1 is constructed by optimized punctured sequence-pair and a certain matrix, such as Hadamard matrix or an orthogonal matrix, where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_Equ9_HTML.gif
(9)
Then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_Equ10_HTML.gif
(10)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq57_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq58_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq59_HTML.gif can be called an optimized punctured ZCZ sequence-pair set. https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq60_HTML.gif is an abbreviation.

3.2. Design of Optimized Punctured ZCZ Sequence-Pair Set

Based on an optimized punctured binary sequence-pair of odd length and a Hadamard matrix, an optimized punctured ZCZPS can be constructed on following steps.

Step 1.

Considering an optimized punctured binary sequence-pair https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq61_HTML.gif of odd length, the length of each sequence is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq62_HTML.gif :
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_Equ11_HTML.gif
(11)

Step 2.

A Hadamard matrix https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq63_HTML.gif (the Hadamard matrix is made up of a set of Walsh sequences) of order https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq64_HTML.gif is used here. https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq65_HTML.gif , the length of each sequence, is equal to the number of the sequences in the matrix. Here, any Hadamard matrix order is possible and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq66_HTML.gif is the row vector of the matrix:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_Equ12_HTML.gif
(12)

Step 3.

Doing bit-multiplication on the optimized punctured binary sequence-pair and each row of the Hadamard matrix B, then sequence-pair set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq67_HTML.gif is obtained,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_Equ13_HTML.gif
(13)

Here, the optimized punctured binary sequence-pairs are of odd lengths and the lengths of Walsh sequence are https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq68_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq69_HTML.gif It is easy to see that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq70_HTML.gif , common divisor of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq71_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq72_HTML.gif is 1, then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq73_HTML.gif . The sequence-pair set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq74_HTML.gif is the optimized punctured ZCZPS and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq75_HTML.gif is the Zero Correlation Zone https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq76_HTML.gif . The length of each sequence in optimized punctured ZCZPS is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq77_HTML.gif that depends on the product of length of optimized punctured sequence-pair and the length of Walsh sequence in Hadamard matrix. The number of sequence-pairs in optimized punctured ZCZPS rests on the order of the Hadamard matrix. The sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq78_HTML.gif in sequence set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq79_HTML.gif and the corresponding sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq80_HTML.gif in sequence set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq81_HTML.gif construct a sequence-pair https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq82_HTML.gif that can be used as a pulse compression code.

The correlation property of the sequence-pairs in optimized punctured ZCZPS is
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_Equ14_HTML.gif
(14)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq83_HTML.gif is the Zero Correlation Zone https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq84_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq85_HTML.gif .

Proof.

( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq86_HTML.gif ) When https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq87_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_Equ15_HTML.gif
(15)
( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq88_HTML.gif ) When https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq89_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_Equ16_HTML.gif
(16)

According to Definition 1, the OPZCZPS constructed by the above method is a ZCZPS.

4. Properties of Optimized Punctured ZCZ Sequence-Pair Set

Considering the optimized punctured ZCZPS constructed by the method mentioned in the last section, the autocorrelation and cross-correlation properties can be simulated and analyzed. For example, the optimized punctured ZCZPS https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq90_HTML.gif is constructed by 31-length optimized punctured binary sequence-pair https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq91_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq92_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq93_HTML.gif (using " https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq94_HTML.gif '' and " https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq95_HTML.gif '' symbols for " https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq96_HTML.gif '' and " https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq97_HTML.gif '') and Hadamard matrix https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq98_HTML.gif of order 4. We follow the three steps presented in Section 3.2 to construct the optimized punctured ZCZPS. The number of sequence-pairs here is 4, and the length of each sequence is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq99_HTML.gif . The first row of each matrix https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq100_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq101_HTML.gif constitute a certain optimized punctured ZCZP https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq102_HTML.gif . Similarly, the second row of each matrix https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq103_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq104_HTML.gif constitute another optimized punctured ZCZ sequence-pair https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq105_HTML.gif , and so on:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_Equ17_HTML.gif
(17)

Here, optimized punctured ZCZ sequence-pairs https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq106_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq107_HTML.gif are studied as two examples in the following parts.

4.1. Autocorrelation and Cross-Correlation Properties

The autocorrelation property and cross-correlation property of 124-length sequence-pairs in the optimized punctured ZCZ sequence-pair set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq108_HTML.gif are shown in Figures 1 and 2.
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_Fig1_HTML.jpg
Figure 1

Periodic autocorrelation property of optimized punctured ZCZPS.

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_Fig2_HTML.jpg
Figure 2

Periodic cross-correlation property of optimized punctured ZCZPS.

From the Figures 1 and 2, the peak autocorrelation sidelobe of ZCZPS and their cross-correlation value are kept as low as zero while the time delay is kept within https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq109_HTML.gif (Zero Correlation Zone). And it is always true that the cross-correlation values of optimized punctured ZCZPS and the autocorrelation sidelobe could be kept as low as zero during ZCZ.

We still have to confess that the energy loss of the proposed codes is no less than 1.7 db due to reference mismatch. However, the perfect periodic ACF and CCF achieved simultaneously during the ZCZ zone and the codes' structure could make up for it. It is known that a suitable criterion for evaluating code of length https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq110_HTML.gif is the ratio of the peak signal mainlobe divided by the peak signal sidelobe (PSR) of their autocorrelation function, which can be bounded by [32]
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_Equ18_HTML.gif
(18)

The only aperiodic uniform phase codes that can reach the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq111_HTML.gif are the Barker codes whose length is equal or less than 13. Considering the periodic sequences, the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq112_HTML.gif -sequences or Legendre sequences could achieve the lowest periodic ACF of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq113_HTML.gif . For non-binary sequences, it is possible to find perfect sequences of ideal ACF. Golomb codes are a kind of two valued (biphase) perfect codes which obtain zero periodic ACF but result in large mismatch power loss. The Ipatov code shows a way of designing code pairs with perfect periodic autocorrelation (the cross-correlation of the code pair) and minimal mismatch loss. In addition, zero periodic autocorrelation function for all nonzero shifts could be obtained by polyphase codes, such as Frank and Zadoff codes. However, for both binary and non-binary periodic sequences, it is not possible for the sequences to have perfect ACF and CCF simultaneously although ideal CCFs could be achieved alone. Comparing with the above codes, the proposed ternary codes could obtain perfect periodic ACF during the ZCZ and the reference sequence is made of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq114_HTML.gif which is much less complicated than other perfect ternary codes such as Ipatvo code. The reference code for Ipatov code is of a three-element alphabet which might not always be integer.

Nevertheless, considering multi targets in the system, multiple peaks of the autocorrelation function of the proposed codes might affect on the range resolution. The range resolution could be limited as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq115_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq116_HTML.gif . Here, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq117_HTML.gif is one bit duration, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq118_HTML.gif is the length of an optimized punctured sequence-pair and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq119_HTML.gif is the length of an optimized punctured ZCZ sequence-pair. In the Figure 1, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq120_HTML.gif . Otherwise, some digital signal processing methods could also be introduced to distinguish the peaks. On the other hand, there may also be the concern that multiple peaks of single transmitting signal reflected from one target may affect determining the main peak of ACF. As a matter of fact, the matched filter here could shift at the period of ZCZ length to track each peak instead of shifting bit by bit after the first peak is acquired. Hence, in this way could it be working more efficiently. Alike the tracking technology in synchronization of CDMA system, checking several peaks instead of only one peak guarantee the precision of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq121_HTML.gif and avoidance of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq122_HTML.gif . In addition, those obtained peaks could be averaged before the detection in order to reduce the effect of random noise in the channel so that the detection performance could be improved.

To sum up, the new code could achieve perfect ACF and CCF in the ZCZ simultaneously according to Figures 1 and 2, and its PSR can be as large as infinite.

4.2. Ambiguity Function

When the transmitted impulse is reflected by a moving target, the reflected echo signal includes a linear phase shift which corresponds to a Doppler shift https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq123_HTML.gif [32]. As a result of the Doppler shift https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq124_HTML.gif , the main peak of the autocorrelation function is reduced. The SNR is degraded and the sidelobe structure is also changed because of the Doppler shift.

The ambiguity function which is usually used to analyze the radar performance within Doppler shift and time delay is defined in [32]:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_Equ19_HTML.gif
(19)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq125_HTML.gif is the time delay between transmitting signal and matched filter, and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq126_HTML.gif is the Doppler shift.

In [33], Periodic Ambiguity Function (PAF) is introduced by Levanon as an extension of the periodic autocorrelation for Doppler shift. And the single-periodic complex envelope is [34]
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_Equ20_HTML.gif
(20)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq127_HTML.gif is one period of the signal.

We are studying sequence-pairs in this research, so we use different codes for transmitting part and receiving part. The single-period ambiguity function for ZCZPS can be rewritten as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_Equ21_HTML.gif
(21)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq128_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq129_HTML.gif is one period of the signal and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq130_HTML.gif is one bit duration. At the same time, when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq131_HTML.gif , (21) can be used to analyze the autocorrelation property within Doppler shift, and when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq132_HTML.gif , (21) can be used to analyze the cross-correlation performance within Doppler shift. Equation (21) is plotted in Figure 3 in a three-dimensional surface plot to analyze the radar performance of optimized punctured ZCZPS within Doppler shift. Here, maximal time delay is 1 unit (normalized to length of the code, in units of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq133_HTML.gif ) and maximal Doppler shift is 5 units for cross-correlation and 3 units for autocorrelation (normalized to the inverse of the length of the code, in units of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq134_HTML.gif ).
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_Fig3_HTML.jpg
Figure 3

Ambiguity function of 124-length ZCZPS: (a) autocorrelation, (b) cross-correlation.

In Figure 3(a), there is relative uniform plateau suggesting low and uniform sidelobes. This low and uniform sidelobes minimize target masking effect in Zero Correlation Zone of time domain, where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq135_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq136_HTML.gif . From Figure 3(b), considering cross-correlation property between any two optimized punctured ZCZ sequence-pairs of the ZCZPS, we can see that the optimized punctured ZCZPS is tolerant of Doppler shift when Doppler shift is not large. When the Doppler shift is zero, or the target is not moving, cross-correlation of our proposed code is zero during ZCZ.

Since synchronizing techniques develop exponentially in the industrial world, time delay between transmitting signal and matched filter can, to some extent, be precisely estimated. Therefore, it is necessary to investigate the property of our proposed code when we have the output of the matched filter at the expected time https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq137_HTML.gif . When https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq138_HTML.gif , the ambiguity function can be expressed as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_Equ22_HTML.gif
(22)
And the Doppler shift performance without time delay is presented in the Figure 4.
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_Fig4_HTML.jpg
Figure 4

Doppler shift of 124-length ZCZPS ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq139_HTML.gif ): (a) autocorrelation (b) cross-correlation.

Figure 4(a) illustrates that without time delay of matched filter but having the Doppler shift less than 1 unit, the autocorrelation value of optimized punctured ZCZPS falls sharply during one unit, and the trend of the amplitude over the whole frequency domain decreases as well. Figure 4(b) shows that there are some convex surfaces in the cross-correlation performance. From Figures 4(a) and 4(b), when Doppler frequencies equal to multiples of the pulse repetition frequency ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq140_HTML.gif ), all the ambiguity values turn to zero except when Doppler frequency is equal to 2 PRF for cross-correlation. That is the same as many widely used pulse compression binary code such as the Barker code. Overall, the ambiguity function performances of optimized punctured ZCZP can be as efficient as conventional pulse compression binary code.

5. Application to Radar System

According to [32], Probability of Detection ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq141_HTML.gif ), Probability of False Alarm ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq142_HTML.gif ) and Probability of Miss ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq143_HTML.gif ) are three probabilities of most interest in the radar system. Note that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq144_HTML.gif . Therefore, we simulated the above three probabilities of using 124-length optimized punctured ZCZ sequence-pair in radar system in this section. The performance of radar system using 124-length P4 code is also studied in order to compare with the performance of optimized punctured ZCZ sequence-pairs of corresponding length. In the simulation model, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq145_HTML.gif times of Monte-Carlo simulation has been run for each SNR value. The Doppler shift frequency is a random variable that is kept less than 1 unit (normalized to the inverse of the length of the code, in units of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq146_HTML.gif ), and the expected peak time of the output of the matched filter is at https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq147_HTML.gif .

From Figure 5, the probabilities of miss target detection https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq148_HTML.gif of the system using 124-length optimized punctured ZCZP are lower than 124-length P4 code especially when the SNR is not high. When SNR is higher than 18 dB, both probabilities of miss targets of the system approach zero. However, the probabilities of miss targets of P4 code fall more quickly than optimized punctured ZCZP.
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_Fig5_HTML.jpg
Figure 5

Probability of miss targets detection: 124-length optimized punctured ZCZ sequence-pair versus 124-length P4 code.

We plotted the detection probability https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq149_HTML.gif versus false alarm probability https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq150_HTML.gif of the coherent receiver. We have simulated the performance at different SNR values. Because of the limited space, we only chose SNR at 12 db and 14 dB. Figure 6 shows performance of 124-length optimized punctured ZCZP and performance of the same length P4 code when the SNR is 12 dB and 14 dB. Within the same SNR value either 12 dB or 14 dB, the detection probabilities of optimized punctured ZCZ sequence-pair are much larger than detection probabilities of P4 code, and meanwhile https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq151_HTML.gif of the first code are also smaller than https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_IEq152_HTML.gif of the latter code. Stating differently, optimized punctured ZCZ sequence-pair has higher target detection probability while keeping a lower false alarm probability. Furthermore, observing Figure 6, 124-length optimized punctured ZCZ sequence-pair even has much better performance at 12 dB SNR than P4 code of corresponding length at 14 dB SNR.
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F254837/MediaObjects/13638_2009_Article_1843_Fig6_HTML.jpg
Figure 6

Probability of detection versus probability of false alarm of the coherent receiver: 124-length optimized punctured ZCZ sequence-pair versus 124-length P4 code.

6. Conclusions

The definition and properties of a set of newly provided ternary codes-ZCZ sequence-pair set were discussed in this paper. Based on optimized punctured sequence-pair and Hadamard matrix, we have investigated a constructing method for a specific ZCZPS-optimized punctured ZCZPS made up of a set of optimized punctured ZCZPs along with studying its properties. The significant advantage of the optimized punctured ZCZPS is the considerably reducedn autocorrelation sidelobe and zero mutual cross-correlation value during ZCZ. According to the radar system simulation results shown in Figures 5 and 6, it is easy to observe that 124-length optimized punctured ZCZPS has better performance than P4 code of the same length when the target is not moving very fast in the system. A general conclusion can be drawn that the optimized punctured ZCZPS consisting of optimized punctured ZCZ sequence-pairs can effectively increase the variety of candidates for pulse compression codes. Because of the ideal cross-correlation properties of optimized punctured ZCZPS, our future work would focus on the application of the optimized punctured ZCZPS in multiple radar systems.

Declarations

Acknowledgments

This work was supported in part by the National Science Foundation under Grants CNS-0721515, CNS- 0831902, CCF-0956438, CNS-0964713, and Office of Naval Research (ONR) under Grant N00014-07-1-0395 and N00014-07-1-1024.

Authors’ Affiliations

(1)
Department of Electrical Engineering, University of Texas at Arlington

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© L. Xu and Q. Liang. 2010

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