Multicarrier constant envelope OFDM signal design for radar applications

¨ ) 64 (2010) 999–1008 Int. J. Electron. Commun. (AEU

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¨) Int. J. Electron. Commun. (AEU journal homepage: www.elsevier.de/aeue

Multicarrier constant envelope OFDM signal design for radar applications Reza Mohseni a,b,, Abbas Sheikhi a, Mohammad Ali Masnadi-Shirazi a a b

Electrical and Computer Engineering Department, Shiraz University, Shiraz, Iran Electrical Engineering Department, Shiraz University of Technology, Shiraz, Iran

a r t i c l e in fo

abstract

Article history: Received 28 May 2009 Accepted 6 October 2009

Orthogonal frequency division multiplexing (OFDM) radar signals have been introduced for high range resolution radars. These signals have prominent properties such as favorable ambiguity function, high bandwidth efficiency, and possibility of use in dual mode radar/communication systems. But the large amplitude fluctuations of the OFDM signal make it susceptible to system nonlinearities. To alleviate this problem, constant envelope OFDM (CE-OFDM) signal has been introduced which combines orthogonal frequency division multiplexing and phase modulation or frequency modulation. Although several works have been reported on OFDM radar signal design, there is no a systematic approach for designing CE-OFDM signals for radar applications. In this paper we will focus on CE-OFDM signal design for radar applications. Two different methods for designing a CE-OFDM signal with favorable ambiguity functions are introduced. The first one is based on modulating a complementary set of sequences on different sub-carriers while the second is based on using a proper single carrier coded signal and then extracting its most similar multicarrier OFDM or CE-OFDM coded signal. & 2009 Elsevier GmbH. All rights reserved.

Keywords: OFDM radar Constant envelope OFDM OFDM-PM OFDM-FM Ambiguity function

1. Introduction Multicarrier modulation is a popular technique for wideband signal design in high range resolution radars. Among which, the orthogonal frequency division multiplexing (OFDM) radar signals, because of their prominent properties such as favorable ambiguity function, high bandwidth efficiency, and possibility of use in dual mode radar/communication systems, have recently been of interest to researchers [1–8]. One of the major problems of multicarrier OFDM signals is the large peak to average power ratio (PAPR). Large power fluctuations in the OFDM signal are affected by nonlinearities in the system including those from power amplifiers. A power back off at the amplifier is usually employed to reduce the OFDM signal distortion and out-of-band spectral leakage. Such a back off not only reduces the radiated power, but also decreases the efficiency of the amplifier considerably. Several researches have been reported to decrease the PAPR of OFDM radar signals that are based on designing appropriate codes and applying subcarrier amplitudes/phase weighting [3,4]. In [9] we have presented another approach to alleviate the undesirable effects due to the high PAPR of OFDM signals. This technique has also been considered for radar systems in [10–13]. The main idea is to combine the orthogonal frequency division multiplexing technique with phase modulation and frequency modulation (OFDM-FM or OFDM-PM, respectively). In these

 Corresponding author at: Electrical Engineering Department, Shiraz University

of Technology, Shiraz, Iran. Tel.:/fax: +987117264121. E-mail address: [email protected] (R. Mohseni). 1434-8411/$ - see front matter & 2009 Elsevier GmbH. All rights reserved. doi:10.1016/j.aeue.2009.10.008

approaches, PM or FM techniques create a constant envelope signal which allow high power amplifiers to operate near saturation levels, thus maximizing the power efficiency. Although several works have been reported on OFDM radar signal design [1–4,14], there is no systematic approach for designing CE-OFDM signals for radar applications. In this paper we will focus on CE-OFDM signal design for radar applications. Two different methods for designing a CE-OFDM signal with favorable ambiguity functions are introduced. The first is based on modulating complementary sets on different sub-carriers. The second proposed method is based on using a proper single carrier coded signal, and the most similar multicarrier OFDM or CEOFDM coded signal is then extracted. In the following sections, the OFDM and the constant envelope OFDM signals are first introduced in Sections 2 and 3, respectively. Then the compression processing schemes of CE-OFDM signals are studied in Section 4, and two methods of designing CEOFDM signals with favorable ambiguity functions are introduced in Section 5. Finally, in Section 6 we have compared our designed CE-OFDM signals with multicarrier OFDM and P4 polyphase single-carrier phase coded signals in the case of their ambiguity function (autocorrelation function), PAPR, and spectral efficiency.

2. OFDM radar signals An M  N OFDM or multicarrier coded radar signal consists of N carriers transmitted simultaneously where each carrier is phase modulated using a sequence of M bits (chip). The carriers are equally spaced, with separation equal to the inverse of the bit

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1000

duration forming orthogonal frequency division multiplexing [2]. The complex envelope of the OFDM signal is therefore " # M N X X an;m expðj2pfn tÞ sðt  ðm  1Þtc Þ ð1Þ xðtÞ ¼ m¼1

where  1; sðtÞ ¼ 0

n¼1

0 r t otc otherwise

ð2Þ

and fn ¼

n1 ; tc

n ¼ 1; 2; . . . ; N

ð3Þ

The string fan;m gM m ¼ 1 in Eq. (1) is the code sequence on n-th carrier and tc is the chip width. Typically, this signal is generated in discrete time by applying the inverse discrete fourier transform (IDFT) to complex valued data sequences. The ambiguity function of the OFDM signal depends on the modulated codes on each carrier signal. Several works carried out on the design of an appropriate code for the OFDM signal can be found in [1–6]. The real and imaginary parts of xðtÞ are R½xðtÞ ¼

M X N X

fR½an;m cosðj2pfn tÞ  I½an;m sinðj2pfn tÞg

m¼1n¼1

 sðt  ðm  1Þtc Þ

I½xðtÞ ¼

M X N X

ð4Þ Fig. 1. (a) The CE-OFDM waveform is derived from mapping the conventional OFDM signal to the unit circle [16]. (b) Instantaneous power of a conventional OFDM signal and a CE-OFDM signal [16].

fI½an;m cosðj2pfn tÞ þ R½an;m sinðj2pfn tÞg

m¼1n¼1

 sðt  ðm  1Þtc Þ

ð5Þ

OFDM signal achieves good spectral efficiency. The frequency spectrum of this baseband signal is almost limited and flat with an effective band width of N=2tc and its frequency sidelobes are less than that of the coded single-carrier signal having a shape similar to ‘‘Sinc’’ function ðsinðpxÞ=pxÞ [1,3].

3. Constant envelope OFDM radar signal The CE-OFDM signal can be viewed as a mapping of the OFDM signal onto the unit circle by angle modulation, as shown in Fig. 1(a). The resulting signal has a constant envelope, thus 0 dB PAPR, as depicted in Fig. 1(b) [15–17]. The baseband representation of the CE-OFDM signal is given as yðtÞ ¼ A ejfðtÞ

ð6Þ

where A is the signal amplitude and the phase signal fðtÞ is proportional to OFDM signal xðtÞ as follows: In CE-OFDM-PM signal, the phase signal fðtÞ is proportional to the OFDM signal that should be real-valued to achieve a constant envelope signal. So if the OFDM signal is real-valued we can choose

fðtÞ ¼ hp xðtÞ

ð7Þ

Otherwise, when the OFDM signal is complex, one choice is to use

fðtÞ ¼ hp R½xðtÞ

ð8Þ

where hp is the modulation index. For CE-OFDM-FM the instantaneous frequency is proportional to the OFDM signal or is proportional to its integral which is required to be real-valued. So if the OFDM signal is real-valued,

Fig. 2. Simplified block diagram of the CE-OFDM signal generator.

we use

fðtÞ ¼ 2phf

Z

t

xðtÞ dt

Otherwise we can use Z t fðtÞ ¼ 2phf R½xðtÞ dt

ð9Þ

ð10Þ

Fig. 2shows a simplified block diagram of the CE-OFDM signal generator. To obtain a real-valued OFDM signal the modulated sequences on different carriers (in a given bit duration) have to satisfy a conjugate symmetry property by assuming N to be an even number, and aNk;m ¼ ak;m ;

k ¼ 1; 2; . . . ; N=2  1

ð11Þ

Thus, the complex envelope of a constant envelope OFDM signal can be expressed by one of the following equations for

¨ ) 64 (2010) 999–1008 R. Mohseni et al. / Int. J. Electron. Commun. (AEU

CE-OFDM-PM: yðtÞ ¼ A exp jhp

M X

N X

fR½an;m cosðj2pfn tÞ

m¼1n¼1

  I½an;m sinðj2pfn tÞg  sðt  ðm  1Þtc Þ

yðtÞ ¼ A exp jhp

"

M X m¼1

aNk;m ¼ ak;m ;

N X

ð12Þ #

!

an;m expðj2pfn tÞ sðt  ðm  1Þtc Þ

n¼1

k ¼ 1; 2; . . . ; N=2  1

ð13Þ

and the following equations for CE-OFDM-FM: yðtÞ ¼ A exp j2phf

Z

t

M X N X

fR½an;m cosðj2pfn tÞ

0 m¼1n¼1

  I½an;m sinðj2pfn tÞg  sðt  ðm  1Þtc Þ dt

yðtÞ ¼ A exp j2phf

aNk;m ¼ ak;m ;

Z

t

M X

0 m¼1

"

N X

ð14Þ #

!

1001

SNR should be greater than the specified value (10 dB), which is in contrast to using the long code to decrease the radar peak power. The second approach which is much better for use in radar systems is shown in Fig. 4. This method is based on using a matched filter that is matched to the transmitted CE-OFDM signal for pulse compression. This method will not confront the problem of threshold effect, but the main problem is to select proper codes for CE-OFDM signals with a possibly appropriate ambiguity function. We have studied this problem and introduced some suitable codes for CE-OFDM radar signals with favorable autocorrelation functions. The designed CE-OFDM signals are compared to OFDM signals which have been introduced in literature. It is noted that there is another degree of freedom in designing CE-OFDM compared to OFDM signal design and that is the modulation index. It can be shown that by increasing the modulation index we can design a CE-OFDM radar signal with very low autocorrelation sidelobes, but this increases the signal bandwidth.

an;m expðj2pfn tÞ sðt  ðm  1Þtc Þ dt

n¼1

k ¼ 1; 2; . . . ; N=2  1

ð15Þ

4. Compression processing schemes for CE-OFDM signals There are two processing approaches at the receiver of a CEOFDM system. The first approach that is shown in Fig. 3 is the same as the conventional method in CE-OFDM communication systems. In this scheme, first, the phase demodulator (which is at the front-end of the receiver or in the signal processing unit) converts the CE-OFDM signal into OFDM, and then OFDM signal processing is applied. Using this scheme in radar, the ambiguity function is the same as that of the real OFDM signal used in the transmitter. So the problem of a proper code selection can be referred to OFDM radar signal design. On the other hand, as it has been shown in [18], the OFDM signal processing or pulse compression can also be implemented as shown in Fig. 3(b), which has lower computational complexity compared to the conventional method of the OFDM pulse compression method and is suitable for use in dual mode radar/communication systems. But the major drawback of this method that makes it useless in radar systems is the threshold effect in which any angle modulation is confronted [19]. Because of this effect the input

5. CE-OFDM radar signal design In Section 3 we introduced four methods for converting an OFDM signal to a CE-OFDM signal based on Eqs. (12)–(15). Designing suitable CE-OFDM signals based on each method needs selection of the proper OFDM code ðan;m Þ and the modulation index (hp or hf ). This section presents two algorithms for designing CE-OFDM signals based on two different approaches for the selection of proper OFDM codes. In both of these methods, after selection of the OFDM codes, the modulation index could be selected according to the bandwidth constraints. It has been shown via simulation that increasing the modulation index will reduce the sidelobe levels of the ambiguity function but it also increases the bandwidth of the resulting CE-OFDM signal. The effect of modulation index on the spectral properties of a CEOFDM signal has also been studied in [11,13].

Fig. 4. The CE-OFDM matched processing.

Fig. 3. (a) First method of CE-OFDM signal processing. (b) Using efficient compression method of OFDM signal in (a) [18].

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5.1. CE-OFDM radar signals based on complementary sets In this section we use the results of [1,2] for selecting proper codes to modulate each sub-carrier. It has been found that lower autocorrelation sidelobes for OFDM signals are obtained when the N sequences are different from each other and constitute a complementary set. A complementary set can be constructed using Popovic’s result which states that ‘‘all the different cyclic time shifted versions of any sequence having an ideal periodic autocorrelation function form a complementary set.’’ Useful polyphase code sequences that exhibit ideal periodic autocorrelaTable 1 Matrix of phase values of all cyclic shift of P4 code of length 5 in degree [1]. Seq.

an;m ¼ expðjfn;m Þ

ð16Þ

where fn;m is the n th phase element of m th code sequence. Phase elements of the P4 code are described by

fm ¼

p M

ðm  1Þ2  pðm  1Þ;

m ¼ 1; 2; . . . ; M

ð17Þ

and for the P3 code where M is even, they are described as

fm ¼

Bit

1 2 3 4 5

tion functions are the P3 and P4 signals [3]. Useful two-phase code sequences with ideal periodic autocorrelation were suggested by Golomb [1,2]. Binary phase sequences (using 03 and 1803 ) with ideal periodic autocorrelations are rare. Barker code of length 4 is an example. We first consider a complementary set based on P3 and P4 signals. Thus modulating sequences, an;m are

1

2

3

4

5

0  144  216  216  144

 144 0  144  216  216

 216  144 0  144  216

 216  216  144 0  144

 144  216  216  144 0

p M

ðm  1Þ2 ;

m ¼ 1; 2; . . . ; M

ð18Þ

For the P4 code and M ¼ 5, the phase (modulo 3603 ) matrix of all the cyclic shifts is given in Table 1 (the first row is P4 code of length 5 and other rows are its cyclic shifts) [1]. The set of complex elements with a uniform magnitude, whose phase is described by Table 1, will remain a complementary set for any reordering of the rows [1]. When one particular order is used as fn;m in Eq. (16) and the conjugate symmetry property of

|χ (τ,ν)|

1

0.5

0 5

8 ν*

Normalized Amplitude

1

6 c

0.6 0.4 0.2

0

4

Mt

0.8

τ /t c

2 0 −5

0 −5

0

5

τ / tc

0

1 0.9

CE−OFDM single carrier

−10 Magnitude (dB)

0.8

|χ (0,ν)|

0.7 0.6 0.5 0.4

−20 −30 −40

0.3

−50

0.2

−60

0.1

−70

0 0

2

4

6 ν *Mtc

8

10

−π

0 Normalized frequency

π

Fig. 5. CE-OFDM-PM signal based on Table 1 and hp ¼ 1:65. (a) Ambiguity function. (b) Autocorrelation function. (c) Cut along Doppler axis. (d) Power spectral density.

¨ ) 64 (2010) 999–1008 R. Mohseni et al. / Int. J. Electron. Commun. (AEU

Eq. (11) is applied to modulate five code sequences on 10 subcarriers, the resulting CE-OFDM signal has the ambiguity function of Fig. 5(a), which shows that this signal exhibits a thumbtack ambiguity function. In this CE-OFDM signal the PM modulation with a modulation index of 1.65 is used. The autocorrelation function of this signal or a cut in the delay axis is also depicted in Fig. 5(b) and the cut in the Doppler axis is shown in Fig. 5(c). The peak sidelobe level (PSL) of the autocorrelation function of this signal is 17:7 dB. Power spectral density (PSD) of this signal is also depicted in Fig. 5(d) and compared with the PSD of a single carrier coded signal with the same range resolution. This figure shows that the spectrum efficiency of this signal is much better than that of the single carrier signals. In the above mentioned CE-OFDM signal, if the FM modulation with a modulation index of 32 MHz is used, the ambiguity and autocorrelation function would be as in Fig. 6. The peak sidelobe level of the autocorrelation function of this signal is PSL ¼  15 dB.

5.2. CE-OFDM radar signals based on similarity property In this section we use another method to construct an OFDM signal and then convert it to a CE-OFDM signal by using proper modulation index. The proposed method for designing an OFDM signal is based on using a proper single carrier coded

|χ (τ, ν)|

1

0.5

0 5

8 6 ν* 4 Mt c

1003

signal and then extracting its most similar multicarrier OFDM coded signal. Sampling the M  N OFDM signal with sampling rate fs ¼ N=tc or at time instants tk ¼ kt c =N; k ¼ 0; 1; 2; . . . leads to     M X N X ktc k ¼ x an;m exp j2p ðn  1Þ s½n  ðm  1ÞN N N m¼1n¼1 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

ð19Þ

IDFT

where window function s½k ¼ sðkt c =NÞ is given by  1; 0 rk r N  1 s½k ¼ 0 else

ð20Þ

So output samples at each time slot (chip width) are N point IDFT of a code sequence that modulates all subcarriers. In general, with sampling rate fs ¼ NL=tc , to have N  L sample of the M  N OFDM signal at each chip width, it is necessary to take the IDFT of length N  L from the vector containing N code on all subcarriers and arrange these samples according to Fig. 7. If L ¼ 1, then the relation between the time domain samples and the subcarrier modulating codes is one to one. This means that for given time domain samples of length M  N (may be samples of a single carrier coded signal) we have an M  N OFDM sequence with the same time domain samples and vice versa. Based on the above discussion, to construct an M  N OFDM signal with proper ambiguity function, first we select a proper single carrier coded signal of length M  N such as Barker, P3, P4 or Huffman sequence and divide it to M subsequence of length N. Then, according to Fig. 8 the N point DFT of each subsequence is computed and N DFT outputs of each set are chosen as the subcarrier codes of the time interval related to that set. In this way an M  N OFDM code is produced whose time domain equivalent signal is the most similar to the single carrier code in the sense of least squares difference. This can be easily shown by the projection theorem for Fourier series [20,21]. In summary, if the selected single carried code is SðtÞ and its time domain samples are represented by Sk where   kT ; k ¼ 0; 1; 2; . . . ; M  N  1; T ¼ Mtc ð21Þ Sk ¼ S MN

0 τ /tc

2 0 −5

M chips

Normalized Amplitude

1

a1,1 a1,2

a1,M

a2,1 a2,2

a2,M

aN,1 aN,2

aN,M

0.8 N subcarrier

0.6

0.4

0.2

0 −5

IDFT 0

IDFT

IDFT

5

τ / tc Time domain samples Fig. 6. CE-OFDM-FM signal based on Table 1 and hf ¼ 32e6. (a) Ambiguity function. (b) Autocorrelation function.

Fig. 7. Relation between OFDM signal samples and modulated codes on subcarriers.

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1004

M chips a1,1 a1,2

a1,M

5

a2,1 a2,2

a2,M

0 −5 −10 dB

N subcarrier

−15 −20 −25

aN,1 aN,2

DFT

aN,M

DFT

−30 −35

DFT

−40 −8

−6

−4

−2

0 τ / tc

2

4

6

8

−8

−6

−4

−2

0

2

4

6

8

Proper code sequence 5

Fig. 8. Designing OFDM signal that is similar to a prescribed single-carrier coded sequence.

0 −5

The designed OFDM code sequence (nearest OFDM signal to SðtÞ) is obtained as follows: 

−10

n ¼ 1; 2; . . . ; N; m ¼ 1; 2; . . . ; M

dB

 rðn  1Þ ; an;m ¼ Sðm1ÞN þ r exp j2p N r¼0 N 1 X

ð22Þ

−15 −20 −25

As an example, we select P4 code sequence of length 64 as a reference and make an 8  8 OFDM signal which is similar to this single carrier signal based on the proposed method. Hence, the selected P4 sequence is divided into eight subsequences of length 8, the 8 point DFT of each subsequence is computed and 8 DFT outputs of each subsequence are chosen as the frequency codes of the related time interval. The autocorrelation function of the designed OFDM signal is depicted in Fig. 9(a). Also, as it has been shown in [2] we can use frequency weighting to reduce the sidelobes of the autocorrelation function. We use the general cosine window described by " #a 2pðn  12Þ wn ¼ a0 þ a1 cos ; n ¼ 1; . . . ; N ð23Þ N where a0 ¼ 0:53836, a1 ¼  0:46164, and a ¼ 0:55. The autocorrelation function of the designed OFDM signal after frequency weighting is shown in Fig. 9(b). As shown, the sidelobe level is reduced to 22 dB, which is 7 dB lower than the 8  8 OFDM signal designed in [2]. The amplitude and phase of the designed OFDM signal (after frequency weighting) are described by Tables 2 and 3, respectively. Fig. 10 shows the autocorrelation and ambiguity function of the CE-OFDM-FM signal obtained by FM modulating the designed 8  8 OFDM signal based on Eq. (9) with a modulation index of 35 MHz. The maximum sidelobe level of this signal is about 18 dB and its ambiguity function has very low sidelobes in the whole delay-Doppler plane. Although the above approach developed based on Fig. 8 can be used for designing an OFDM signal and then the designed OFDM signal can be converted to a CE-OFDM signal using one of Eqs. (12)– (15), by selecting a proper modulation index, we have modified the proposed method to design CE-OFDM signals directly. The modified approach is developed based on Fig. 11. After selecting a proper single carrier coded signal, first phase or frequency

−30 −35 −40 τ / tc Fig. 9. OFDM signal designed using similarity property and based on P4 sequence of length 64. (a) Autocorrelation function. (b) Autocorrelation function after frequency weighting.

Table 2 Matrix of amplitude values of 8  8 OFDM signal based on P4 signal. Freq.

1 2 3 4 5 6 7 8

Bit 1

2

3

4

5

6

7

8

0.54 0.77 1.13 2.01 5.57 4.04 1.37 0.64

0.64 0.76 0.93 1.23 2.01 5.12 3.33 0.98

0.98 0.9 0.92 1.02 1.23 1.84 4.21 2.38

2.38 1.37 1.09 1.01 1.02 1.13 1.52 3.02

3.02 3.33 1.66 1.19 1.01 0.93 0.93 1.09

1.09 4.21 4.04 1.81 1.19 0.92 0.77 0.67

0.67 1.52 5.12 4.4 1.81 1.09 0.76 0.55

0.55 0.93 1.84 5.57 4.4 1.66 0.9 0.54

demodulator is used to demodulate this signal (by selecting proper modulation index) and this demodulated signal is then used as a reference for designing the most similar multicarrier OFDM signal according to the above method. Based on this approach the designed OFDM signal is similar to the phase or frequency of the selected single carrier coded signal. Thus, after modulation (PM or

¨ ) 64 (2010) 999–1008 R. Mohseni et al. / Int. J. Electron. Commun. (AEU

M chips

Table 3 Matrix of phase values of 8  8 OFDM signal based on P4 signal in degree. Freq.

Bit

1 2 3 4 5 6 7 8

1005

1

2

3

4

5

6

7

8

 11 11 34 54 44  116  63  35

144 168  168  145  125  135 63 116

 63  35  11 11 34 54 44  116

63 116 144 168  168  145  125  135

44  116  63  35  11 11 34 54

 125  135 63 116 144 168  168  145

34 54 44  116  63  35  11 11

 168  145  125  135 63 116 144 168

a1,1 a1,2

a1,M

a2,1 a2,2

a2,M

aN,1 aN,2

aN,M

N subcarrier

DFT

DFT

DFT

|χ (τ, ν)|

1 Phase/frequency Demodulator 0.5

0 Proper code sequence of length MN 8

8 6 ν *M

tc

0 τ /tc

4 2 0

Fig. 11. Designing CE-OFDM signal that is similar to a prescribed single-carrier coded sequence.

Then the modulated code on each subcarrier is obtained by finding the nearest OFDM signal to S0 ðtÞ) as follows:

−8

an;m ¼

1

  rðn  1Þ 0 Sðm1ÞN n ¼ 1; 2; . . . ; N; þ r exp j2p N r¼0 N 1 X

m ¼ 1; 2; . . . ; M

Normalized Amplitude

ð25Þ 0.8

As an example we select the P3 code sequence of length 64 as a reference and make a CE-OFDM-PM signal which is similar to it (with hp ¼ 5). The designed signal has autocorrelation and ambiguity functions as shown in Fig. 12. This approach can be used to convert any code sequence of good autocorrelation function to a CE-OFDM radar signal. As it was shown in this section, the proposed CE-OFDM signals are appropriate pulse compression techniques for radar applications and lead to near ideal ambiguity functions.

0.6

0.4

0.2

0 −8

−6

−4

−2

0 τ / tc

2

4

6

8

6. Comparison between CE-OFDM and other signals Fig. 10. CE-OFDM-FM signal based on Tables 2 and 3, and hf ¼ 35e6. (a) Ambiguity function. (b) Autocorrelation function.

FM) the resulting CE-OFDM signal is similar to the reference single carrier coded signal. Assume that the selected single carried coded signal is SðtÞ with time domain samples of Sk as in Eq. (21), and we want to generate its nearest CE-OFDM-PM signal. Based on the above discussion, first the normalized phase demodulated stream of this signal should be extracted: Sk0 ¼

1

angleðSk Þ

fD maxðangleðSk ÞÞ

ð24Þ

Our designed CE-OFDM signals have very good ambiguity functions. It will be interesting to compare their properties with those of other signals of the same range resolution or bandwidth. In this section we compare the designed CE-OFDM signals with multicarrier OFDM and P4 polyphase single-carrier phase coded signals in the case of their ambiguity function (autocorrelation function), PAPR, and bandwidth efficiency. The ambiguity function demonstrates the expected resolutions and sidelobe levels in both delay and Doppler domain and also defines the Doppler sensitivity. The PAPR illustrate the amplitude variations (effects on transmitter power efficiency) and spectral efficiency shows the amount of out of band radiated power, which is very important when many similar radar units must coexist in physical proximity, e.g. in automotive radar applications.

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1006

Normalized Amplitude

1

|χ (τ, ν)|

0.8 0.6 0.4

1 0.8 0.6 0.4 0.2 0 5

8

0.2

6 ν* M

tc

0 −8

−6

−4

−2

0 τ / tc

2

4

6

0 τ /tc

4 2

8

0 −5 Fig. 13. Ambiguity function of a 5  5 OFDM signal designed by Levanon [1].

1

0.5 |χ (τ, ν)|

|χ (τ, ν)|

1

0

0.5

8

8 6 Mt

ν*

c

0

4

τ /tc

2 0 −8

Fig. 12. CE-OFDM-PM signal based on P3 signal, and hp ¼ 5. (a) Autocorrelation function. (b) Ambiguity function.

6.1. Comparison with OFDM signals It has been shown that a well-designed OFDM signal exhibits thumbtack ambiguity function with low sidelobe levels in both range and Doppler and also high Doppler sensitivity [1–4]. This is the same as that of the designed CE-OFDM signals. As an example, the ambiguity function of a 5  5 OFDM signal designed by Levanon [1] is depicted in Fig. 13. One of the main differences between an OFDM and a CE-OFDM signal is that a CE-OFDM signal exhibits a constant amplitude, while a multicarrier OFDM signal has a varying amplitude. If the maximum allowable transmit power is limited by the saturation level of the power amplifier, an OFDM radar system must have a back off to accommodate the PAPR of the transmitted waveform. In comparison, constant envelope systems do not require any back off. Thus, the constant envelope systems have the capability to transmit roughly PAPR dB more energy for a given power amplifier than an OFDM system utilizing an identical power amplifier. This is referred to as power gain in the literature [17]. For example, PAPR of the 5  5 OFDM signal designed by Levanon [1] is 4.7. This means that a CE-OFDM signal has approximately 6 dB power gain relative to this OFDM signal. It has also been shown that CE-OFDM is not as spectrally efficient as OFDM. CE-OFDM with a high value of modulation index results in a Gaussian shaped spectrum similar to wideband FM [15]. The spectral property of this signal is studied in [13].

0 10 8

20 ν*

6

10

Mt

c

4 2 0

−20

0 −10 τ /t c

Fig. 14. Ambiguity function of a 25 element P4 signal.

Compared to OFDM, CE-OFDM is shown to have a slightly broader spectral mainlobe but faster frequency roll off at higher frequency.

6.2. Comparison with P4 single-carrier phase coded signal Fig. 14 shows the ambiguity function for the 25 element P4 pulse ðN ¼ 25Þ. Compared with Figs. 5, 6, 10, and 12 the ambiguity function of the CE-OFDM pulse does not exhibit the ridge seen in the ambiguity function of the P4 pulse (also typical of LFM). Zooming on the ambiguity function of CE-OFDM signals will reveal that there is no rapid increase of the sidelobe level with a small Doppler shift. Also, cut in zero delay of the CE-OFDM ambiguity function which is depicted in Fig. 5(c) demonstrates a ‘‘Sinc’’ shape with a narrow main lobe and relatively low sidelobes. This means that the Doppler sensitivity of CE-OFDM (and OFDM) signals is much higher than the P4 signal and inversely the P4 signal is much more Doppler tolerant than CEOFDM signals. When the target Doppler is unknown, using highly Doppler-tolerant waveforms allows for simplified receiver hardware with negligible degradation in detection performance, but if we want to estimate the target Doppler, highly Doppler sensitive signals such as CE-OFDM with a bank of matched filters should be used.

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In the case of PAPR, both P4 and CE-OFDM signals have a constant envelope and hence 0 dB PAPR. From a spectral-width point of view, it can be shown that CEOFDM signals exhibit a more efficient spectral usage compared with the P4 signal. The power spectral density of P4 resembles a ‘‘Sinc’’ function. The first null is at f ¼ 1=tc and the spectrum peak sidelobe level is 13 dB. Such a long-tailed spectrum may violate spectral emission regulations, can cause interference to neighboring radars, and may be too wide for the next radio-frequency (RF) stages-the antenna. But for CE-OFDM pulse as depicted in Fig. 5(d) the spectrum peak sidelobe level is much lower. One of the advantages of OFDM and CE-OFDM signals compared with single carrier signals is the capability of using in dual use radar/communication systems [11]. These systems have many practical applications such as in synthetic aperture radar (SAR) [8] (to relay radar-obtained imagery to ground terminals with a low-cost mechanism for real-time analysis), aeronautical telemetry (to provide command guidance and a means for monitoring vehicle status), and netted radar systems (exchange each radar detection information with the other nodes, allowing for powerful joint target detection and tracking algorithms) [11]. Also, it has been shown that multicarrier modulation based radar systems perform significantly better than single carrier modulation based radar systems for target detection [22]. This could be another important advantage of CE-OFDM as a multicarrier signal compared with a P4 signal.

7. Conclusion In this paper constant envelope OFDM signals have been considered to alleviate undesirable effects due to high PAPR of OFDM radar signals. Two different approaches have been presented to design a CE-OFDM radar signal with favorable ambiguity and autocorrelation functions. The first method is based on modulating a complementary set of sequences on different sub-carriers while the second method is based on using a simple mathematical operation to convert a given single-carrier coded signal to its most similar OFDM or CE-OFDM signal. This method can be used to convert all single-carrier coded signals such as Barker, Frank, or Huffman coded signals to a multicarrier OFDM or CE-OFDM coded signal with a proper ambiguity function. It has been shown that the designed signals have favorable characteristics as a radar signal. Comparison with OFDM and P4 signals show that the designed CE-OFDM signals have several advantages. As a P4 signal, CE-OFDM is a digitally phase modulated signal with a fixed envelope, but unlike P4, CE-OFDM exhibits thumbtack ambiguity function and uses the spectrum more efficiently. Like OFDM, it is a multicarrier signal capable of using a dual use radar/communication system and also improves the target detection performance, but unlike OFDM, it has a constant envelope and could be transmitted efficiently. References [1] Levanon N. Multifrequency complementary phase-coded radar signal. IEE Proc Radar Sonar Navig 2000;147:276–84. [2] Levanon N, Mozeson E. Multicarrier radar signal-pulse train and cw. IEEE Trans AES 2002;38:707–20. [3] Levanon N, Mozeson E. Radar signals. New York: Wiley; 2004. [4] Mozeson E, Levanon N. Multicarrier radar signals with low peak-to-mean envelope ratio. IEE Proc Radar Sonar Navig 2003;150:71–7. [5] Franken G, Nikookar H, Gendern PV. Doppler tolerance of ofdm-coded radar signals. In: Proceedings of European radar conference, September 2006. p. 108–11.

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[6] Singh SP, Rao K. Pulse train of multicarrier complementary phase coded radar signal for favourable autocorrelation and ambiguity function. In: Proceedings of CIE international conference on radar, India, January 2005. p. 76–80. [7] Ruggiano M, Genderen P. Wideband ambiguity function and optimized coded radar signals. In: Proceedings of the fourth European radar conference, Munich, Germany, October 2007. p. 142–5. [8] Garmatyuk D, Schuerger J, Morton YT, Binns K, Durbin M, Kimani J. Feasibility study of a multi-carrier dual-use imaging radar and communication system. In: Proceedings of the fourth European radar conference, Munich, Germany, October 2007. p. 194–7. [9] Mohseni R, Sheikhi A, Shirazi MM. Constant envelope ofdm signals for radar applications. In: Proceedings of IEEE radar conference 2008, Italy, May 2008. p. 453–7. [10] Stralka JP, Meyer GGL. Constant envelope orthogonal frequency-division multiplexing phase modulation for radar pulse compression. In: Proceedings of information sciences and systems conference (CISS), March 2007. p. 558. [11] Thompson S, Stralka JP. Constant envelope ofdm for power-efficient radar and data communications. In: Proceedings of fourth international waveform design and diversity (WDD) conference, Orlando, FL, February 2009. p. 291–5. [12] Stralka JP. Applications of orthogonal frequency-division multiplexing (ofdm) to radar. Dissertation, Department of Electrical and Computer Engineering, Johns Hopkins University, Baltimore; 2008. [13] Stralka JP, Meyer GGL. Constant envelope ofdm phase modulated radar waveforms. In: Adaptive sensor array processing workshop (ASAP 2007), MIT Lincoln Lab., Lexington, MA, June 2007. [14] Sebt M, Sheikhi A, Nayebi M. Orthogonal frequency-division multiplexing radar signal design with optimised ambiguity function and low peak-toaverage power ratio. IET Radar Sonar Navig 2009;3:122–32. [15] Thompson S. Constant-envelope ofdm phase modulation. Dissertation, University of California, San Diego; 2005. [16] Thompson S, Proakis J, Zeidler J. Noncoherent reception of constant envelope ofdm in flat fading channels. In: Proceedings of IEEE 16th international symposium on personal, indoor and mobile radio communications, PIMRC 2005, vol. 1, September 2005. p. 517–52. [17] Thompson S, Proakis J, Zeidler J. Constant-envelope binary ofdm phase modulation. In: Proceedings of IEEE MILCOM 2003, October 2003. p. 621–6. [18] Mohseni R, Sheikhi A, Shirazi MM. Compression of multicarrier phase-coded radar signals based on discrete fourier transform (dft). Prog Electromagn Res C 2008;5:93–117. [19] Carlson A, Crilly P, Rutledge J. Communication systems. New York: McGrawHill; 2002. [20] Sebt M. Code design in multi-frequency radars based on ofdm signal. Diploma thesis, Sharif University of Technology, Tehran, Iran, September 2007 [in Farsi]. [21] Browder A. Mathematical analysis: an introduction. Berlin: Springer; 2001. [22] Prasad N, Shameem V, Desai U, Merchant S. Improvement in target detection performance of pulse coded doppler radar based on multicarrier modulation with fast fourier transform (fft). IEE Proc Radar Sonar Navig February 2004;151:11–7.

Reza Mohseni was born in Jahrom, Iran, in 1980. He studied electrical engineering at the Shiraz University, Shiraz, Iran, and he got his M.Sc. and Ph.D. in communications engineering from the same university in 2004 and 2009, respectively. He is currently with Shiraz University of Technology, Shiraz, Iran. His current research interest lies in the field of radar signal processing and signal design.

Abbas Sheikhi was born in Shiraz, Iran, in 1971. He received B.Sc. degree in 1993, from Shiraz University in electronics engineering, Shiraz, Iran, M.Sc. in 1995 in biomedical engineering and Ph.D. in 1999 in communications engineering both from Sharif University of Technology, Tehran, Iran, all with first class honors. Since 1999, he has been with the Electrical and Electronics Engineering Department, Shiraz University, where he is now an associated professor. He has published more than 80 papers in scientific journals and conferences. His current research interest lies in the field of statistical signal processing in radar systems.

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Mohammad Ali Masnadi-Shirazi was born in Shiraz, Iran, in 1955. He received the B.S. and M.S. degrees in electrical engineering from Shiraz University, Shiraz, Iran, in 1974 and 1984, respectively, and the Ph.D. degree in electrical and computer engineering from the University of New Mexico, Albuquerque, in 1990. He was teaching at Shiraz College of Electronics from 1974 to 1984, and a teaching assistant in the Department of Electrical and Computer Engineering at the University of New Mexico from 1986 to 1990. He was a postdoctoral researcher at Scrips Institution of Oceanography, University of California, San Diego, from 1990 to 1992. In 1992 he joined the Department of

Electrical Engineering at Shiraz University where he is now a Professor. During the years 2001–2003 he was a visiting professor in the Department of Electrical Engineering at the University of Texas, Arlington. His research interests are in the areas of systems and statistical signal processing with applications to optimal filtering, communications, radar, sonar and speech data.