Almost perfect sequences applied for ionospheric oblique backscattering detection



[1] Pseudorandom sequences are often used in radio systems; however, the nonzero out-of-phase autocorrelation of many binary sequences induces range sidelobes which significantly reduce the echo signal-to-noise ratio (SNR). In this paper, the use of almost perfect sequences, exhibiting zero out-of-phase autocorrelation except one value in the middle is examined with reference to common m sequences and perfect sequence. The ambiguity functions demonstrate that it is possible to use the almost perfect sequences for ranging without sidelobes and that their Doppler measurement performance is similar to m sequence of the same length. This is an important result for ionospheric oblique backscattering detection where the echoes are superposed and where range sidelobes can submerge the main lobes of weak signals. The 124-bit almost perfect sequence and the 127-bit m sequence are applied to the Wuhan Ionospheric Oblique Backscattering Sounding System for sequence testing. The test results have proven that the almost perfect sequence exhibits a higher echo SNR for the same detection conditions.

1. Introduction

[2] Phase coding techniques include biphase and polyphase coding. They use digital methods to develop a compressed pulse having low sidelobes and permit to reduce the peak transmitting power of the ionospheric sounders and radar systems [Bibl and Reinisch, 1978; Reinisch, 1986; Huang and MacDougall, 2005]. Compared with biphase coding, polyphase coding produces lower sidelobe levels, but it is more sensitive to Doppler and its coding and decoding systems are more complex [Golomb and Scholtz, 1965; Somaini and Ackroyd, 1974]. Therefore, the practical biphase coding has been widely applied to radio systems.

[3] In binary phase coding, the phase of any pulse has one of two possible values. The values 0 and π are popular choices that have the widest possible separation (since phase is modulo 2π) and generally lead to the best performance. In order to get a unique echo after pulse compression, one requires the phase coding sequence having three perfect randomness properties, including perfect balance property, run property and autocorrelation property [Golomb, 1982], thereinto, the autocorrelation property is the most important and determines the sequence performance. The poor autocorrelation property would produce the sidelobes, which raise the noise level and induce the false echoes.

[4] The binary pseudorandom sequences with zero out-of-phase autocorrelation coefficients are perfectly suited for phase coding radio system and we call them perfect sequences [Golomb, 1992; Freedman et al., 1995; Jungnickel and Pott, 1999]. The detection signals coded by perfect sequences have no extended range sidelobes theoretically. A perfect sequence of period n, written as a row vector B = [b0, b1, …bi …, bn−1], bi ∈(− 1, + 1), has the autocorrelation function shown as follows:

equation image

In this expression, E denotes the energy of vector B and bi+τ is the shifted vector of bi. However, in the case of (−1, +1) sequences of period n, it has been proven that there is no perfect sequence with period 4 < n ⩽ 12100 [Baumert, 1971]. However, detection without range sidelobes by binary coding is still possible and the complementary code is a very good solution [Golay, 1961].

[5] In this paper we describe another biphase coding technique (almost perfect sequence) to eliminate range sidelobes. Almost perfect sequence exhibiting zero out-of-phase autocorrelation except one value in the middle could replace perfect sequence to sound without sidelobes and so the problem that the superposed sidelobes submerge the mainlobes of the weak echoes in ionospheric oblique backscattering detection can be solved. We give an algebraic explanation of almost perfect sequences and then compare them with perfect sequence and m sequences by ambiguity function and real detection. The Wuhan ionospheric oblique backscattering sounding system (WIOBSS) developed by Ionospheric Laboratory is used to test the sequences [Chen et al., 2007, 2009].

2. Almost Perfect Sequences

[6] Earlier, long almost perfect sequences were put forward as complex periodic sequences such that all out-of-phase autocorrelation coefficients are zero except one [Wolfmann, 1992; Pott and Bradley, 1995]. Let (s0, s1, ⋯, sn−1) be an almost perfect sequence, that is, si+n = si for every i∈(1, n). The three valued autocorrelation function of the sequence is shown as follows:

equation image

where period n is a multiple of 4, R(0) is the in-phase autocorrelation coefficient and R(τ) (τ ≠ 0(mod n)) is the out-of-phase autocorrelation coefficient.

[7] Unlike m sequences produced expediently and promptly by employing a shift register with feedback, almost perfect sequences are hunted with an exhaustive computer search by two “miracle configurations” stemming from the perfect cyclic difference sets [Wolfmann, 1992; Wah, 1998].

equation image
equation image

Equation (3) is the characteristic formulas and equation (4) shows that the symbols in the left half of almost perfect sequences are complementary to the symbols in the right half, respectively, except one pair. Besides sn/2 = sn = 0, another symmetry can be observed for canonical almost perfect sequences:

equation image

All symbols of a canonical almost perfect sequence can be directly determined by equations (3), (4), and (5). Since not all the nonlinear equations yield a binary solution, some periods n (multiple of 4) do not have almost perfect sequence. Some typical almost perfect sequences and the binary perfect sequence of period 4 hunted by us are listed in Table 1.

Table 1. Some Almost Perfect Sequences
PeriodAlmost Perfect Sequences in Hexadecimal Form

3. Sequence Analysis

3.1. Ambiguity Function

[8] The autocorrelation function is generally used to examine the range information, or evaluate the peak sidelobes level (PSL) of a sequence. However, the correlation function only describes the amplitude similarity. The Doppler-shifted echoes are complex signals and described as the envelope signals u(t) exp(−j2πfdt). In order to joint delay and Doppler estimation, the natural generalization of the correlation process is the complex ambiguity function.

[9] Ambiguity function describes the response of a particular delay-Doppler resolution cell of radar to the echo of a point target, as the delay and Doppler of the echo vary. Radar performance in terms of capability to resolve target and clutter scatterance in delay and Doppler dimensions can be assessed by direct examination of the ambiguity function surface in the delay-Doppler ambiguity plane. Echo signal-to-clutter power ratio can be calculated for specified radar and target geometries by integrating the product of ambiguity function and the clutter and echo power distributions over all delay and Doppler domain where echoes or clutter are present. Therefore the ambiguity function is generally used to design radar waveform. The most obvious property of ambiguity function is that when variable fd = 0, we obtain a crosscut of ambiguity function along the time axis, which is the delay autocorrelation function; when τ = 0, we obtain a crosscut along the zero time delay axis, which is the Doppler autocorrelation function.

[10] Ambiguity function, originally put forward for radar applications by Woodward [1980], has been generalized for the wideband waveforms by various works. The expression of ambiguity function is shown as follows,

equation image

In this expression, u(t) and u*(tτ) are complex envelopes of the examined waveform, variables τ and fd are the time lag and frequency offset parameters, respectively, to be searched simultaneously for the values that cause A(τ, fd) to peak.

[11] The envelop expression of a pulse is defined by

equation image

where Tp is the pulse width, and the expression of phase coded pulse trains can be written in this form,

equation image

where Ci is the element of (+1, −1) pseudorandom sequence, P is the sequence period, N is the number of consequently transmitted pulse trains, Tr1 is the pulse repetition period and Tr2 is the repetition period of pulse train. Substituting (8) into (6), we obtain the ambiguity function of phase coded pulse trains

equation image


equation image

and Ci+∣S is the shifted vector of Ci.

[12] Let fd = 0, the delay autocorrelation function can be obtained,

equation image


equation image

Let τ = 0 and S = Q = 0, the Doppler autocorrelation function in the central banding can be obtain,

equation image

where χ1(0, fd) = Sa(fd · Tp). Expression (11) shows that the Doppler autocorrelation function is independent of the sequence autocorrelation feature.

3.2. Sequence Comparison

[13] The three ambiguity plots shown in Figures 1a–1c come from the pulse trains modulated by the perfect sequence [+1, +1, −1, +1], the m sequence of period 7 and the almost perfect sequence of period 8, respectively. The coded pulse train with 1s pulse width and 20% duty cycle is continuously transmitted four times. We perform a section of the ambiguity plots along the zero Doppler axes to get the delay autocorrelation plots and along the zero delay axes to get the Doppler autocorrelation plots.

Figure 1.

Comparison of the ambiguity plots of (a) the perfect sequence of period 4, (b) the m sequence of period 7, and (c) the almost perfect sequence of period 8. (left) The ambiguity plot, (top right) the delay autocorrelation plot, and (bottom right) the Doppler autocorrelation plot.

Figure 1.


Figure 1.


[14] The physical volumes of the three ambiguity plots are distributed on many bandings parallel to the fd axis. Many peaks are arranged on the parallel bandings and the distribution of the peaks is determined by waveform parameters. The region between two bandings exhibits the delay detecting performance of the selected waveform. As shown in Figure 1a, the unambiguous delay detection range (UDLDR) of perfect sequence is equal to region width Tr2 − 2Tp and the delay resolution is equal to the half-power width Tp of the centrally located main peak. Compared with the ambiguity plot of m sequence in Figure 1b, perfect sequence with no autocorrelation sidelobes between the bandings has lower noise lever and will not induce the false echoes. However, as mentioned above, the sequence shown in Figure 1a is the only perfect sequence hunted nowadays. The almost perfect sequence has only one huge sidelobe in the middle of the region between the bandings as shown in Figure 1c. If the maximum delay detection range of a radio system is set less than (Tr2 − 2Tp)/2, almost perfect sequences can be used to detect without range sidelobes. Many Doppler ambiguity peaks are arranged in each banding with 2/NTr2 width and 1/Tr2 spacing. There are N − 2 small Doppler sidelobes between the Doppler ambiguity peaks, composing the clutter in the bandings. The Doppler detection performance is determined by the waveform parameters Tr2 and N, but not the phase coding sequences.

[15] How do the autocorrelation sidelobes reduce the detecting performance of a radio system? It will be illuminated by m sequences. Given the m sequence S:{xii = 1, 2, …, P}, which is applied for phase coding (where xi is the element of the sequence and P is the sequence period.), one period of the recorded baseband signal can be expressed as Ak · Sequation image, where Ak is the amplitude and τk is the phase shifting of the pulse train. When a period of the pulse train is received, the mainlobe amplitude output by the pulse compressor is PAk, the sidelobe level is Ak and the mainlobe-to-sidelobe ratio is P. When several echo pulse trains of the same or different delay are received simultaneously, the recorded baseband signal can be expressed as

equation image

where K is the echo number. The pulse compressor output

equation image
equation image

Expression (13) indicates the correlation coefficient of the superposed echo sequences. While τ = τm, m ∈[1, K], the mainlobe amplitude of the mth echo is

equation image

and the sidelobe level is

equation image

Given ξm = equation image, the mainlobe to sidelobe ratio is

equation image

When ηm ≤ 1, we get

equation image

The calculated result (18) indicates that when the multipath effect happens, several superposed echo pulse trains are recorded, the mainlobe of each compressed echo will decrease and the decreasing value is the sum of the other echo amplitude equation imageAkAm, while the sidelobe level increases of the same value. If the ratio of the echo amplitude to the sum of all echo amplitude is no more than 2/(P + 1) as show in expression (18), the echo mainlobe is submerged by the superposed sidelobes.

[16] In brief, the nonperfect autocorrelation of m sequences and other pseudorandom sequences will induce the superposed sidelobes by multipath to reduce the echo peaks and rise up the noise level and so the SNR falls greatly. In ionospheric oblique backscattering detection, the mainlobes of the weak backscattered echoes will be weaken and even submerged by the powerful sidelobes of the ground echoes and the vertical incident echoes. As shown in Figure 1c, there is only one huge sidelobe between two adjacent bandings, therefore, if the selected maximum detection delay is no more than (Tr2 − 2Tp)/2, almost perfect sequences can be used to detect without range sidelobes. The Doppler autocorrelation plots of the three sequences indicate that the Doppler measurement performance of binary pseudorandom sequences is interrelated to the pulse train period and the coherent integration time, but not to the code form.

4. Sequence Testing

4.1. Testing System

[17] WIOBSS is used to examine the almost perfect sequences. It is a HF sky wave over-the-horizon radar developed for ionospheric research and HF channel management. Its schematic block diagram is shown in Figure 2. The monostatic and software-controlled system using log-periodic antenna for both transmission and reception transmits the phase coded pulse trains with long coherent integration time for good sensitivity. In order to prevent receiver from saturation when transmitting, the high-speed and high-isolation T/R switches as well as some attenuators are used in the receiver. The transmitting and receiving channels are built on two VXI (VME extension for instrumentation) C size modules for the sake of favorable electromagnetic compatibility and maneuverability and they share one 40 MHz Oven-Controlled Crystal Oscillator (OCXO) as the reference frequency. The system controller is a Field-Programmable Gate Array (FPGA) circuit and control of the whole system is programmed from it. The controller stores the pseudorandom sequences, outputs the digital baseband signal to the Quadrature Digital Upconverter (QDU) to produce the phase coded waveform, controls the T/R switch using the transceiving time sequence, commands the receiver to record echoes at the operating frequency and so on. A Digital Signal Processor (DSP) is used as pulse compressor to process the recorded echoes and then transfer the compressed data to computer for further processing and storage. The Global Positioning System (GPS) is used to provide position information and calibrate the system clock.

Figure 2.

Schematic block diagram of WIOBSS.

[18] The coded pulse trains are transmitted with regular pulse interval. The receiver is continuous operating and several gain controllers within it are used to avoid self-interference. The inphase and quadrature components of the received echoes are sampled with 24 kHz sampling frequency. The sampled data are input to pulse compressor for channel impulse response (CIR). The continuously recorded CIRs compose bitemporal ionospheric response (BTIR) [Bello, 1963; Chong et al., 2000]. BTIR is converted into channel scattering function (CSF) in the frequency domain along the time axis by fast Fourier transform (FFT). The characters of the return, diffuse, multimode and multipath composition are also usually evident from an examination of CSF [Gaarder, 1968; Kay and Doyle, 2003].

4.2. Closed-Loop System Testing Observations

[19] In the closed-loop system experiment, the excitation signal from the QDU of WIOBSS is directly input into the receiver. This experiment has shielded the radio system from the external electromagnetic interference to make it easy to test the sequence ranging performance quantitatively and qualitatively. The m sequences of period 63 and 127 and the almost perfect sequences of period 64 and 124 are used to code the pulse trains with 83 μs pulse width, 20% duty cycle and 11.2 MHz operating frequency. The impulse responses of the four sequences after pulse compression are plotted in Figure 3 and some parameters obtained from the impulse responses are listed in Table 2.

Figure 3.

Impulse response of closed-loop system using (a) the 63-bit m sequence, (b) the 64-bit almost perfect sequence, (c) the 127-bit m sequence, and (d) the 124-bit almost perfect sequence.

Table 2. Comparison of Different Sequencesa
 63-Bit m Sequence64-Bit Almost Perfect Sequence127-Bit m Sequence124-Bit Almost Perfect Sequence
  • a

    Ambiguity function of perfect sequence and its sections, ambiguity function of m sequence and its sections, and ambiguity function of almost perfect sequence and its sections.

Pulse compression gain (dB)35.98736.12436.12441.868
Mainlobe amplitude (dB)114.256114.323120.264120.109
Maximum sidelobe amplitude (dB)78.61347.75078.65951.000
MSR(dB)35.643 41.605 
SNR (dB) 66.573 69.019

[20] Due to the very low noise level in the closed-loop system, the sidelobes of m sequences obviously distribute along the range bins as shown in Figures 3a and 3c, while the almost perfect sequences of period 64 and 124 have no sidelobes on the range bins as shown in Figures 3b and 3d. Therefore, it is obvious that the almost perfect sequences have much higher SNR. In Table 2, the pulse compression gain of the sequence of period P is defined as GPC = 20lg P. The mainlobe amplitude of the four sequences is proportional to the product of the pulse impression gain and the transmitting power as shown by expression (15). The two m sequences have the same powerful sidelobes, the level of which is determined by the transmitting power, but not the pulse compression gain as shown by expression (16). Therefore, the maximum SNR gained by the radio system using m sequence will not exceed the mainlobe-to-sidelobe ratio (MSR), which is defended as

equation image

where m ≠ 0; ρ0 is the mainlobe amplitude and ρm is the sidelobe level. The MSR of m sequence is approximately equal to the pulse compression gain as shown in Table 2. The two almost perfect sequences have no sidelobe on the shown bins obtained higher SNR.

[21] The biphase coded signal, including the pulse train modulated by almost perfect sequence, is Doppler sensitive. Figure 4 shows the Doppler effect of the pulse train coded by the almost perfect sequences of period 64, 88, 108, 124, 148, and 168. The pulse width and the duty cycle of the pulse train are 41.66 μs and 10%, respectively. Figure 4a indicates that the mainlobe amplitude, which is directly proportional to the sequence period, runs down slowly with the increase of the Doppler shift. The longer the sequence period is, the more obvious the downward trend is. Due to the Doppler sidelobes, the noise level climbs linearly and quickly in Figure 4b. The longer the sequence is, more quickly the noise level rises. Therefore, the SNR decreases greatly when the Doppler value increases. Figure 4c displays the measured SNR of the almost perfect sequences with different Doppler values and we find that the shorter sequence shows better detection performance with large Doppler shift. So when the long pseudorandom sequences are applied for the moving-target detection, it is necessary to compensate the Doppler shift to improve the measurement performance [Yuping et al., 2002; Yu and White, 2007].

Figure 4.

The Doppler effect on (a) mainlobe amplitude, (b) noise level, and (c) SNR, when different almost perfect sequences are used.

4.3. Backscatter Detection Observation

[22] The echo SNR in ionospheric backscatter detection is not only determined by the performance of the radio system, but also related to the ionospheric condition, the path attenuation, the ground scattering characteristic and so on. It is necessary to carefully design the experiment to compare the different sequences for the same conditions. The 124-bit almost perfect sequence and the 127-bit m sequence are programmed in WIOBSS and can be easily and rapidly selected for detection. The peak transmitting power is set on 200 W unchanged. The experiment was carried out at noon, 8 January 2007. At first the almost perfect sequence were used for detection and the coded pulse trains with 83 μs pulse width and 20% duty cycle were continuously transmitted 128 times. And then the m sequence was applied with the same waveform parameters. The two sequences were alternately used thrice within a minute and then three echo data of almost perfect sequence and three echo data of m sequence had been recorded. After pulse compression and FFT, the echo data are converted into the CSF with 12.5 km range resolution and 0.15 Hz Doppler resolution. The three CSFs of each sequence are averaged to weaken the ionospheric time-varying effect. The experiments were performed with 6.8 MHz, 10.4 MHz, and 11.228 MHz operating frequencies and the average CSF plots as well as their side views and sections of the three operating frequencies are displayed in Figure 5, 6, and 7.

Figure 5.

The echoes of 6.8 MHz operating frequency. (a) A side view of CSFs, (b) CSF of m sequence, (c) CSF of almost perfect sequence, (d) echo spectrums from 350 km, and (e) echo spectrums from 594 km. The data of m sequence are shown in blue lines, and the data of almost perfect sequence are shown in red lines.

Figure 6.

The echoes of 10.4 MHz operating frequency. (a) A side view of CSFs, (b) CSF of m sequence, (c) CSF of almost perfect sequence, (d) echo spectrums from 963 km, and (e) echo spectrums from 1031 km. The data of m sequence are shown in blue lines, and the data of almost perfect sequence are shown in red lines.

Figure 7.

The echoes of 11.228 MHz operating frequency. (a) A side view of CSFs, (b) CSF of m sequence, (c) CSF of almost perfect sequence, (d) echo spectrums from 1031 km, and (e) echo spectrums from 1100 km. The data of m sequence are shown in blue lines, and the data of almost perfect sequence are shown in red lines.

[23] Compared with the 127-bit m sequence, the pulse compression gain of the 124-bit almost perfect sequence is ΔGPC = 20log equation image = 0.21dB lower. Therefore, the compressed echo amplitude of the almost perfect sequence will not be higher than that of the m sequence. The echo SNR is also one of the most important parameters to weigh the detection performance of a radio system. The CSF plots of the recorded data are all displayed in SNR form. Figures 5b, 6b, and 7b displays the average CSFs of the 127-bit m sequence and Figures 5c, 6c, and 7c display that of the 124-bit almost perfect sequence. The recorded echo delay, delay spread, Doppler shift and spread shown in the two CSF plots are identical; therefore, it is possible to compare the performance of the two sequences effectively. The side view of the CSF is displayed in Figures 5a, 6a, and 7a; the data of m sequence are shown in blue lines and the data of almost perfect sequence are shown in red lines. The huge sidelobe of the 124-bit almost perfect sequence appears behind 3875 km, so the shown echoes within 1500 km are free of range sidelobes and the measured SNR is higher than that of the 127-bit m sequence. The echo spectrums of two chosen ranges are displayed in Figures 5d and 5e, Figures 6d and 6e, and Figures 7d and 7e. The Doppler peaks of the two sequences have the same frequency shift and spread and the peaks of the almost perfect sequence are higher.

5. Conclusion

[24] Almost perfect sequences are also a kind of binary pseudorandom sequence and meet the Golomb's three randomness postulates for binary sequences [Golomb, 1982]. Therefore, the sequences can be applied to the radio system as other pseudorandom sequences. The period of almost perfect sequences is an integral multiple of 4 and that of m sequences is 2n − 1, so more almost perfect sequences can be chosen in a limited band of periods. The most dramatic feature is that the sequences have zero out-of-phase autocorrelation except one value in the middle, making it possible to detect without range sidelobes as perfect sequence does. Due to ranging without sidelobes, the echo SNR could be enhanced to a certain degree. The examining results also show that almost perfect sequences have better ranging performance. However, unlike m sequences expediently produced by a shift register, extreme search by computer is the only way to obtain almost perfect sequences. Along with the linear increase of the period, the hunting time is elongated exponentially.


[25] This research is supported by the National Natural Science Foundation (40804042) and the Post Doctor Foundation of China (20070420919). The authors are grateful to Paul Cannon for a critical reading of the manuscript and for his suggested changes.