Good Code Sets from Complementary Pairs via Discrete Frequency Chips

It is shown that replacing the sinusoidal chip in Golay complementary code pairs by special classes of waveforms that satisfy two conditions, symmetry/anti-symmetry and quazi-orthogonality in the convolution sense, renders the complementary codes immune to frequency selective fading and also allows for concatenating them in time using one frequency band/channel. This results in a zero-sidelobe region around the mainlobe and an adjacent region of small cross-correlation sidelobes. The symmetry/anti-symmetry property results in the zero-sidelobe region on either side of the mainlobe, while quasi-orthogonality of the two chips keeps the adjacent region of cross-correlations small. Such codes are constructed using discrete frequency-coding waveforms (DFCW) based on linear frequency modulation (LFM) and piecewise LFM (PLFM) waveforms as chips for the complementary code pair, as they satisfy both the symmetry/anti-symmetry and quasi-orthogonality conditions. It is also shown that changing the slopes/chirp rates of the DFCW waveforms (based on LFM and PLFM waveforms) used as chips with the same complementary code pair results in good code sets with a zero-sidelobe region. It is also shown that a second good code set with a zero-sidelobe region could be constructed from the mates of the complementary code pair, while using the same DFCW waveforms as their chips. The cross-correlation between the two sets is shown to contain a zero-sidelobe region and an adjacent region of small cross-correlation sidelobes. Thus, the two sets are quasi-orthogonal and could be combined to form a good code set with twice the number of codes without affecting their cross-correlation properties. Or a better good code set with the same number codes could be constructed by choosing the best candidates form the two sets. Such code sets find utility in multiple input-multiple output (MIMO) radar applications.


Background
Good aperiodic codes are characterized by a narrow mainlobe and small sidelobes.Smaller autocorrelation peak sidelobes reduce the probability of false alarm, while a narrower mainlobe enhances the range resolution.Such properties are desirable in certain communications applications like preamble synchronization and in most radar applications.
Let [] be a code of length .Its aperiodic autocorrelation (ACF) can be represented by, In the -domain, if []  ⇔ () then, its ACF can be represented by,
where (  )  ⇔ () and   represents the number of samples in ().ACF of   () can be represented by,   () =   ()  ( −1 ) () = (   )()( −  )( −1 ) () = (   )  () () =   () + (   )  () where   () = ()( −1 ) represents the ACF of the highly sampled chip and () is given in Equation (2).It can be observed that the mainlobe and the sidelobes are spread by the ACF of the chip (  ()).Some of the well-known good aperiodic codes are the Barker codes, Frank codes [1] and waveforms based on Costas arrays [2][3][4].Barker codes are biphase codes that have unity magnitude peak sidelobes.However, the longest known Barker code of odd length is of length 13 which gives a peak sidelobe ratio of .Frank codes are polyphase codes obtained by concatenating the rows of a discrete Fourier transform (DFT) matrix.Thus, Frank codes are available for a variety of code lengths and achieve high peak sidelobe ratios for long code lengths.However, as the code length increases the number of distinct phases also increases.Costas arrays are completely classified to date up to order 27, and are known to exist of orders up to 200.They are widely used in radar and sonar applications because of their thumbtack ambiguity function.The codes presented in this paper are based on biphase complementary code pair that are available at a variety of code lengths.
One of the widely used approaches to suppress the ACF sidelobes of a waveform/code is via windowing whereby the peak sidelobe level is reduced, but at the cost of increasing the mainlobe width.Mismatched filters like the ones introduced in [5][6][7][8] could also be used to reduce ACF sidelobes at the cost of a small loss in signal-to-noise ratio.In [9], it is shown that complete sidelobe cancellation for a class of aperiodic codes is possible using additive-multiplicative processing of the matched filter (MF) output.However, the non-linear processing involved in these mismatched filters incurs additional computational cost and suffer from degraded performance in high noise environments.Although it is impossible for individual aperiodic codes to have zero sidelobes, Golay complementary code pair [10], polyphase complementary codes [11] and complementary code sequences [12] achieve complete sidelobe cancellation via addition of their autocorrelations.Figure 1 shows the transmission and reception of a Golay complementary code pair (say  [𝑛] and [], of length   ).[] is transmitted at frequency  1 (using  1 () =  2 1  ) and [] at  2 (using  2 () =  2 2  ).At the receiver, the code pair are received in their own matched filters.Addition of their MF outputs (  () and   ()) results in complete cancellation of the sidelobes.However, frequency selective fading results in unequal attenuation of the two codes.This results in reemerging sidelobes due to inexact cancellation of the non-zero sidelobes of the code pair, as shown in Figure 2 for   = 8.Since the sidelobes of the code pair are not small, any inexact cancellation results in reemerging sidelobes which is highly undesirable.In Figure 2,   (),   () and () represent the fading absent case and   ′(),   ′() and ′() represent the fading present case.For the fading present case, () and () are attenuated by 0.95 and 0.85 respectively.Galati et al. [13] also discusses in detail this effect of unequal attenuation on the complementary code pairs.Liu et al. [14] describes z-complementary code pairs that achieve a zero-correlation zone around the mainlobe and minimum possible sidelobe peaks outsize the zero domain.However, these codes also require two frequencies or channels which makes them vulnerable to the aforementioned effects of fading.Tang et al. [15] and Li [16] describe (loosely synchronized) LS and (large area) LA codes for (quasi-synchronous code division multiple access) QS-CDMA systems.These codes also achieve a zero interference zone in both the autocorrelation and the cross-correlations, but the peak sidelobes outside the zero-sidelobe region are not small.In this paper, we introduce codes constructed from biphase Golay complementary code pairs that are available at a variety of lengths.These codes have a zero-sidelobe domain on either side of the mainlobe, while the sidelobe peaks outside the zero domain are very small.Since these codes use the same frequency band, they are not affected by the effects of fading.This concatenated code has a region of zero-sidelobes on either side of the mainlobe and an adjacent region of small cross-correlations.Due to the symmetry/anti-symmetry property, the two chips have identical ACF.This results in exact sidelobe cancellation of the complementary code pair, creating a zero-sidelobe region on either side of the mainlobe.The cross-correlation peak () between () and (−) is represented by, Since () and (−) are quazi-orthogonal in the convolution,  is small compared to the mainlobe peak 2  .This property makes the adjacent region of cross-correlation sidelobes small.Discrete frequency-coding waveforms (DFCW) based on linear frequency modulation (LFM, also known as a chirp signal) and piecewise LFM (PLFM) [17] waveforms satisfy both the symmetry/anti-symmetry and quasi-orthogonality conditions.Hence, they are used as chips for the complementary code pair.A discrete frequency-coding waveform, say (), is the sum of  contiguous sub-pulses of the same duration Δ, but not necessarily the same frequency.
In this paper, it is also shown that a quasi-orthogonal set of symmetrical/anti-symmetrical DFCW waveforms could be used as chips with the same complementary code pair to result in a good code set with a zero-sidelobe region.Several good code sets are constructed by changing the slopes/chirp rates of the DFCW chips based on LFM and PLFM waveforms.The code sets constructed with DFCW chips based on LFM waveforms occupy different bandwidths but have smaller peak cross-correlations, while the code sets constructed with DFCW chips based on PLFM waveforms occupy the same bandwidth but have slightly larger peak cross-correlations.Thus, two good code sets could be constructed from the complementary code pair and their mates using the same DFCW waveforms as chips.Since these two good code sets will be quasi-orthogonal to each other, they could be combined to form a larger good code set with double the number of codes in the set without affecting the cross-correlation properties.Instead of using all the codes in the two sets, it is shown that a better good code set with significantly reduced cross-correlation peaks could be constructed by choosing the best candidates form the two sets.

Previous Research
In [25,26], continuous frequency LFM and PLFM waveforms were used as chips for [] and  [𝑛].The current manuscript uses the same code structure as in [25,26], but DFCW waveforms based on LFM and PLFM waveforms are used as chips for the complementary code pair.These codes find utility in more modern digital radar systems like the ones discussed in [17][18][19].In addition, this paper also introduces a method of doubling the number of codes in the set without affecting the cross-correlation properties or constructing a better good code set with significantly smaller cross-correlations than the ones introduced in [25,26].This is achieved by constructing good code sets from a complementary code pair and their mates, while using the same DFCW waveforms as chips.This is explained in detail in Section 7.

Paper Structure
Section 2 explains the two properties of symmetry/anti-symmetry and quasi-orthogonality for the discrete frequency-coding waveform chips based on LFM and PFLM waveforms.Construction of the proposed codes using DFCW chips is explained in Section 3. Doppler properties of the proposed codes are discussed in Section 4. Invariance of the zero-sidelobe region under frequency selective fading is demonstrated in Section 5. Section 6 shows the construction of good code sets.Section 7 shows the method of doubling the number of codes in the set or constructing a better good code set followed by conclusion in Section 8.
Delaying    () by  and adding it to    () results in, Since  2 = 1 and   () +   () = 2  ,   (   )  () +  2   (   )  () = 2    (), Equation ( 17) becomes () contains the mainlobe ( 2    () ) that is spread by the chip ACF (i.e.,   () ), a zero-sidelobe region on either side of the mainlobe and an adjacent region of cross-correlation sidelobes (i.e., the first and the last terms in Equation ( 19)).If  = 1, then () and ( −1 ) = ( −1 ) represent symmetrical waveforms.If  = −1, then () and ( −1 ) = −( −1 ) represent anti-symmetrical waveforms.For both the symmetry (  = 1 ) and anti-symmetry (  = −1 ) conditions, ACF of (), i.e.,   () = ()( −1 ) is identical to the ACF of (), i.e.,   () =  2 ( −1 )() =   ().As a result, exact sidelobe cancellation of the complementary code pair is achieved resulting in the zero-sidelobe region on either side of the mainlobe as shown in Equation (19).If () and ( −1 ) are also quasi-orthogonal in the convolution sense, i.e., their cross-correlation peak (as defined in Equation ( 10)) is small compared to the mainlobe peak (2  ), then the first and the last terms in Equation ( 19) are small.It should be noted that using a sinusoidal chip for the complementary code pair will also result in a zero-sidelobe region around the mainlobe as it satisfies the symmetry/anti-symmetry condition.However, a sinusoidal chip is not quasi-orthogonal to its symmetrical/anti-symmetrical mirror image.Thus, the proposed codes with sinusoidal chip achieves zero-sidelobe region but the cross-correlation sidelobes outside the zero domain are large which is undesirable.Hence, any waveform that satisfies the two conditions, symmetry/anti-symmetry and quasi-orthogonality, could be used as chips for the complementary code pair in the proposed code design.
DFCWs based on LFM and PLFM waveforms are constructed by discretizing their instantaneous frequencies.These will be referred to as Discrete Frequency LFM (DF-LFM) and Discrete Frequency PLFM (DF-PLFM) waveforms, respectively.The DF-LFM and DF-PLFM waveforms satisfy the symmetry/anti-symmetry and the quasi-orthogonality conditions required for the chips in the proposed code design.The following three types of DFCWs are used as chips for the complementary code pair in the proposed code design: 1. DF-LFM waveforms.2. DF-PLFM waveforms comprised of two up-chirp segments.3. DF-PLFM waveforms comprised of an up-chirp followed by a down-chirp as in [21].
The DF-PLFM waveforms comprised of two up-chirp segments can be represented by, where The DF-PLFM waveforms comprised of an up-chirp and a down-chirp segment can be represented by, Figure 3. shows the plots of instantaneous frequencies of the aforementioned DFCW chips.Let   () and   () represent the instantaneous frequencies of () and () , respectively.  () ,   () and  , () represent the ACF of (), ACF of () and cross-correlation between () and () respectively.They can be represented as shown in Equation (4).Since () and () are symmetrical mirror images of each other, their ACFs are identical, i.e.,   () =   (), as shown in the plots in Figure 4. Since () and () are also quasi-orthogonal, max  | , ()| is very small as shown in their plots in Figure 4.For these plots,  = 0.24 and  = 32. closes to 0 results in a DF-LFM/DF-PLFM waveform that is close to a pure sinusoid, while  close to 0.5 results in a very sharp slope.
In one path,  ′ () is passed into the matched filter (MF) of () implemented as a cascade of two MFs, one for the digital code ([]) and the other for the chip (()), as shown in Figure 6.
In Equation (37), 2    () is the mainlobe of () (  () is the ACF of the DFCW chip, as shown in the plots in Figure 4).2   ≤  ≤ 3   −  and 3   +  ≤  ≤ 4   are the zero-sidelobe regions on either side of the mainlobe. , () and  , () (defined in Equation (32)), respectively, represent the small cross-correlation sidelobes.Let  represent the peak cross-correlation sidelobe.Since,  , () and  , () are mirror images of each other, The table in Figure 7 shows the plots of |  ′ , ( − 2  )|, |  ′ , ()| and |()|, all normalized by 2  and in the absence of noise, for   = 8 and  = 16 for the three DFCW chips in Column 1. 3D-plots of the corresponding cross-correlation peaks () for a range of   and  are shown in Column 2. Clearly, the ACF plots show the zero-sidelobe region around the mainlobe and the adjacent region of small cross-correlation sidelobes.From the 3D-plots of  it can be observed that  values are very small, and decrease in magnitude as the code length   and the number of discrete frequencies  in the DF-LFM/DF-PLFM chips increase.In addition, the  values are slightly larger for the codes constructed using the DFCW chips based on PLFM waveforms compared to the  values for the codes constructed using DFCW chips based on LFM waveform, but still very small (≈ −30 dB).
If −() and () are concatenated and transmitted at a second frequency band and received using a receiver similar to the one described in Figure 6, then the final output of this code can be represented by, Clearly, adding Equations ( 37) and (39) results in complete cancellation of the cross-correlation sidelobes.Thus, achieving complete zero-sidelobes on either side of the mainlobe.This is shown in the plots of (), ′() and () + ′() for   = 8 and  = 16 in Figure 8. Frequency selective fading between the two bands could result in inexact cancellation of the cross-correlation sidelobes that are adjacent to the zero-sidelobe region.However, since the cross-correlation sidelobes are very small, any inexact cancellation due to fading will not create undesirable high magnitude sidelobes.).This scheme results in a transmission without gaps, while still using the same frequency band as in the previous scheme.Fishler et al. [18] and Deng [19] describe this scheme using continuous LFM and PLFM chips.However, for this interlaced code, the zero-sidelobes and the small cross-correlation sidelobes in the final output are interlaced as well.Doppler shift.As a result, ACF of the complementary code pair do not align with each other.Thus, the proposed codes do not have good Doppler properties.One way to improve the Doppler properties could be to use triangular FM based waveforms as chips for the complementary code pair in the proposed code structure.Another method to improve the Doppler properties could be to use the approach discussed in [27] to construct Doppler resilient Golay complementary code pairs.

Effect of Frequency Selective Fading
In the presence of frequency selective fading, the higher frequencies within the frequency sweep ( = ( − 1)ΔΔ) of the DF-LFM/DF-PLFM waveforms could be slightly attenuated compared to the lower frequencies.However, autocorrelations of the DF-LFM/DF-PLFM waveforms remain identical, since they are symmetrical.Hence, the zero-sidelobe region is not affected.However, the adjacent region of small cross-correlation sidelobes could vary slightly depending on the attenuation due to fading.However, its effect on the error in the location of the mainlobe is negligible as will be shown in the simulation results later in this section.
In the presence of frequency selective fading, let the lower and the higher frequency components of the DF-LFM/DF-PLFM chips be attenuated by  1 and  2 respectively, where  2 <  1 .The DF-LFM chips (  () and   ()) in the presence of fading can be represented by, The DF-PLFM chips comprised of two up-chirp segments in the presence of fading can be represented by, The DF-PLFM chips comprised of an up-chirp and a down-chirp segment in the presence of fading can be represented by, For the DF-LFM waveforms, all the discrete frequencies 0 ≤  ≤  1 − 1, are considered as the lower frequency component which are attenuated by  1 , while the discrete frequencies  1 ≤  ≤  − 1 will be considered as the higher frequency component which are attenuated by  2 .For the DF-PLFM waveforms, all the discrete frequencies 0 ≤  ≤  Figure 11 shows the matched filter outputs,   ′ , () and   ′ , (), and their sum () (given by Equations ( 30), ( 33) and (37), respectively), in the presence of frequency selective fading.Clearly, the zero-sidelobe region on either side of the mainlobe is not affected by fading.

|R(t)|
Root mean square error () in the location of the mainlobe is used as a metric to measure of effectiveness in sidelobe cancellation in the presence of frequency selective fading and noise.Location of the mainlobe of () is given by, where () is the final output, given by Equation (37). in the location of the mainlobe can be represented by, where  is the number of trials,  is the actual timing of the mainlobe,  ̂ is the estimated timing of the mainlobe for the ith trial in the presence of noise and  is the chip duration. vs. signal-to-noise ratio (SNR) plots for the proposed codes are shown in Figure 12.SNR is given by 20 log 10 2   2 , where 2  is the code energy and  2 represents the noise power.SNR is varied by changing  2 and  = 1000 iterations are performed for each SNR value. 1 and  2 are randomly chosen such that 0.6 ≤  1 ,  2 ≤ 1.From these plots it can be observed that the RMSE curve with fading stays very close to the  in the absence of fading.This is true for the DF-LFM and the DF-PLFM chips.
As discussed in the Introduction, complete sidelobe cancellation can be achieved using a complementary code pair.However, it requires transmission and reception of the complementary code pair at two frequencies/channels as shown in Figure 1.In the presence of frequency selective fading, the cancellation of sidelobes is inexact which results in increasing the probability of false alarm.This can be seen in the plots of  vs. SNR in Figure 13 for a Golay complementary code pair (, ) of length   = 16 and 64. is attenuated by  1 and  is attenuated by  2 such that 0.6 ≤  1 ,  2 ≤ 1.It can be observed that frequency selective fading results in increasing the RMSE in the location of the mainlobe.The proposed codes are resistant to the effect of frequency selective fading, as shown in the  plots in Figure 12.
Figure 13.Plot of root mean square error () vs. signal-to-noise ratio (SNR) for complementary code pair using sinusoidal chip.

Good Code Sets from Complementary Pair Using DF-LFM and DF-PLFM Waveforms as Chips
A good code set is constructed by changing the slopes of the instantaneous frequency (shown in the plots in Figure 3) of the DF-LFM/DF-PLFM waveforms used as chips for the same complementary code pair.The parameter   , as shown in the Equations ( 20)- (25), is varied carefully to result in DF-LFM and DF-PLFM chips with different slopes.The resulting good code set has a zero domain on either side of the mainlobe and an adjacent region of small sidelobes.The cross-correlation between any code pair in the set is very small.The lth code in the set can be represented by, , and   () and   () represent the two symmetrical DF-LFM/DF-PLFM waveforms.
() and   () for the good code sets from complementary pairs using DF-LFM chips can be represented by,   () and   () for the good code sets from complementary pairs using DF-PLFM chips comprised of two up-chirps can be represented by, (52)   () and   () for the good code sets from complementary pairs using DF-PLFM chips comprised of an up-chirp and a down-chirp are given by, The receiver for each code is as shown in Figure 6.The ACFs of all the codes in the set are as shown in the plots in Figure 7. Thus, the autocorrelation sidelobe peak (  ) of the ℎ code in the set is given by, where    ,  is the same as  , which is defined in Equation (38) and shown in the plots in Figure 7. Maximum cross-correlation peak () and average cross-correlation peak () are used as measures to compare the different good code sets introduced in this paper.Good code sets from complementary pairs using DF-PLFM waveforms comprised of two up-chirps as chips  = 0.24 and  = 32.

𝑀𝐶𝐶𝑃 = max
Good code sets from complementary pairs using DF-PLFM waveforms comprised of an up-chirp and a down-chirp as chips  = 0.24 and  = 32 From the plots in Figure 14 it can be observed that the good code sets constructed using DF-LFM chips have better  and  properties compared to those constructed using  (Column 1) Plots of  and  vs.  for the good code set with twice the number of codes (Column 2) Plots of  and  vs.  for the good codes set with better cross-correlation properties Good code sets from complementary pairs and their mates using DF-LFM waveforms as chips Good code sets from complementary pairs and their mates using DF-PLFM waveforms comprised of two up-chirps as chips  If a better good code set with the smaller  and  is required, then the best candidates from the two sets could be selected.This could be achieved by using only half the slope options for the DFCW waveforms used as chips for the complementary code pair and its mates i.e., the DF-LFM/DF-PLFM waveforms with  = 0, 2, … ,  − 1 in Equation ( 52)-(57).This results in a good code set with significantly better  and  values, as shown in the plots in column 2 of the table in Figures 16 and 17.These are significantly smaller than the  and the  values in Figure 14 for the same number of codes in the set.Thus, constructing the proposed codes from complementary code pair and their mates, while using the same chips for both pairs, offers the aforementioned flexibility of choosing between a better good code set and a larger good code set.
Table 1 shows the average autocorrelation and average cross-correlation values for the Deng's polyphase [28] good code set and the Deng's discrete frequency [20] set designs.The autocorrelation and cross-correlation sidelobe peaks of the codes introduced in this paper are smaller, as shown in the plots in Figures 7 and 17 respectively.The proposed codes constructed via concatenating biphase Golay complementary code pairs and using discrete frequency chips results in a zero-sidelobe domain and an adjacent region of very small cross-correlation sidelobes.In [28], it is shown that the average autocorrelation sidelobe and cross-correlation peaks decrease with increase in code length and approach ( 1 √ ) for large .Their discrete frequency good code set [20] being wideband, achieve much smaller sidelobe peaks.In the case of the good code set design proposed in this paper, for a given code length (  ), increasing the number of discrete frequencies () in the chips results in a better code set.

Conclusions
In this paper, it is shown that, by replacing the sinusoidal chips in the complementary code pair with waveforms that satisfy two conditions, symmetry/anti-symmetry and quazi-orthogonality in the convolution sense, allows for concatenating them in the time domain using one frequency band/channel.This results in a zero sidelobe domain around the mainlobe and an adjacent region of small cross-correlation sidelobes.Symmetry/anti-symmetry property results in the zero-sidelobe region, while the quasi-orthogonality property makes the adjacent region of cross-correlation sidelobes small.Discrete frequency coding waveform (DFCW) based on LFM and PLFM waveforms are used as chips, since they satisfy both the symmetry/anti-symmetry and quasi-orthogonality conditions.Since, frequency selective fading does not affect the symmetry/anti-symmetry property, the zero-sidelobe region is resistant to the effects of fading.A good code set with a zero-sidelobe region is then constructed by varying the slopes of the DFCW chips, while using the same complementary code pair.Such good code sets are constructed using a set of quasi-orthogonal

Figure 1 .
Figure 1.Transmitter (Tx) and receiver (Rx) of a Golay complementary code pair.

Figure 2 .
Figure 2. (a-c) Plots of matched filter outputs in the absence of frequency selective fading; and (d-f) plots of matched filter outputs in the presence of fading.

1 . 2 .
The Proposed Codes Based on Complementary Pairs via Discrete Frequency Chips Given any complementary code pair [] and [] , [] + [ −   ] is constructed by concatenating [] and [] in the time domain using one frequency band/channel.The chips used for [] and [] satisfy the following two conditions: 1.They are symmetrical/anti-symmetrical mirror images of each other, i.e., () with [] and (−)/−(−) (or (−), where  = 1 or −1) with [].2. They are quasi-orthogonal in the convolution sense.

Figure 3 .
Figure 3. Plots of instantaneous frequencies of (a) discrete frequency linear frequency modulation (DF-LFM) chips, (b) discrete frequency piecewise linear frequency modulation (DF-PLFM) chips comprised of two up-chirp segments and (c) DF-PLFM chips comprised of an up-chirp and a down-chirp segment.

Figure 5 .
Figure 5. Block diagram showing the construction of the proposed code using Golay complementary code pair with DFCW chips.

Figure 9 .
Figure 9. Plots of 3D-ambiguity function in (a) linear scale and (b) in dB scale of the proposed codes constructed using DF-LFM chips.

Figure 12 .
Figure 12.Plots of root mean square error () vs. SNR for the proposed codes.
,  () is the normalized cross-correlation between the code pair {, } in the set.The table (Column 2) in Figure14shows the plots of  and  for each set for different values of ,   and .(Column 1) Plots of    (),    () vs. time (   () are shown in solid lines and   () in lines with dashes) (Column 2) Plots of  and  vs.  for different values of   and  Good code sets from complementary pairs using DF-LFM waveforms as chips  = 0.24 and  = 32

Table 1 .
Table containing the average autocorrelation and cross-correlation peaks for the proposed set in comparison with Deng's polyphase and discrete frequency code sets.