aerospace
Article
Good Code Sets from Complementary Pairs via
Discrete Frequency Chips
Ravi Kadlimatti 1, * and Adly T. Fam 2
1
2
*
Advanced Wireless Systems Research Center, State University of New York at Oswego,
Oswego, NY 13126, USA
Department of Electrical Engineering, University at Buffalo, The State University of New York,
Buffalo, NY 14260, USA; [email protected]
Correspondence: [email protected]; Tel.: +1-716-491-8985
Academic Editors: Yan (Rockee) Zhang and Mark E. Weber
Received: 22 March 2017; Accepted: 3 May 2017; Published: 7 May 2017
Abstract: 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.
Keywords: discrete frequency-coding waveform; linear FM; chirp; piecewise LFM; complementary
code pair; symmetry; anti-symmetry; good code sets; MIMO radar
1. Introduction
1.1. 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 x [n] be a code of length N. Its aperiodic autocorrelation (ACF) can be represented by,
N −1
R[n] =
∑
x [k] x ∗ [n + k]
(1)
k =0
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Z
In the Z-domain, if x [n] ⇔ X (z). then, its ACF can be represented by,
Z
R [ n ] ⇔ R ( z ) = X ( z ) X z −1 = N + S ( z )
(2)
where N and S(z) represent the mainlobe/peak and the sidelobes respectively.
Let p(t) = e j2π f p t , 0 ≤ t ≤ T, represent a sinusoidal chip. x [n] modulates p(t) resulting in x p (t).
N −1
x p (t) =
∑
x [n] p(t − nT )
(3)
n =0
Its ACF can be represented by,
R x (t) =
Z t+ T
If x p (t) is sampled at intervals of Ts =
t
x p (τ ) x ∗p (t + τ )dτ
(4)
1
, then in the Z-domain, x p (mTs ) for m = 0, 1, 2, . . . ,
fs
( N − 1) Ns , can be represented by,
Z
x p (mTs ) ⇔ X p (z) = X z Ns P(z)
Z
(5)
where p(mTs ) ⇔ P(z) and Ns represents the number of samples in P(z). ACF of X p (z) can be
represented by,
R x ( z ) = X p ( z ) X p z −1
(6)
R x (z) = X z Ns P(z) X z− Ns P z−1
(7)
R x (z) = R z Ns R p (z)
(8)
R x (z) = NR p (z) + S z Ns R p (z)
(9)
where R p (z) = P(z) P z−1 represents the ACF of the highly sampled chip and R(z) is given in
Equation (2). It can be observed that the mainlobe and the sidelobes are spread by the ACF of the
chip (R p (z)).
Some of the well-known good aperiodic codes are the Barker codes, Frank codes [1] and
waveforms based on Costas arrays [2–4]. 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
1
a peak sidelobe ratio of 13
. 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 [5–8] 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
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complementary code pair [10], polyphase complementary codes [11] and complementary code
sequences [12] achieve complete sidelobe cancellation via addition of their autocorrelations. Figure 1
code pair [10], polyphase complementary codes [11] and complementary code sequences [12] achieve
shows the transmission and reception of a Golay complementary code pair (say [ ] and [ ], of
complete sidelobe cancellation via addition of their autocorrelations. Figure 1 shows the transmission
length
). [ ] is transmitted at frequency
(using ( ) =
) and [ ] at
(using
and reception of a Golay complementary code pair (say a[n] and b[n], of length Ng ). a[n] is transmitted
( )=
). At the receiver, the code pair are received in their own matched filters. Addition of
at frequency f 1 (using c1 (t) = e j2π f1 t ) and b[n] at f 2 (using c2 (t) = e j2π f2 t ). At the receiver, the code
their MF outputs ( ( ) and
( )) results in complete cancellation of the sidelobes. However,
pair are received in their own matched filters. Addition of their MF outputs (R a (t) and Rb (t)) results
frequency selective fading results in unequal attenuation of the two codes. This results in
in complete cancellation of the sidelobes. However, frequency selective fading results in unequal
reemerging sidelobes due to inexact cancellation of the non-zero sidelobes of the code pair, as shown
attenuation of the two codes. This results in reemerging sidelobes due to inexact cancellation of the
in Figure 2 for
= 8. Since the sidelobes of the code pair are not small, any inexact cancellation
non-zero sidelobes of the code pair, as shown in Figure 2 for Ng = 8. Since the sidelobes of the
results in reemerging sidelobes which is highly undesirable. In Figure 2,
( ),
( ) and ( )
code pair are not small, any inexact cancellation results in reemerging sidelobes which is highly
represent the fading absent case and
′( ),
′( ) and ′( ) represent the fading present case. For
undesirable. In Figure 2, R a (t), Rb (t) and R(t) represent the fading absent case and R a 0(t), Rb 0(t) and
the fading present case, ( ) and ( ) are attenuated by 0.95 and 0.85 respectively. Galati et al.
R0(t) represent the fading present case. For the fading present case, a(t) and b(t) are attenuated by
[13] also discusses in detail this effect of unequal attenuation on the complementary code pairs. Liu
0.95 and 0.85 respectively. Galati et al. [13] also discusses in detail this effect of unequal attenuation
et al. [14] describes z-complementary code pairs that achieve a zero-correlation zone around the
on the complementary code pairs. Liu et al. [14] describes z-complementary code pairs that achieve
mainlobe and minimum possible sidelobe peaks outsize the zero domain. However, these codes also
a zero-correlation zone around the mainlobe and minimum possible sidelobe peaks outsize the zero
require two frequencies or channels which makes them vulnerable to the aforementioned effects of
domain. However, these codes also require two frequencies or channels which makes them vulnerable
fading. Tang et al. [15] and Li [16] describe loosely synchronized (LS) and large area (LA) codes for
to the aforementioned effects of fading. Tang et al. [15] and Li [16] describe loosely synchronized (LS)
quasi-synchronous code division multiple access (QS-CDMA) systems. These codes also achieve a
and large area (LA) codes for quasi-synchronous code division multiple access (QS-CDMA) systems.
zero interference zone in both the autocorrelation and the cross-correlations, but the peak sidelobes
These codes also achieve a zero interference zone in both the autocorrelation and the cross-correlations,
outside the zero-sidelobe region are not small. In this paper, we introduce codes constructed from
but the peak sidelobes outside the zero-sidelobe region are not small. In this paper, we introduce codes
biphase Golay complementary code pairs that are available at a variety of lengths. These codes have
constructed from biphase Golay complementary code pairs that are available at a variety of lengths.
a zero-sidelobe domain on either side of the mainlobe, while the sidelobe peaks outside the zero
These codes have a zero-sidelobe domain on either side of the mainlobe, while the sidelobe peaks
domain are very small. Since these codes use the same frequency band, they are not affected by the
outside the zero domain are very small. Since these codes use the same frequency band, they are not
effects of fading.
affected by the effects of fading.
Ng
0
a [n ]
a (t )
am (t )
2N g
a* (T - t )
am (t )+n (t )
e - j 2πfc t
e j 2πf1 t e j 2πfc t
BPF1
R a (t )
Matched
filter
0
R (t )
Ng
0
b [n ]
b (t )
bm (t )
e j 2πf2 t e j 2πfc t
Tx
b* (T - t )
bm (t )+n (t )
e - j 2πfc t
Channel
BPF2
Matched
filter
R b (t )
Rx
Figure 1. Transmitter (Tx) and receiver (Rx) of a Golay complementary code pair.
Figure 1. Transmitter (Tx) and receiver (Rx) of a Golay complementary code pair.
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44 of
Ng
Ng
2Ng
0
0
0
Ng
(a) Ra(t)
0
Ng
(b) Rb(t)
0
0
(d) Ra'(t)
(c) R(t)
2Ng
(e) Rb'(t)
(f) R'(t)
Figure 2. (a–c) Plots of matched filter outputs in the absence of frequency selective fading; and (d–f)
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.
plots of matched filter outputs in the presence of fading.
1.2. The
The Proposed
Proposed Codes
Codes Based
Based on
on Complementary
Complementary Pairs
1.2.
Pairs via
via Discrete
Discrete Frequency
Frequency Chips
Chips
Given any
any complementary
complementary code
code pair
pair a[[n]] and
and b[[n]], , a[n
by
[ ] ]+
Given
+ b[[n −
− ND ]] is
is constructed
constructed by
concatenating a[n[ ] ]and
time
domain
using
oneone
frequency
band/channel.
TheThe
chips
used
for
concatenating
andb[n][ in] the
in the
time
domain
using
frequency
band/channel.
chips
used
afor
b
n
satisfy
the
following
two
conditions:
[n] and
[
]
[ ] and [ ] satisfy the following two conditions:
1.
2.
They
They are
are symmetrical/anti-symmetrical
symmetrical/anti-symmetricalmirror
mirrorimages
imagesofofeach
eachother,
other,i.e.,
i.e., (c()t)with
with a[[n]] and
−1)
c(−
c(−)t)(or
(or γc(−
t),where
where γ==11 or
or −
1) with b[[n].].
(−t)/−
)/ −(−
(−),
They
They are
are quasi-orthogonal
quasi-orthogonal in
in the
the convolution
convolution sense.
sense.
This concatenated code has a region of zero-sidelobes on either side of the mainlobe and an
This concatenated code has a region of zero-sidelobes on either side of the mainlobe and
adjacent region of small cross-correlations. Due to the symmetry/anti-symmetry property, the two
an adjacent region of small cross-correlations. Due to the symmetry/anti-symmetry property,
chips have identical ACF. This results in exact sidelobe cancellation of the complementary code pair,
the two chips have identical ACF. This results in exact sidelobe cancellation of the complementary
creating a zero-sidelobe region on either side of the mainlobe. The cross-correlation peak (
)
code pair, creating a zero-sidelobe region on either side of the mainlobe. The cross-correlation peak
between ( ) and (− ) is represented by,
(CCP) between c(t) and γc(−t) is represented by,
( ) ∗ (− + )
R t+ T
(10)
=max
γc(τ )c∗ (−
2 t + τ )dτ CCP = max t
(10)
t 2Ng
Since ( ) and (− ) are quazi-orthogonal in the convolution,
is small compared to the
mainlobe
2 .γc
This
makes the adjacent
of cross-correlation
sidelobes
small.
Sincepeak
c(t) and
in theregion
convolution,
CCP is small
compared
to the
(−tproperty
) are quazi-orthogonal
mainlobe
peakfrequency-coding
2Ng . This property
makes the(DFCW)
adjacent based
regionon
of cross-correlation
small.
Discrete
waveforms
linear frequency sidelobes
modulation
(LFM,
(DFCW)
based on
frequency
modulation
also Discrete
known asfrequency-coding
a chirp signal) waveforms
and piecewise
LFM (PLFM)
[17]linear
waveforms
satisfy
both the
(LFM,
also known as a chirp
and piecewise
LFM (PLFM)
[17]they
waveforms
both
symmetry/anti-symmetry
and signal)
quasi-orthogonality
conditions.
Hence,
are used satisfy
as chips
for the
symmetry/anti-symmetry
conditions.
Hence, they
used
complementary
code pair.and
A quasi-orthogonality
discrete frequency-coding
waveform,
say are
( ),
is as
thechips
sumfor
ofthe
complementary
code pair.
A discrete
frequency-coding
waveform,
saythe
c(tsame
sum of N contiguous
), is the
contiguous
sub-pulses
of the
same duration
Δ , but not
necessarily
frequency.
sub-pulses of the same duration ∆T, but not necessarily the same frequency.
1
[ ]
( ) = N −1
(11)
1 √Δ j2π f [n]∆Wtn
c(t) = √
(11)
∑e
∆T n=0
[
]Δ
where Δ
is the smallest possible frequency offset between two frequencies,
is the
where
∆W of
is the smallest
possible
[ ] ∈ {0,offset
frequency
nth sub-pulse
andfrequency
1, 2, …between
, − 1}.two frequencies, f [n]∆W is the frequency
of theDFCW
nth sub-pulse
f [n] in
∈ {MIMO
0, 1, 2,radars
. . . , N[18]
− 1}and
. Orthogonal Netted Radar Systems (ONRS)
sets findand
utility
DFCW
find utility
MIMO radars
[18] andspatial
Orthogonal
Netted
Radar good
Systems
(ONRS)
[19,20]
that sets
improve
radarinperformance
through
diversity.
Several
DFCW
sets[19,20]
have
that improve
been
proposed,radar
such performance
as the ones in through
[20–22]. spatial diversity. Several good DFCW sets have been
proposed,
as the
in [20–22].
In thissuch
paper,
it ones
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
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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.
1.3. Construction of a Good Code Set from the Mates of the Compelemntary Code Pair Resulting in Doubling
the Number of Codes in the Set or a Better Good Code Set with Significaltly Smaller Cross-Correlations
Given a complementary code pair, a[n] and b[n], there exists the complementary code pair:
b −n + Ng and − a −n + Ng , such that sum of the cross-correlation of a[n] with b −n + Ng and
that of b[n] with − a −n + Ng results in complete sidelobe cancellation. The code pair: b −n + Ng
and − a −n + Ng , are called as the mates [23,24] of the complementary code pair: a[n] and b[n].
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.
1.4. Previous Research
In [25,26], continuous frequency LFM and PLFM waveforms were used as chips for a[n] and b[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 [17–19]. 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.
1.5. 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.
2. Symmetrical/Anti-Symmetrical DFCW Chips That Are Quasi-Orthogonal
Let a(t) and b(t) be constructed by using c(t), for 0 ≤ t ≤ T, and γc(−t) (where γ = 1 or −1) as
chips for the Golay complementary code pair {a[n], b[n]} of length Ng , respectively. a(t) and b(t) can
be represented as shown in Equation (3).
Z
Z
If a[n] ⇔ A(z) and b[n] ⇔ B(z), then
Z
a(mTs ) ⇔ Ac (z) = A z Nc C (z)
(12)
Z
b(mTs ) ⇔ Bc (z) = γB z Nc C z−1
(13)
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where a(mTs ) and b(mTs ) represent the sampled versions of a(t) and b(t), respectively, Nc is the
Z
number of samples in C (mTs ) ⇔ C (z) and m ∈ {0, ±1, ±2, . . . , ±∞ }.
Ac (z) + z− D Bc (z) represents a code constructed by concatenating Ac (z) and Bc (z) with a gap D
in between. At the receiver, this code is passed into two matched filters: Ac z−1 which is the matched
filter of Ac (z), and Bc z−1 which is the MF of Bc (z). The cross-correlation of Ac (z) + z− D Bc (z) with
Ac z−1 results in,
R Ac (z) = Ac (z) + z− D Bc (z) Ac z−1
R Ac (z) = A z Nc C (z) A z− Nc C z−1 + γz− D B z Nc C z−1 A z− Nc C z−1
R Ac (z) = R A z Nc Rc (z) + γz− D R B,A z Nc C2 z−1
(14)
(15)
(16)
where R A (z) = A(z) A z−1 represents the autocorrelation of a[n], Rc (z) = C (z)C z−1 represents the
chip ACF (C z−1 is the MF of C (z)), R B,A (z) = B(z) A z−1 represents the cross-correlation of b[n]
with a[n] and γC2 z−1 represents the cross-correlation of γC z−1 with C z−1 .
Similarly, the cross-correlation of Ac (z) + z− D Bc (z) with Bc−1 (z) can be represented by,
R Bc (z) = γR A,B z Nc C2 (z) + γ2 z− D R B z Nc Rc (z)
(17)
where R A,B (z) = A(z) B z−1 represents the cross-correlation of a[n] with b[n], R B (z) = B(z) B z−1
represents the ACF of b[n] and γC2 (z) is the cross-correlation of C (z) with γC (z).
Delaying R Ac (z) by D and adding it to R Bc (z) results in,
Rs (z) = z− D R Ac (z) + R Bc (z)
(18)
Since γ2 = 1 and R A (z) + R B (z) = 2Ng , R A z Nc Rc (z) + γ2 R B z Nc Rc (z) = 2Ng Rc (z),
Equation (17) becomes
Rs (z) = γR A,B z Nc C2 (z) + 2Ng z− D Rc (z) + γz−2D R B,A z Nc C2 z−1
(19)
Rs (z) contains the mainlobe (2Ng Rc (z)) that is spread by the chip ACF (i.e., Rc (z)), 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 γ = 1, then C (z) and γC z−1 = C z−1 represent
symmetrical waveforms. If γ = −1, then C (z) and γC z−1 = −C z−1 represent anti-symmetrical
waveforms. For both the symmetry (γ = 1) and anti-symmetry (γ = −1) conditions, ACF of C (z),
i.e., Rc (z) = C (z)C z−1 is identical to the ACF of γC (z), i.e., Rγc (z) = γ2 C z−1 C (z) = Rc (z).
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 C (z) and γC z−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 (2Ng ), 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
Aerospace 2017, 4, 28
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proposed code design. The following three types of DFCWs are used as chips for the complementary
code pair in the proposed code design:
1.
2.
3.
DF-LFM waveforms.
DF-PLFM waveforms comprised of two up-chirp segments.
DF-PLFM waveforms comprised of an up-chirp followed by a down-chirp as in [21].
Let u(t) and d(t) = u∗ (−t) represent a DF-LFM/DF-PLFM waveform and its symmetrical mirror
image respectively. The DF-LFM waveforms can be represented by,
u(t) = √
d(t) = √
where f [n] = n, ∆W =
1
N −1
∆T
n =0
e j2π f [n]∆Wtn
(20)
e j2π f [ N −1−n]∆Wtn
(21)
∑
1
N −1
∆T
n =0
∑
1
T
, ∆T =
, n∆T ≤ tn ≤ (n + 1)∆T and T. is the time-duration of the
∆T
N
DFCW chip.
The DF-PLFM waveforms comprised of two up-chirp segments can be represented by,
u(t) =









d(t) =









=
where f 1 [n]
round
2k ( N −1)n
N
(1−k)2n
)
N −1
N −1
2
1
√
∑ e j2π f1 [n]∆Wtn
∆T n=0
N −1
√1
∑ e j2π f2 [ N −1−n]∆Wtn
∆T
n= N
2
(22)
N −1
2
√1
∑ e j2π f2 [n]∆Wtn
∆T n=0
N −1
√1
∑ e j2π f1 [ N −1−n]∆Wtn
∆T
n= N
2
(23)
for n
=
0, 1, . . . ,
N
2
− 1, 0
<
k
≤
0.5 and
f 2 [n] = round ( N − 1)(1 −
for n = N2 , N2 + 1, . . . , N − 1.
The DF-PLFM waveforms comprised of an up-chirp and a down-chirp segment can be
represented by,

N −1
2


1
j2π f 1 [n]∆Wtn

√
 ∆T ∑ e
n =0
u(t) =
(24)
N −1

j2π f 2 [n]∆Wtn

√1
e

∑
 ∆T
n= N
2
d(t) =









N −1
2
N
√1
∑ e j2π f2 [ 2 −1−n]∆Wtn
∆T n=0
N −1
√1
∑ e j2π f1 [ N −1−n]∆Wtn
∆T
n= N
2
(25)
Figure 3 shows the plots of instantaneous frequencies of the aforementioned DFCW chips. Let f u (t)
and f d (t) represent the instantaneous frequencies of u(t) and d(t), respectively. Ru (t), Rd (t) and
Ru,d (t) represent the ACF of u(t), ACF of d(t) and cross-correlation between u(t) and d(t) respectively.
They can be represented as shown in Equation (4). Since u(t) and d(t) are symmetrical mirror images
of each other, their ACFs are identical, i.e., Ru (t) = Rd (t) as shown in the plots in Figure 4. Since u(t)
and d(t) are also quasi-orthogonal, max Ru,d (t) is very small as shown in their plots in Figure 4.
t
Aerospace 2017, 4, 28
8 of 24
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8 of 24
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
For these plots, k = 0.24 and N = 32. k closes to 0 results in a DF-LFM/DF-PLFM waveform that is
sharp
slope.
Aerospace
2017, 4, 28
8 of 24
close to a pure sinusoid, while k close to 0.5 results in a very sharp slope.
(N-1)ΔW
(N-1)ΔW
(N-1)ΔW
(N-1)ΔW
Instantaneous frequency [Hz]
Instantaneous frequency [Hz]
(N-1)ΔW
(N-1)ΔW
00
00
NΔT
NΔT
00
00
(N/2)ΔT
(a)Time
Time[s]
[s]
(a)
00
00
NΔT
NΔT
(N/2)ΔT
(N/2)ΔT
(c)
(c)Time
Time[s][s]
(b) Time
Time [s]
(b)
[s]
instantaneous frequencies
frequencies of
(a) discrete
Figure3.3. Plots
Plots of
of instantaneous
frequencies
of (a)
Figure
discrete frequency
frequency linear
linear frequency
frequencymodulation
modulation
(DF-LFM)
chips;
discrete
frequency
piecewise
linear
frequency
modulation
chips,
(b)
(DF-LFM) chips, (b) discrete frequency piecewise linear frequency modulation (DF-PLFM)
(DF-PLFM) chips
chips
oftwo
twoup-chirp
up-chirpsegments
segmentsand
and(c)
(c)DF-PLFM
DF-PLFMchips
chipscomprised
comprisedof
anup-chirp
up-chirpand
and
comprised of
of
two
up-chirp
segments
and
(c)
DF-PLFM
aa
comprised
chips
comprised
ofofan
an
up-chirp
and
a
down-chirp
segment.
down-chirp
segment.
down-chirp segment.
DF-LFM waveform
DF-LFM waveform
0
0
-20
-20
-60
-60
0
0
-20
-20
-60
-60
0
0
-20
|Ru(t)|
|Ru(t)|
-0.5T
-0.5T
0
0 [s]
(a) Time
(a) Time [s]
0.5T
0.5T
|Rd(t)|
|Rd(t)|
-0.5T
-0.5T
0
(b) Time
0 [s]
(b) Time [s]
0.5T
0.5T
-0.5T
-0.5T
0
(c) Time
0 [s]
0.5T
-20
-60
-60
|Ru,d(t)|
|Ru,d(t)|
0.5T
(c) Time [s]
DF-PLFM waveform comprised of two up-chirp segments
0
DF-PLFM
waveform comprised of two up-chirp segments
|Ru(t)|
-20
0
|Ru(t)|
-20
-60
-60
0
-0.5T
-0.5T
-20
0
0
(a) Time [s]
0
(a) Time [s]
0.5T
0.5T
|Rd(t)|
-20
-60
-60
0
-0.5T
-0.5T
-20
0
0
(b) Time [s]
0
(b) Time [s]
0.5T
0.5T
|Ru,d(t)|
|Ru,d(t)|
-20
-60
-60
|Rd(t)|
-0.5T
-0.5T
0
(c) Time [s]
0
(c) Time [s]
0.5T
0.5T
DF-PLFM waveform comprised of an up-chirp and a down-chirp
Figure 4. Cont.
Cont.
Figure
Figure 4.
4. Cont.
NΔT
NΔT
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9 of 24
9 of 24
DF-PLFM waveform comprised of an up-chirp and a down-chirp
0
|Ru(t)|
-20
-60
-0.5T
0
(a) Time [s]
0
0.5T
|Rd(t)|
-20
-60
-0.5T
0
(b) Time [s]
0
0.5T
|Ru,d(t)|
-20
-60
-0.5T
0
(c) Time [s]
0.5T
Figure
4. (a,b)
Figure 4.
(a,b) Autocorrelation
Autocorrelation and
and (c)
(c) cross-correlation
cross-correlation of
of the
the discrete
discrete frequency
frequency linear
linear frequency
frequency
modulation
(DF-LFM)
and
discrete
frequency
piecewise
linear
frequency
modulation
modulation (DF-LFM) and discrete frequency piecewise linear frequency modulation (DF-PLFM)
(DF-PLFM)
waveforms.
waveforms.
3.
and DF-PLFM
DF-PLFM Waveforms
Waveforms
3. The
The Proposed
Proposed Code
Code Based
Based on
on Complementary
Complementary Pair
Pair Using
Using DF-LFM
DF-LFM and
as
Chips
as Chips
Let
andb[n[] ]represent
representa Golay
a Golay
complementary
code
pair
of length
. Let
( ) and
Let a[[n]] and
complementary
code
pair
of length
Ng . Let
a(t) and
b(t)
( ) represent
the waveforms
obtained
by modulating
( ) (Equation
(20),
(22) or
(24))
represent
the waveforms
obtained
by modulating
u(t) (Equation
(20), (22)
or (24))
with
a[nwith
] and d[(t])
and ( ) (Equation
(21),
(23)
or (25))
with [ ], respectively.
(Equation
(21), (23) or
(25))
with
b[n], respectively.
Ng
[ ] ( − ( − 1) )
(26)
(26)
b(t) = (∑
) )
) =b[n]d([t −
] ( n−−(1)−T1)
(27)
(27)
a(t) =
( )=
∑ a [ n ] u ( t − ( n − 1) T )
n =1
Ng
n =1
where T =
d(t(). ).
= N∆T
Δ is the duration of u((t))and
and
byby
concatenating
a(t) and
b(t) with
delay
) +b t(−−2N
A new
new code,
code,s(t() )==a(t() +
2 g T ,),isisconstructed
constructed
concatenating
( ) and
( ) awith
a
of
2N
T
between
them,
as
shown
in
Figure
5.
s
t
is
then
carrier
(
f
)
modulated
before
transmission,
(
)
c
delay g of 2
between them, as shown in Figure 5. ( ) is then
carrier ( ) modulated before
resulting in smresulting
5. The
signal
can be represented
by,represented by,
(t), as shown
transmission,
in in
( Figure
), as shown
intransmitted
Figure 5. The
transmitted
signal can be
sm (t) = ( a)(=t) +(b) t+− 2N−g 2T
e j2π f c t
(28)
(28)
Figure 6 shows the block diagram of the receiver. The received signal, ( ) = ( ) + ( ), is
Figure 6 shows the block diagram of the receiver. The received signal, x (t) = sm (t) + n(t),
first carrier demodulated and then bandpass filtered (BPF), giving ( ) as
shown in Figure 6. Since
is first carrier demodulated and then bandpass filtered (BPF), giving x 0 (t) as shown in Figure 6.
both ( ) and ( ) occupy the same bandwidth, only one BPF is used. ~ (0, 2) represents
Since both a(t) and b(t) occupy the same bandwidth, only one BPF is used. n ∼ N 0, σ represents
additive white Gaussian noise.
additive white Gaussian noise.
( )= ( )+
− 2
++ ( )
(29)
x 0 (t) = a(t) + b t − 2Ng T + +n(t)e− j2π f c t
(29)
In one path, ( ) is passed into the matched filter (MF) of ( ) implemented as a cascade of
two MFs,
one
for the
code (into
[ ])the
and
the other
for the
chip
as shown in as
Figure
6.
In one
path,
x 0 (tdigital
matched
filter
(MF)
of (a(t() )),
implemented
a cascade
of
) is passed
two MFs, one for the digital code (a[n])(and
the
) = the(other
) + for
− chip
2 (u(+t)), as
( )shown in Figure 6. (30)
,
,
R x0 ,aof
R a (t), +
t − 2Ng Tthe+cross-correlation
n a (t)
(t) =
(R
) b,a
where
represents
of ( ) with (30)
( )
( ) represents the ACF
( ),
and
( ) represents the filtered noise process,
~ (0,
).
where R a (t) represents the ACF of a(t), Rb,a (t) represents the cross-correlation of b(t) with a(t) and
n a (t) represents the filtered noise process, n a ∼ N 0, Ng σ2 .
Aerospace 2017, 4, 28
Aerospace 2017, 4, 28
Aerospace 2017, 4, 28
R a (t) =
10 of 24
Ng
Ng
∑ ∑
∗
a[n] a Ng − m
n =1 m =1
,
[ ]
[ ]
( )=
Ng, ( N)g=
∗
∗
t+ T
Z
t
10 of 24
10 of 24
∗
u(τ − (n − 1) T )u (t + τ + (m − 1) T )dτ
−
−t+ T
Z
( − ( − 1) ) ∗∗( + + (
( − ( − 1) ) ( + + (
− 1) )
− 1) )
(31)
(32)
(32)
d(τ − (n − 1) T )u∗ (t + τ + (m − 1) T )dτ
(32)
Rb,a (t) = ∑ ∑ b[n] a Ng − m
t
In another path,
(
)
is
passed
into
the
MF
of
( ) which is also implemented as a cascade of
n
=
1
m
=
1
In another path, ( ) is passed into the MF of ( ) which is also implemented as a cascade of
MF of [ ] and that of
0 ( t ) ((is ).
MFInofanother
[ ] and
thatxof
).passed into the MF of b(t) which is also implemented as a cascade of MF
path,
)= , ( )+
− 2
+ ( )
(33)
of b[n] and that of d(t).
, (
−2
+ ( )
(33)
, ( )+
, ( )=
R x0 ,b (t) =
R a,b (t) +
Rb t − 2Ng T + nb (t)
(33)
where
( ) with ( ), which can be represented as
, ( ) represents the cross-correlation of
(
)
where
represents
the
cross-correlation
of
(
)
with
(
),
which
can
be
represented
as
where
R a,bin(t,Equation
a(t)ofwith
t), which
can be represented
asnoise
shown
) represents
shown
(17),the cross-correlation
( ) represents theof
ACF
( )b(and
( ) represents
the filtered
shown in Equation (17),
( ) represents the ACF of ( ) and
( ) represents the filtered noise
in process,
Equation (17),
the ACF of b(t) and n a (t) represents the filtered noise process,
~ (0,Rb (t) represents
).
process, ~ (0,
).
nb ∼ N 0, Ng σ2 .
∗
Figure
5. 5.
Block
diagram
the proposed
proposedcode
codeusing
usingGolay
Golay
complementary
Figure
Block
diagramshowing
showingthe
theconstruction
construction of the
complementary
Figure 5. Block diagram showing the construction of the proposed code using Golay complementary
code
pair
with
DFCW
code
pair
with
DFCWchips.
chips.
code pair with DFCW chips.
1
1
1
x (t )
x (t )
t
e -- jj 22πfc
e πfc t
x' (t )
x' (t )
a* [Ng-n ]
a* [Ng-n ]
BPF
BPF
Digital
Digital
code
MF
code MF
b* [Ng-n ]
b* [Ng-n ]
Digital
Digital
code
MF
code MF
u* (T - t )
u* (T - t )
DFCW
DFCW
chip
MF
chip MF
d* (T - t )
d* (T - t )
DFCW
DFCW
chip
MF
chip MF
( )
( )
1
0.1
0.1
z -- 22NNggTT
z
Delay
Delay
block
block
1
1
1
1
1
1
R x’,a (t )
R x’,a (t )
(b )
(b )
(c )
(c )
1
0.1
0.1
0.1
0
0.1
0
R (t )
R (t )
1
0.1
0.1
R x’,b (t )
R x’,b (t )
Figure6.
6. Receiver block
Figure
block diagram.
diagram.
Figure 6. Receiver
Receiver block
diagram.
Delaying
( ) by 2
(as shown in Figure 6) and adding it to
( ) results in,
Delaying
) by
shown
Figure
andadding
addingitittotoR x0 ,b,,(t() )results
resultsin,
in,
Delaying
R x0 ,a (,,t)( by
2N2g T (as(as
shown
in in
Figure
6)6)and
( )= , ( )+
+
+
+
− 2
−2 + ,
−4
−
2
+
(( )) (34)
( )= , ( )+
+ ,
+
−2
−4
−2
(34)(34)
R(t) = R a,b
(t) + R a t − 2Ng−T2 + Rb+ t − 2N
g T + Rb,a t − 4Ng T + n a t − 2Ng T + nb ( t )
Since,
Since,
Since,
( ), − ≤ ≤
2
( ), − ≤ ≤
( ) + (( ) = 2
(35)
( ) + ( 2N
) =g Ru (t)0,
(35)
, −T
ℎ≤t≤T
0, ℎ
R a (t) + Rb (t) =
(35)
0, chip.
otherwise
( ) = ( ) represents the ACF of the DFCW
where
( ) = ( ) represents the ACF of the DFCW chip.
where
where Ru (t) = R(d ()t=
chip.
) represents
( ) + 2 the ACF
( − of
2 the) DFCW
+
−4
+
−2
+ ( )
(36)
( −2
) + ,,
−4
+
−2
+ ( )
( ) = ,, ( ) + 2
(36)
Thus, the final output, ( ), (excluding the noise terms) consists of,
Thus, the final output, ( ), (excluding the noise terms) consists of,
Aerospace 2017, 4, 28
11 of 24
R(t) = R a,b (t) + 2Ng Ru t − 2Ng T + Rb,a t − 4Ng T + n a t − 2Ng T + nb (t)
Aerospace 2017, 4, 28
(36)
11 of 24
Thus, the final output, R(t), (excluding the noise terms) consists of,
( ) = ( ) represents
where
the ACF of the DFCW chip.
(t), 0 ≤ t < 2Ng T

 R(a,b−
( ) = , ( ) + 2
2
)+ ,
−4
+
−2


0,
2N
g T ≤ t < 3Ng T − T

Thus, the final output,
noise
of,+ T
R(t) =( ), (excluding
2Ng Ru (t),the
3N
T ≤ tconsists
< 3Ng T
g T −terms)


 0, 3Ng T + T, (≤),t0<
≤ 4N
<g2T



Rb,a (t), 0,
4N
2 g T ≤≤ t <
< 6N
3 g T−
( )= 2
( ), 3
−
≤ <3
+
( )
(36)
(37)
+
(37)
In Equation (37), 2Ng Ru (t) is the mainlobe
(t) (R
0, 3 of R+
≤ u (<t)4is the ACF of the DFCW chip, as shown
in the plots in Figure 4). 2Ng T ≤ t ≤ 3Ng T ,−( T), 4and 3N
≤ gT
<+
6 T ≤ t ≤ 4Ng T are the zero-sidelobe
regions on either side of the mainlobe. R a,b (t) and Rb,a (t) (defined in Equation (32)), respectively,
( ) is the mainlobe of ( ) ( ( ) is the ACF of the DFCW chip, as
In Equation (37), 2
represent the small cross-correlation sidelobes. Let CCP represent the peak cross-correlation sidelobe.
shown in the plots in Figure 4). 2
≤ ≤3
−
and 3
+ ≤ ≤4
are the
Since, R a,b (t) and Rb,a (t) are mirror images of each other,
zero-sidelobe regions on either side of the mainlobe.
(
)
and
(
)
(defined
in
Equation
(32)),
,
,
sidelobes.
respectively, represent the small cross-correlation
Let
represent
the
peak
R a,b (t) are mirror
images of each other,
(38)
CCP
=
max
cross-correlation sidelobe. Since,
(
)
and
(
)
,
, t 2N
g
, ( )
(38)
The table in Figure 7 shows the plots of =Rmax
x 0 ,a t −
2 2Ng T , R x0 ,b (t) and | R(t)|, all normalized
by 2Ng and in the absence of noise, for Ng = 8 and N = 16 for the three DFCW chips in Column
−2
|, | , ( )| and | ( )|, all normalized
The table in Figure 7 shows the plots of | ,
1. 3D-plots of the corresponding cross-correlation
peaks (CCP) for
a range of Ng and N are shown
by 2
and in the absence of noise, for
= 8 and = 16 for the three DFCW chips in Column 1.
in Column 2. Clearly, the ACF plots show the zero-sidelobe region around the mainlobe and the
3D-plots of the corresponding cross-correlation peaks (
) for a range of
and
are shown in
adjacent region of small cross-correlation sidelobes. From the 3D-plots of CCP it can be observed
Column 2. Clearly, the ACF plots show the zero-sidelobe region around the mainlobe and the
that CCP
values
areofvery
small,
and decrease
in magnitude
the code
Ng and
the number
adjacent
region
small
cross-correlation
sidelobes.
From theas3D-plots
of length
it can
be observed
of discrete
frequencies
N
in
the
DF-LFM/DF-PLFM
chips
increase.
In
addition,
the
CCP
values are
that
values are very small, and decrease in magnitude as the code length
and the number
slightly
larger for
the codes constructed
using the DFCW
based
on PLFMthe
waveforms
compared
of discrete
frequencies
in the DF-LFM/DF-PLFM
chipschips
increase.
In addition,
values
are
to theslightly
CCP values
chips based
LFM
but still very
largerfor
forthe
thecodes
codesconstructed
constructedusing
usingDFCW
the DFCW
chips on
based
onwaveform,
PLFM waveforms
to the
values for the codes constructed using DFCW chips based on LFM waveform,
smallcompared
(≈−30 dB).
but
still
very
small
(≈
−30
dB).
If − a(t) and b(t) are concatenated
and transmitted at a second frequency band and received
If
−
(
)
and
(
)
are
and in
transmitted
a second
frequency
band
using a receiver similar to theconcatenated
one described
Figure 6,atthen
the final
output
of and
this received
code can be
using a receiver similar to the one described in Figure 6, then the final output of this code can be
represented by,

represented by,
− R a,b (t), 0 ≤ t < 2Ng T





≤T < 2
0,
2Ng T ≤ t−< 3N
, ( g),T0−

0,
2
≤
−g T + T
R0(t) =
(39)
2Ng Ru (t), 3Ng T − T ≤< t3< 3N


(
),
3
−
≤
<
3
+
2
)
′(
=

0,
3N
T
+
T
≤
t
<
4N
T
(39)
g
g



0, 3
+ ≤ <4
− Rb,a (t), 4N
g T ≤ t < 6Ng T
−
,
( ), 4
≤ <6
Column 1
Column 2
DF-LFM Chips
0
|Rx',a(t-2NgT)|
-20
0
2NgT
0
3NgT
(a) Time [s]
4NgT
|Rx',b(t)|
-20
-70
2NgT
0
3NgT
(b) Time [s]
4NgT
-70
2NgT
3NgT
(c) Time [s]
-20
-45
-30
-40
-50
-50
|R(t)|
-20
-40
-10
CCPa,b [dB]
-70
-35
-60
16
64
32
N
4NgT
Figure 7. Cont.
Figure 7. Cont.
-55
32
64
128
8
16
Ng
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12 of 24
12 of 24
Aerospace 2017, 4, 28
12 of 24
0
|R
(t-2N T)|
|R
(t-2N T)|
-28
g
DF-PLFM chipsx',a comprised
of two up-chirp segments
0
-20
2NgT
0
-70
3NgT
(a) Time [s]
2NgT
-20
0
3NgT
(a) Time [s]
x',a
|R
4NgT
2NgT
-70
-20
x',b
|R
-70
3NgT
(b) Time [s]
2NgT
0
3NgT
(b) Time [s]
x',b
(t)|
(t)|
4NgT
4NgT
|R(t)|
2NgT
-70
3NgT
[s]
2NgT (c) Time
3NgT
(c) Time [s]
-30
-10
-32
0
-34
-10
-36
-20
-30
-20
-38
-40
-30
-40
-40-50
-50-60
|R(t)|
-20
-70
0
4NgT
-20
0
g
CCPa,b [dB]
-70
-30
-28
CCPa,b [dB]
-20
-60
16
64
16
32
32
32
64
64
4NgT
128
N
128
N
4NgT
8
16
8
16
32
Ng
Ng
-32
-34
-36
-38
-40
-42
64
-42
-44
-44
-46
-46
-48
-48
DF-PLFM chips comprised of an up-chirp and a down-chirp
DF-PLFM chips comprised of an up-chirp and a down-chirp
0
|R
0
-20
0
-20
-70
0
3NgT
2NgT (a) Time
3NgT
[s]
(a) Time [s]
-70
0
0
-20
-20
-70
g
x',a
-28
(t-2N T)|
g
-30
4NgT
|R
(t)|
x',b
3NgT
3NgT
Time
(b) (b)
Time
[s][s]
2NgT
3NgT
3NgT
(c) Time [s]
-10
-34
4NgT
4NgT
-36
-20
-38
-30-30
-40-40
-50-50
-60-60
161632
-30
-32
-34
-20
|R(t)|
|R(t)|
2NgT
-32
-10
|Rx',b(t)|
2NgT
2NgT
0
0
4NgT
-20
-70
-70
2NgT
-28
(t-2N T)|
CCPa,b [dB]
-70
x',a
|R
CCPa,b [dB]
-20
64
32
N
4NgT
4NgT
32
64 64
128
N
8
16
128
8
16
Ng
32
Ng
-36
-38
-40
-40
-42
-42
64
-44
-44
-46
-46
-48
-48
(c) Time [s]
Figure 7. Column 1: (a–c) Plots in dB of the normalized matched filter (MF) outputs. Column 2: Plots
Figure 7. Column
Column 1:
matched
filter
(MF)
outputs.
Column
2: Plots
of
Figure
1: (a–c)
(a–c)Plots
PlotsinindB
dBofofthe
thenormalized
normalized
matched
filter
(MF)
outputs.
Column
2: Plots
of the corresponding cross-correlation sidelobe peaks (
) vs. code length ( ) vs. number of
thethe
corresponding
cross-correlation
sidelobe
peakspeaks
(CCP) (vs. code
discreteof
) vs. of
number
of
corresponding
cross-correlation
sidelobe
) vs.length
code (N
length
g ) vs. (number
discrete frequencies in the DF-LFM/DF-PLFM chips ( ).
frequencies
in the DF-LFM/DF-PLFM
chips (N).
discrete
frequencies
in the DF-LFM/DF-PLFM
chips ( ).
Clearly, adding Equations (37) and (39) results in complete cancellation of the cross-correlation
Clearly, adding
Equations
(37) and
and
(39)results
resultson
complete
cancellation
the
cross-correlation
Equations
(37)
(39)
inincomplete
ofofthe
cross-correlation
sidelobes.
Thus, achieving
complete
zero-sidelobes
either
sidecancellation
of
the mainlobe.
This
is shown in
sidelobes.
Thus,
achieving
complete
zero-sidelobes
either
ofFigure
the
mainlobe.
is shown
( )+
the plots
of achieving
( ), ′( ) complete
and
′( ) for
=on
8on
and
= side
16ofinthe
8. Frequency
selective
sidelobes.
Thus,
zero-sidelobes
either
side
mainlobe.
ThisThis
is shown
in thein
( could
fading
result
in 8inexact
of in
the8.
cross-correlation
sidelobes
the
plots
′( two
) Rand
= 8Ncancellation
and
16
Figure
8. Frequency
selective
plots
of Rof
t), R(0(),t)the
and
t) + R
0() t+
N) gfor
=
and
= 16 in=Figure
Frequency
selective
fading
(between
(bands
) for′(
thatbetween
are
totwo
thecould
zero-sidelobe
region.
However,
sidelobes
are sidelobes
very
fading
bands
couldinresult
in
inexact since
cancellation
of the cross-correlation
between
theadjacent
twothe
bands
result
inexact
cancellation
ofthe
thecross-correlation
cross-correlation
sidelobes
that are
small,
any
inexact
cancellation
due
to
fading
will
not
create
undesirable
high
magnitude
sidelobes.
adjacent
to the zero-sidelobe
region. However,
since thesince
cross-correlation
sidelobes
are veryare
small,
that
are adjacent
to the zero-sidelobe
region. However,
the cross-correlation
sidelobes
very
any inexact
cancellation
due0to fading
notwill
create
high magnitude
sidelobes.
small,
any inexact
cancellation
due to will
fading
notundesirable
create undesirable
high magnitude
sidelobes.
|R(t)|
-20
0
|R(t)|
-20
-70
-70
2NgT
0
-20
2NgT
0
-20-70
2NgT
0
-70
-20
4NgT
3NgT
(a) Time [s]
4NgT
|R'(t)|
|R'(t)|
2NgT
0
-70-20
3NgT
(a) Time [s]
2NgT
Figure
8. Plots in dB of: (a)
-70
2NgT
3NgT
(b) Time [s]
4NgT
3NgT
(b) Time [s]
3NgT
(c) Time [s]
| ( )|
; (b)
3NgT
(c) Time [s]
4NgT
4NgT
|
( )
|
|R(t)+R'(t)|
|R(t)+R'(t)|
; and (c)
4NgT
( )
| ( )
( )|
.
| ( )
| ( )|
| 0(t) |
( )|
Figure8.8.Plots
PlotsinindB
dBof:
of:(a)
(a)| R(t)| ; ;(b)
(b)| R | ; ;and
and(c)
(c)| R(t)+ R0(t)| ..
Figure
Ng
Ng
2Ng
Aerospace
Aerospace2017,
2017,4,4,28
28
13
13ofof24
24
(i.e., t
Thecomplementary
complementary
code
could
also
be interlaced
in the
time(i.e.,
domain
b T −T +
The
code
pairpair
could
also be
interlaced
in the time
domain
a Tt +
).
− ). This
scheme
results in a transmission
without
whilethe
still
using
the sameband
frequency
This scheme
results
in a transmission
without gaps,
while gaps,
still using
same
frequency
as in
the
previous
scheme.
Fishler
et al. [18]
and Deng
this[19]
scheme
usingthis
continuous
band
as in the
previous
scheme.
Fishler
et al. [19]
[18] describe
and Deng
describe
scheme LFM
using
and
PLFM
chips.
However,
for
this
interlaced
code,
the
zero-sidelobes
and
the
small
cross-correlation
continuous LFM and PLFM chips. However, for this interlaced code, the zero-sidelobes and the
sidelobes
in the final output
are interlaced
as output
well. are interlaced as well.
small cross-correlation
sidelobes
in the final
4.4.Doppler
DopplerProperties
Properties
Figure
Figure 9a,b
9a,b shows
shows the
the 3D
3D ambiguity
ambiguity function
function (AF)
(AF) of
of the
theproposed
proposed code
code constructed
constructed using
using
DF-LFM
chips
for
N
=
8
and
N
=
16
as
a
function
of
the
normalized
Doppler
frequency
f
T.
The
AF
g
d
DF-LFM chips for
= 8 and = 16 as a function of the normalized Doppler frequency
. The
can
be
represented
by,
AF can be represented by,
Z∞
R(t, f ) =
s (τ )s∗ (t − ∗τ )dτ
( ) ( − )
( , ) =f
−∞
(40)
(40)
where
≤≤t ≤≤3N
wheres f (t() )==s((t))e j2π f d t , for
, for0 0
3 g T, ,isisthe
thetransmitted
transmitted code
code Doppler
Doppler shifted by f d , T isisthe
the
duration
durationof
ofthe
theDF-LFM/DF-PLFM
DF-LFM/DF-PLFM chips and
and s((t))isisthe
theproposed
proposedcode.
code.
Figure
Figure9.9.Plots
Plotsof
of3D-ambiguity
3D-ambiguityfunction
functionin
in(a)
(a)linear
linearscale
scaleand
and(b)
(b)in
indB
dBscale
scaleof
ofthe
theproposed
proposedcodes
codes
constructed
using
DF-LFM
chips.
constructed using DF-LFM chips.
canbe
beobserved
observedthat
thatDoppler
Dopplershift
shiftadversely
adverselyaffects
affectsthe
thecancellation
cancellation of
ofthe
thesidelobes
sidelobesof
ofthe
the
ItItcan
complementarycode
codepair,
pair,resulting
resulting
the
disappearance
zero-sidelobe
domain
either
complementary
inin
the
disappearance
of of
thethe
zero-sidelobe
domain
on on
either
sideside
of
of the
mainlobe.
The ACF
mainlobes
the symmetrical
chips
shift in opposite
under
the
mainlobe.
The ACF
mainlobes
of the of
symmetrical
chips shift
in opposite
directionsdirections
under Doppler
Doppler
a result,
of the complementary
pair with
do not
align
with
eachthe
other.
Thus,
shift.
As ashift.
result,As
ACF
of theACF
complementary
code pair docode
not align
each
other.
Thus,
proposed
the proposed
codes
doDoppler
not have
good Doppler
way to properties
improve the
Doppler
codes
do not have
good
properties.
One wayproperties.
to improveOne
the Doppler
could
be to
properties
could
to use
triangularas
FM
based
as chips for
thepair
complementary
codecode
pair
use
triangular
FMbe
based
waveforms
chips
forwaveforms
the complementary
code
in the proposed
in the proposed
structure.
Another
improve could
the Doppler
properties
coulddiscussed
be to use
structure.
Anothercode
method
to improve
the method
Dopplerto
properties
be to use
the approach
the
approach
discussed
in [27]
to construct
Doppler resilient
Golay
complementary code pairs.
in
[27]
to construct
Doppler
resilient
Golay complementary
code
pairs.
5.5.Effect
Effectof
ofFrequency
FrequencySelective
SelectiveFading
Fading
In
of of
frequency
selective
fading,
the higher
frequencies
withinwithin
the frequency
sweep
Inthe
thepresence
presence
frequency
selective
fading,
the higher
frequencies
the frequency
(W
=
N
N
−
1
∆W∆T)
of
the
DF-LFM/DF-PLFM
waveforms
could
be
slightly
attenuated
compared
(
)
sweep ( = ( − 1)Δ Δ ) of the DF-LFM/DF-PLFM waveforms could be slightly attenuated
to
the lowertofrequencies.
However, However,
autocorrelations
of the DF-LFM/DF-PLFM
waveforms
remain
compared
the lower frequencies.
autocorrelations
of the DF-LFM/DF-PLFM
waveforms
identical,
since they
are they
symmetrical.
Hence, the
zero-sidelobe
region is not
affected.
remain identical,
since
are symmetrical.
Hence,
the zero-sidelobe
region
is notHowever,
affected.
the
adjacent
of region
small of
cross-correlation
sidelobes
could vary
depending
on the
However,
theregion
adjacent
small cross-correlation
sidelobes
couldslightly
vary slightly
depending
on
attenuation
due todue
fading.
However,
its effect
the error
in the
location
the mainlobe
negligibleis
the attenuation
to fading.
However,
itson
effect
on the
error
in theoflocation
of the ismainlobe
as
will be shown
simulation
laterresults
in this later
section.
negligible
as willinbethe
shown
in the results
simulation
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
and
respectively, where
<
. The DF-LFM chips ( ( ) and ( )) in the presence of fading can be represented by,
Aerospace 2017, 4, 28
14 of 24
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 α1 and α2 respectively, where α2 < α1 . The DF-LFM
chips (u f (t) and d f (t)) in the presence of fading can be represented by,
u f (t) =







N1 −1
√α1
∑ e j2π f [n]∆Wtn
∆T n=0
N −1
√α2
∑ e j2π f [n]∆Wtn
∆T n= N
(41)
1
d f (t) =
N −1− N1




√α2
∆T



√α1
∑ e j2π f [ N −1−n]∆Wtn
∆T n= N − N
1
∑
e j2π f [ N −1− N1 −n]∆Wtn
n =0
N −1
(42)
The DF-PLFM chips comprised of two up-chirp segments in the presence of fading can be
represented by,

N −1
2


 √α1 ∑ e j2π f1 [n]∆Wtn

∆T n=0
u f (t) =
(43)
N −1

j2π f 2 [ N −1−n]∆Wtn

√α2
e

∑

∆T
d f (t) =









n= N
2
N −1
2
√α2
∑ e j2π f2 [n]∆Wtn
∆T n=0
N −1
√α1
∑ e j2π f1 [ N −1−n]∆Wtn
∆T
n= N
2
(44)
The DF-PLFM chips comprised of an up-chirp and a down-chirp segment in the presence of
fading can be represented by,
u f (t) =
N −1
2
√α1
∑ e j2π f1 [n]∆Wtn
∆T n=0
N −1
√α2
∑ e j2π f2 [n]∆Wtn
∆T
n= N
2
(45)
N −1
2
N
√α2
∑ e j2π f2 [ 2 −1−n]∆Wtn
∆T n=0
N −1
√α1
∑ e j2π f1 [ N −1−n]∆Wtn
∆T
n= N
2
(46)









d f (t) =









For the DF-LFM waveforms, all the discrete frequencies 0 ≤ n ≤ N1 − 1, are considered as the
lower frequency component which are attenuated by α1 , while the discrete frequencies N1 ≤ n ≤ N − 1
will be considered as the higher frequency component which are attenuated by α2 . For the DF-PLFM
waveforms, all the discrete frequencies 0 ≤ n ≤ N2 − 1 are considered as the lower frequency
component and N2 ≤ n ≤ N − 1, the higher frequency component. Figure 10 shows the autocorrelations
of the DF-LFM chips with and without fading, i.e., u f (t) and d f (t) and u(t) and d(t) for N = 16,
N1 = 12, Ng = 16, α1 = 0.9792 and α2 = 0.8470 (α1 and α2 are randomly
generated).
Although,
fading results in reducing the mainlobe peaks of the autocorrelations, Ru f (t) and Rd f (t) (as seen in
the plots), they remain identical.
Aerospace 2017, 4, 28
Aerospace 2017, 4, 28
15 of 24
15 of 24
( ) for
= 16 ,
= 12 ,
= 16 ,
= 0.9792 and
= 0.8470 (
and
are randomly
(
)
for
=
16
,
=
12
,
=
16
,
=
0.9792
and
=
0.8470
(
and
are
randomly
generated). Although, fading results in reducing the mainlobe peaks of the autocorrelations,
Aerospace 2017, 4, 28
15 of 24
generated).
Although,
fading
results
in
reducing
the
mainlobe
peaks
of
the
autocorrelations,
( )| and |
( )| (as seen in the plots), they remain identical.
|
( )| and |
( )| (as seen in the plots), they remain identical.
|
0
0
0
0
-20
-20
-20
|Ru(t)|
|Ru(t)|
f
-60
-60
-60
-60
0
0
-20
-20
|Ru (t)|
f
|Ru (t)|
-20
-0.5T
0
0.5T
-0.5T (a) Time
0 [s] 0.5T
(a) Time [s]
-0.5T
0
0.5T
-0.5T (c) Time
0 [s] 0.5T
(c) Time [s]
0
0
-20
|Rd(t)|
|Rd(t)|
|Rd (t)|
f
|Rd (t)|
-20
f
-60
-60
-60
-60
-0.5T
0
0.5T
-0.5T (b) Time
0 [s] 0.5T
(b) Time [s]
-0.5T
0
0.5T
-0.5T (d) Time
0 [s] 0.5T
(d) Time [s] ( )| and |
( )|.
Figure 10. (a,b) Plots in dB of | ( )| and | ( )|; and (c,d) plots of |
Figure 10. (a,b) Plots in dB of | Ru (t)| and | Rd (t)|; and (c,d) plots of Ru f (t) and Rd f (t).
( )| and |
( )|.
Figure 10. (a,b) Plots in dB of | ( )| and | ( )|; and (c,d) plots of |
Figure 11 shows the matched filter outputs,
( ) (given
, ( ) and
, ( ), and their sum
Figure 11
shows
matched
filter
R x0 ,apresence
and
their
sum
R(t) ((given
by
(,t)(, and
11(30),
shows
the
matched
filteroutputs,
outputs,
andRofx0 ,b
), and
their
sum
)Clearly,
(given
,( t() )
by Equations
(33)the
and
(37), respectively),
in the
frequency
selective
fading.
Equations
(30),
(33)
and
(37),
respectively),
selective
by
(30),
(33)
and
(37),
respectively),
inthe
thepresence
presence
offrequency
frequency
selective fading. Clearly,
theEquations
zero-sidelobe
region
on
either
side of the in
mainlobe
is not of
affected
by fading.
the
the zero-sidelobe
zero-sidelobe region
region on
on either
either side
side of
of the
the mainlobe
mainlobeisisnot
notaffected
affectedby
byfading.
fading.
0
|Rx',a(t-2NgT)|
|Rx',a(t-2NgT)|
0
-20
-20
-70
-70
0
2NgT
2NgT
3NgT
(a) 3NgT
Time [s]
4NgT
4NgT
|Rx',b(t)|
|Rx',b(t)|
(a) Time [s]
0
-20
-20
-70
-70
0
2NgT
2NgT
3NgT
(b) 3NgT
Time [s]
4NgT
4NgT
|R(t)|
|R(t)|
(b) Time [s]
0
-20
-20
-70
-70
2NgT
2NgT
3NgT
(c) 3NgT
Time [s]
4NgT
4NgT
(c) Time [s]
,
Figure 11. Plots in dB of: (a)
; (b)
R x0 ,a, (t−2Ng T ); (b)
Figure
11.
Plots
in
dB
of:
(a)
Figure
11.
Plots
in
dB
of:
(a)
; (b)
selective fading.
Ng
selective
selective fading.
fading.
,
( )
| ( )|
; and (c)
, in the presence of frequency
| ( )|
| R(t)|
; ; and
of frequency
frequency
and (c)
(c) 2Ng , , in
in the
the presence
presence of
( )
R x,0 ,b (t)
Ng
Root mean square error (
) in the location of the mainlobe is used as a metric to measure of
Root
mean
square
error
(
)
of
mainlobe
is used
used
as aa metric
metric
to measure
measure
of
effectiveness
sidelobe
in location
the
presence
frequency
selective
fading
and noise.
Root meaninsquare
errorcancellation
(RMSE) in
in the
the
location
of the
the of
mainlobe
is
as
to
of
effectiveness
in
sidelobe
cancellation
in
the
presence
of
frequency
selective
fading
and
noise.
Location of the
mainlobecancellation
of ( ) is given
effectiveness
in sidelobe
in the by,
presence of frequency selective fading and noise. Location
Location
of
the
mainlobe
of
(
)
is
given
by,
( )
of the mainlobe of R(t) is given by,
̂ = argmax
(47)
R(t) 2( )
̂ = argmax
(47)
t̂ = argmax
(47)
2Ng 2
t
where ( ) is the final output, given by Equation (37).
where R((t))isin
isthe
thefinal
finaloutput,
output,
given
Equation
(37).
where
given
bybyEquation
the
location
of the
mainlobe
can be(37).
represented
by,
represented by,
by,
RMSE in the location of the mainlobe can be represented
1
RMSE =
T
1
K
!0.5
K
∑
i =1
t̂i − t
2
(48)
Aerospace 2017, 4, 28
16 of 24
Aerospace 2017, 4, 28
=
.
1 1
16 of 24
(̂ − )
(48)
actual
timing
of of
thethe
mainlobe,
t̂i is ̂ the
estimated
timing
of the
is the
estimated
timing
of
where K isisthe
thenumber
numberofoftrials,
trials,t isisthe
the
actual
timing
mainlobe,
mainlobe
for
the
ith
trial
in
the
presence
of
noise
and
T
is
the
chip
duration.
the mainlobe for the ith trial in the presence of noise and
is the chip duration.
RMSE vs.
vs. signal-to-noise
signal-to-noise ratio
ratio (SNR)
(SNR) plots
plots for
for the
the proposed
proposed codes are shown in Figure 12.
DF-LFM chips under frequency selective fading
1
10
Ng=16, N=32,
Without fading
Ng=16, N=32,
With fading
0
10
-1
RMSE
10
-2
10
-3
10
-4
10
-5
10
-4
-2
0
2
4
SNR [dB]
6
8
10
12
DF-PLFM chips comprised of two up-chirp segments under frequency selective fading
1
10
0
Ng=16, N=32,
Without fading
-1
Ng=16, N=32,
With fading
10
10
RMSE
-2
10
-3
10
-4
10
-5
10
-6
10
-4
-2
0
2
4
SNR [dB]
6
8
10
12
DF-PLFM chips comprised of an up-chirp and a down-chirp segment under frequency selective fading
10
10
RMSE
10
10
10
10
10
10
1
Ng=16, N=32,
Without fading
0
Ng=16, N=32,
With fading
-1
-2
-3
-4
-5
-6
-4
-2
0
2
4
SNR [dB]
6
8
10
12
Figure 12.
12. Plots
Plots of
of root
root mean
mean square
square error
error (RMSE)
(
) vs.
Figure
vs. SNR
SNR for
for the
the proposed
proposed codes.
codes.
SNR is given by 20 log 2Ng, where 2
is the code energy and
represents the noise power.
SNR is given by 20 log10 σ2 , where 2Ng is the code energy and σ2 represents the noise power.
andK ==1000
1000iterations
iterations
are
performed
each
SNR
value.
and
SNR
SNR is
is varied
varied by
by changing
changing σ2 and
are
performed
forfor
each
SNR
value.
α1 and
α2
are
randomly
chosen
such
that
0.6
≤
,
≤
1.
From
these
plots
it
can
be
observed
that
the
are randomly chosen such that 0.6 ≤ α1 , α2 ≤ 1. From these plots it can be observed that the RMSE
curve
with stays
fadingvery
stays
very
to thein the absence
in the absence
ofThis
fading.
This
trueDF-LFM
for the
curve with
fading
close
to close
the RMSE
of fading.
is true
foristhe
DF-LFM
and
the
DF-PLFM
chips.
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
Aerospace 2017, 4, 28
17 of 24
Aerospace 2017, 4, 28
17 of 24
code
as shown
code pair
pair at
attwo
twofrequencies/channels
frequencies/channels as
shown in
in Figure
Figure 1.
1. In
In the
the presence
presence of
of frequency
frequency selective
selective
fading,
the
cancellation
of
sidelobes
is
inexact
which
results
in
increasing
the
probability
of falseofalarm.
fading, the cancellation of sidelobes is inexact which results in increasing the probability
false
This
canThis
be seen
plots
of RMSE
for a Golay
complementary
code pair (A,
B)
alarm.
can in
bethe
seen
in the
plots ofvs. SNR in
vs.Figure
SNR 13
in Figure
13 for
a Golay complementary
code
of
length
N
=
16
and
64.
A
is
attenuated
by
α
and
B
is
attenuated
by
α
such
that
0.6
≤
α
,
α
≤
1.
g
2 that
1
1such
pair ( , ) of length
= 16 and 64.
is attenuated
by
and
is 2attenuated by
It0.6
can
be
observed
that
frequency
selective
fading
results
in
increasing
the
RMSE
in
the
location
of
the
≤ ,
≤ 1. It can be observed that frequency selective fading results in increasing the RMSE in
mainlobe.
Theofproposed
codes The
are resistant
to codes
the effect
frequency
selective
as shown
in the
the location
the mainlobe.
proposed
areof
resistant
to the
effectfading,
of frequency
selective
RMSE
plots
in
Figure
12.
fading, as shown in the
plots in Figure 12.
1
10
Without fading, Ng=16
With fading, Ng=16
0
10
Without fading, Ng=64
-1
With fading, Ng=64
RMSE
10
-2
10
-3
10
-4
10
-5
10
-4
-2
0
2
4
6
8
SNR [dB]
10
12
14
16
Figure13.
13.Plot
Plotof
ofroot
rootmean
meansquare
squareerror
error(RMSE)
(
) vs.
vs. signal-to-noise
signal-to-noise ratio
ratio (SNR)
(SNR) for
for complementary
complementary
Figure
codepair
pairusing
usingsinusoidal
sinusoidalchip.
chip.
code
6. Good Code Sets from Complementary Pair Using DF-LFM and DF-PLFM Waveforms as Chips
6. 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
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
the plots in Figure 3) of the DF-LFM/DF-PLFM waveforms used as chips for the same complementary
complementary code pair. The parameter , as shown in the Equations (20)–(25), is varied carefully
code pair. The parameter γl , as shown in the Equations (20)–(25), is varied carefully to result in DF-LFM
to result in DF-LFM and DF-PLFM chips with different slopes. The resulting good code set has a
and DF-PLFM chips with different slopes. The resulting good code set has a zero domain on either
zero domain on either side of the mainlobe and an adjacent region of small sidelobes. The
side of the mainlobe and an adjacent region of small sidelobes. The cross-correlation between any code
cross-correlation between any code pair in the set is very small. The lth code in the set can be
pair in the set is very small. The lth code in the set can be represented by,
represented by,
sl (t) =(a)l (t) + (bl) t − 2Ng T
(49)
=
+ ( −2
)
(49)
Ng
Ng
) =∑∑ a[n]u[ (]t −( (n−−
( 1−
1) )u,l (tand
( )
whereal (t() =
where
, and
represent
) T1)
), bl)(,t) =( )∑=b∑
[n]dl (t[−] (n(−−1)(T )−
) and d(l ()t) and
l
n
=
1
n
=
1
represent the two symmetrical DF-LFM/DF-PLFM waveforms.
the two symmetrical DF-LFM/DF-PLFM waveforms.
( ) and ( ) for the good code sets from complementary pairs using DF-LFM chips can be
ul (t) and dl (t) for the good code sets from complementary pairs using DF-LFM chips can be
represented
by,
represented by,
1 N −1
ul (t() )= √ 1 ∑ e j2π f l[[n]]∆Wtn
(50)
= ∆T
(50)
n
=
0
√Δ
dl (t) = √
where f l [n] = round
2γl ( N −1)n
N
(
)
1
N −1
∆T
1
n =0
∑
e j2π f l [ Nl −1−n]∆Wtn
]
[
( )=
√Δ
for n = 0, 1, 2, . . . , N − 1, γl = k +
(51)
(l −1)(1−3k)
,
L −1
(
)(
)
(51)
l = 0, 1, 2, . . . , L − 1,
1
kwhere
∈ (0, 0.5[], ]∆W
and n∆T ≤ for
tn ≤ (n=+0,11,
)∆T.
= = ∆T
2, … , − 1 ,
= +
, = 0, 1, 2, … , − 1 ,
u
t
and
d
t
for
the
good
code
sets
from
complementary
pairs
using
DF-PLFM
chips comprised
(
)
(
)
l
l
and Δ ≤ ≤ ( + 1)Δ .
∈ (0, 0.5], Δ =
of two up-chirps 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,
Aerospace 2017, 4, 28
18 of 24
ul (t) =









dl (t) =









N −1
2
1
√
∑ e j2π f l,1 [n]∆Wtn
∆T n=0
N −1
√1
∑ e j2π f l,2 [ N −1−n]∆Wtn
∆T
n= N
2
(52)
N −1
2
√1
∑ e j2π f l,2 [n]∆Wtn
∆T n=0
N −1
√1
∑ e j2π f l,1 [ N −1−n]∆Wtn
∆T
n= N
2
(53)
= round 2γl ( NN−1)n for n = 0, 1, 2, . . . , N2 − 1, γl = k + (l −1L)(−11−2k) ,
(1−γ )2n
f l,2 [n] = round ( N − 1) 1 − N −l 1
for n = N2 , N2 + 1, . . . , N − 1 and k ∈ (0, 0.5].
ul (t) and dl (t) for the good code sets from complementary pairs using DF-PLFM chips comprised
of an up-chirp and a down-chirp are given by,
where f l,1 [n]
ul (t) =
N −1
2
√1
∑ e j2π f l,1 [n]∆Wtn
∆T n=0
N −1
√1
∑ e j2π f l,2 [n]∆Wtn
∆T
n= N
2
(54)
N −1
2
√1
∑ e j2π f l,2 [ Nl −1−n]∆Wtn
∆T n=0
N −1
√1
∑ e j2π f l,1 [ N −1−n]∆Wtn
∆T
n= N
2
(55)









dl (t) =









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 (ASPl ) of the lth code in the set
is given by,
ASPl = CCPal ,bl
(56)
where CCPal ,bl is the same as CCPa,b which is defined in Equation (38) and shown in the plots
in Figure 7.
Maximum cross-correlation peak (MCCP) and average cross-correlation peak (ACCP) are used
as measures to compare the different good code sets introduced in this paper.
MCCP = max
max Rsl ,sk (t)
(57)
l, k =1,2,..,L
t
∑lL=1 ∑kL=l max Rsl ,sk (t)
t
!
ACCP =
L
2
(58)
where Rsl ,sk (t) is the normalized cross-correlation between the code pair {l, k} in the set.
The table (Column 2) in Figure 14 shows the plots of MCCP and ACCP for each set for different
values of N, Ng and L.
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19 of 24
19 of 24
(Column 1) Plots of
( ),
( ) vs. time
( ( ) are shown in solid lines and ( ) in
lines with dashes)
(Column 2) Plots of
and
different values of
and
vs.
for
Good code sets from complementary pairs using DF-LFM waveforms as chips
0
0
N=64, Ng=8
-10
-20
-30
-40
4
5
0
0
NΔT
= 0.24 and
MCCP [dB]
0
-10
7
5
6
(c) L
0
N=128, Ng=16
-30
6
(b) L
-30
-40
4
N=64, Ng=16
5
N=128, Ng=8
-20
8
-20
-40
4
= 32
6
(a) L
N=64, Ng=8
-10
ACCP [dB]
N=128, Ng=8
ACCP [dB]
MCCP [dB]
Instantaneous frequency [Hz]
(N-1)ΔW
7
8
N=64, Ng=16
-10
N=128, Ng=16
-20
-30
-40
4
8
7
5
6
(d) L
7
8
Good code sets from complementary pairs using DF-PLFM waveforms
comprised of two up-chirps as chips
-20
-30
-40
4
0
(N/2)ΔT
= 0.24 and
NΔT
MCCP [dB]
0
0
-10
5
6
(a) L
7
N=128, Ng=16
-30
6
(b) L
N=128, Ng=8
-20
-30
5
6
(c) L
0
N=64, Ng=16
5
N=64, Ng=8
-10
-40
4
8
-20
-40
4
= 32.
N=128, Ng=8
ACCP [dB]
MCCP [dB]
Instantaneous frequency [Hz]
-10
0
N=64, Ng=8
ACCP [dB]
0
(N-1)ΔW
7
8
N=64, Ng=16
-10
N=128, Ng=16
-20
-30
-40
4
8
7
5
6
(d) L
7
8
Good code sets from complementary pairs using DF-PLFM waveforms
comprised of an up-chirp and a down-chirp as chips
0
-10
-10
-20
N=64, Ng=8
-30
-40
4
N=128, Ng=8
5
0
(N/2)ΔT
= 0.24 and
NΔT
= 32
MCCP [dB]
0
0
ACCP [dB]
0
6
(a) L
7
-20
-40
4
N=64, Ng=16
N=128, Ng=16
5
6
(b) L
N=128, Ng=8
-30
5
0
-10
-30
N=64, Ng=8
-20
-40
4
8
ACCP [dB]
MCCP [dB]
Instantaneous frequency [Hz]
(N-1)ΔW
7
8
6
(c) L
7
8
N=64, Ng=16
-10
N=128, Ng=16
-20
-30
-40
4
5
6
(d) L
7
Figure 14.
14. Plots
Plots of
of instantaneous
instantaneousfrequencies,
frequencies,i.e.,
i.e.,f (t() and
) and
) time
vs. time
in Column
1;(a,b)
and Plots
(a,b)
Figure
f dl (t) (vs.
in Column
1; and
ul
Plots
of
maximum
cross-correlation
peaks
(
)
and
(c,d)
average
cross-correlation
peaks
(
of maximum cross-correlation peaks (MCCP) and (c,d) average cross-correlation peaks (ACCP) vs.)
) and
number
of discrete
frequencies
vs. number
of codes
), various
for various
values
of code
length
number
of codes
(L), (for
values
of code
length
(Ng )( and
number
of discrete
frequencies
(N)
(in )Column
in Column
2.
2.
8
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20 of 24
From the plots in Figure 14, it can be observed that the good code sets constructed using DF-LFM
chips have better MCCP and ACCP properties compared to those constructed using DF-PLFM chips.
However, the time bandwidth products of all the codes in the sets constructed using DF-PLFM chips is
the same (i.e., N ( N − 1)∆W∆T), while those of the codes in the set constructed using DF-LFM chips
are different. In addition, for a given complementary code pair code length (Ng ), the cross-correlation
sidelobe peaks could be reduced by increasing the number of discrete frequencies (N) in the chips.
7. Construction of a Good Code Set from the Mates of a Complementary Code Pair
Consider the complementary code pair: B z−1 and − A z−1 . The sum of the cross-correlation
of A(z) with B z−1 and that of B(z) with − A z−1 will be,
A(z) B(z) − B(z) A(z) = 0
(59)
Thus, the code pair: B z−1 and − A z−1 , are orthogonal to the complementary code pair: A(z)
and B(z). Such code pairs are called complementary code pair mates, as described in [23,24].
Let S1 (z) = A z Nc C (z) + z−D B z Nc γC z−1 and S2 (z) = B z− Nc C (z) − z− D A z− Nc γC z−1
be the codes constructed by concatenating the complementary code pair mates (A(z), B(z) and B z−1 ,
− A z−1 ) using C (z) and its symmetrical/anti-symmetrical mirror image (γC z−1 , where γ = ±1)
as chips, as described in Section 2. Their autocorrelations can be represented by,
RS1 (z) = γR A,B z Nc C2 (z) + 2Ng z− D Rc (z) + γz−2D R B,A z Nc C2 z−1
RS2 (z) = γR A,B z− Nc C2 (z) + 2Ng z− D Rc (z) − γz−2D R B,A z− Nc C2 z−1
(60)
(61)
Both the autocorrelations consist of a zero-sidelobe region on either side of the mainlobe and
an adjacent region of small cross-correlation sidelobes, as shown in Equation (37) in Section 2.
Their autocorrelation sidelobe peaks (i.e., CCP, given by (38)) vary with N and Ng as shown in
the plots in Figure 7. The cross-correlation between S1 (z) and S2 (z) can be represented by,
RS1 ,S2 (z) = A z Nc C (z) + z− D B z Nc γC z−1
B z Nc C z−1 − z D A z Nc γC z1
(62)
Simplifying Equation (62) results in,
RS1 ,S2 (z) = − A2 z Nc C2 (z) + 0 + γz−2D B2 z Nc C2 z−1
(63)
Clearly, the cross-correlation consists of a zero-sidelobe region and an adjacent region of small
cross-correlation sidelobes. Thus, S1 (z) and S2 (z) are quazi-orthogonal in the convolution sense.
The zero-sidelobe domain in the cross-correlation is due to the symmetry/anti-symmetry property of
the chips. Since, the symmetry/anti-symmetry property is immune to frequency selective fading, the
zero-sidelobe domain in the cross-correlation is also immune to frequency selective fading. Figure 15
show the autocorrelations S1 (z) and S2 (z) and their cross-correlation. For these plots, S1 (z) and
S2 (z) are constructed using DF-LFM chips with N = 8 and Ng = 16. These plots clearly show
the zero-sidelobe domain in the cross-correlation between the proposed code constructed using
a complementary code pair and the one constructed using its mate.
Varying the slopes of the DFCW chips in both S1 (z) and S2 (z), as shown in Section 6, results in two
good code sets. Since these two sets are quasi-orthogonal to each other, they could be combined to form
a larger set with twice the number of codes, while maintaining the same cross-correlation properties
as those of the individual sets. This can be seen in the plots of maximum cross-correlation peaks
(MCCP) and average cross-correlation peaks (ACCP) in Column 1 of the table in Figures 16 and 17.
When compared to the plots in Figure 14, clearly the cross-correlation properties remain the same,
while doubling the number of the codes in the set.
Aerospace 2017, 4, 28
Aerospace 2017,
2017, 4,
4, 28
28
Aerospace
21 of 24
21 of
of 24
24
21
00
-20
-20
|R (z)|
(z)|
|R
S1
S1
-70
-70
2NgT
2NgT
3NgT
3NgT
(a)Time
Time [s]
[s]
(a)
4NgT
4NgT
3NgT
3NgT
(b)Time
Time [s]
[s]
(b)
4NgT
4NgT
3NgT
3NgT
(c) Time
Time [s]
[s]
(c)
4NgT
4NgT
00
-20
-20
-70
-70
2NgT
2NgT
00
-20
-20
-70
-70
2NgT
2NgT
|RS2(z)|
(z)|
|R
S2
|RS1,S2(z)|
(z)|
|R
S1,S2
Figure15.
15.Plots
Plotsinin
indB
dBof:
of:(a)
(a)RS1 ((z());
(b)RS2 (((z)););
and(c)
(c)RS1 ,S, 2 (((z)),),, for
for N =
and Ng =
==16
16
using
Figure
;);(b)
; and
= 888 and
16using
using
Figure
15.
Plots
dB
of:
(a)
(b)
and
(c)
for
,
DF-LFM
chips.
DF-LFM
chips.
DF-LFM chips.
12
14
16
12 L
14
16
(a)
(a) L
N=64, Ng=16
N=64, Ng=16
N=128, Ng=16
N=128, Ng=16
10
10
12
12 L
(b)
(b) L
14
14
16
16
-40
-40 8
8
0
0
-10
-10
-20
-20
-30
-30
-40
-40 8
8
10
10
12
14
16
12 L
14
16
(c)
(c) L
N=64, Ng=16
N=64, Ng=16
N=128, Ng=16
N=128, Ng=16
10
10
12
12 L
(d)
(d) L
14
14
16
16
0
0
-10
-10
-20
-20
-30
-30
-40
-40
-50
-50 4
4
0
0
-10
-10
-20
-20
-30
-30
-40
-40
-50
-50 4
4
N=64, Ng=8
N=64, Ng=8
N=128, Ng=8
N=128, Ng=8
5
5
6
7
8
6L
7
8
(a)
(a) L
N=64, Ng=16
N=64, Ng=16
N=128, Ng=16
N=128, Ng=16
5
5
6
6 L
(b)
(b) L
7
7
0
-10
-10
-20
-20
-30
-30
-40
-40
-50
-50 4
4
0
0
-10
-10
-20
-20
-30
-30
-40
-40
-50
-50 4
4
N=64, Ng=8
N=64, Ng=8
N=128, Ng=8
N=128, Ng=8
ACCP
ACCP
MCCP
MCCP
N=64, Ng=8
N=64, Ng=8
N=128, Ng=8
N=128, Ng=8
ACCP
ACCP
10
10
0
-10
-10
-20
-20
-30
-30
MCCP
MCCP
-40
-40 8
8
0
0
-10
-10
-20
-20
-30
-30
-40
-40 8
8
ACCP
ACCP
N=64, Ng=8
N=64, Ng=8
N=128, Ng=8
N=128, Ng=8
0
-10
-10
-20
-20
-30
-30
ACCP
ACCP
MCCP
MCCP
MCCP
MCCP
(Column 1)
1) Plots
Plots of
of
and
vs.
(Column 2)
2) Plots
Plots of
of
and
vs.
(Column
and
vs.
(Column
and
vs.
for the
the good
good code
code set
set with
with twice
twice the
the number
number of
of
for the
the good
good codes
codes set
set with
with better
better
for
for
codes
cross-correlation
properties
codes
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-LFM waveforms
as chips
0
0
0
8
8
5
5
6
7
8
6L
7
8
(c)
(c) L
N=64, Ng=16
N=64, Ng=16
N=128, Ng=16
N=128, Ng=16
5
5
6
6L
(d)
(d) L
7
7
8
8
Good code sets from complementary pairs and their mates using DF-PLFM waveforms
Good code sets from complementary
pairs
their mates
using DF-PLFM waveforms
comprised of
twoand
up-chirps
as chips
comprised of two up-chirps
as chips
0
0
0
0
12
(a)
12 L
10
L
N=64,(a)
Ng=16
14
16
14
16
N=128, Ng=16
10
10
12
(b)
12 L
(b) L
14
16
14
16
-30
-40
8
-40
8
0
10
10
12
(c)
12 L
14
16
14
16
N=128, Ng=16
10
10
12
(d)
12 L
(d) L
-30
-40
5
5
0
-10
N=64,
NgN=16
N=128,
=16
g
-20
-30
-30
-40
8
-40
8
N=128, Ng=8
-20
-30
-40
-50
4
-50
04
(c) L
N=64, Ng=16
-10
-20
-10
-20
14
16
14
16
ACCP
ACCP
MCCP
MCCP
-20
-30
0
-10
N=64,
NgN=16
N=128,
=16
g
-20
-30
-30
-40
8
-40
8
N=128, Ng=8
N=128, Ng=16
-20
-30
-30
-40
-40
-50
4
-50
4
5
5
6
(b)
6L
(b) L
7
8
7
8
N=64,
NgN=8
N=128,
=8
g
-10
-20
N=128, Ng=8
-20
-30
-30
-40
-40
-50
4
-50
04
6
7
8
(a) L
6
7
8
(a) LN=64, N =16
g
N=64,
NgN=16
N=128,
=16
g
-10
-20
N=64, Ng=8
0
-10
N=64,
NgN=8
N=128,
=8
g
5
5
0
-10
ACCP
ACCP
-10
-20
10
-10
-20
N=64, Ng=8
0
-10
N=64,
NgN=8=8
N=128,
g
MCCP
MCCP
0
-10
MCCP
MCCP
N=128, Ng=8
-20
-30
-30
-40
8
-40
08
N=64, Ng=8
0
-10
ACCP
ACCP
-10
-20
N=64, Ng=8
N=64,
NgN=8
N=128,
=8
g
ACCP
ACCP
MCCP
MCCP
0
-10
6
7
8
(c) L
6
7
8
(c) L
N=64, Ng=16
N=64,
NgN=16
N=128,
=16
g
-10
-20
N=128, Ng=16
-20
-30
-30
-40
-40
-50
4
-50
4
Figure 16. Plots of (a,b)
vs.
and (c,d)
vs. .
Figure 16. Plots of (a,b)
and (c,d)
Figure
16. Plots of (a,b) MCCPvs.
vs. L and
(c,d) ACCP vs.
vs. L..
5
5
6
(d)
6 L
(d) L
7
8
7
8
Aerospace 2017, 4, 28
22 of 24
Aerospace 2017, 4, 28
22 of 24
(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-PLFM waveforms
comprised of an up-chirp and a down-chirp as chips
-40
8
-10
14
16
10
0
N=64, Ng=16
N=128, Ng=16
12
(c) L
14
16
-20
-40
8
-40
8
-50
4
16
10
12
(d) L
14
16
7
-30
-50
4
8
5
0
6
(c) L
7
8
N=64, Ng=16
-10
N=128, Ng=16
-30
-40
14
6
(a) L
N=64, Ng=16
-20
-30
12
(b) L
5
-10
N=128, Ng=16
N=128, Ng=8
-20
-40
0
-30
10
ACCP
MCCP
-30
-50
4
N=64, Ng=16
-10
-20
-20
ACCP
-40
8
N=64, Ng=8
-10
N=128, Ng=8
-40
MCCP
-30
12
(a) L
N=128, Ng=8
0
N=64, Ng=8
-10
-20
-30
10
0
N=64, Ng=8
-10
ACCP
MCCP
N=128, Ng=8
-20
0
MCCP
0
N=64, Ng=8
ACCP
0
-10
N=128, Ng=16
-20
-30
-40
5
6
(b) L
7
8
-50
4
5
6
(d) L
7
8
Figure 17. Plots of (a,b)
vs.
and (c,d)
vs. L.
Figure 17. Plots of (a,b) MCCP vs. L and (c,d) ACCP vs. L.
If a better good code set with the smaller
and
is required, then the best
If a better
good
code
withbethe
smaller
and
ACCPbyisusing
required,
then
best
candidates
from
the two
setsset
could
selected.
ThisMCCP
could be
achieved
only half
thethe
slope
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.,
options
for the DFCW waveforms
the
complementary
code pair
its in
mates
the DF-LFM/DF-PLFM
waveformsused
with as=chips
0, 2, …for
, −
1 in
Equations (52)–(57).
This and
results
a
i.e.,good
the DF-LFM/DF-PLFM
waveforms
0, 2, . . . , values,
L − 1 inasEquations
This
results2 in
code set with significantly
betterwith l =and
shown in(52)–(57).
the plots in
column
of thecode
tablesetinwith
Figures
16 and 17.
These
are significantly
than
the in the and
a good
significantly
better
MCCP
and ACCP smaller
values, as
shown
plotsthe
in column
values
in Figure
14 for the
same17.
number
codes
in the set. smaller
Thus, constructing
the proposed
2 of
the table
in Figures
16 and
Theseofare
significantly
than the MCCP
and thecodes
ACCP
fromin
complementary
codesame
pair and
their of
mates,
while
using
same
chips for both
offers codes
the
values
Figure 14 for the
number
codes
in the
set. the
Thus,
constructing
thepairs,
proposed
aforementioned
flexibility
of choosing
between
betterusing
good the
codesame
set and
a larger
good
codeoffers
set. the
from
complementary
code pair
and their
mates, awhile
chips
for both
pairs,
Table 1 shows
the average
autocorrelation
average
values
for the
Deng’s
aforementioned
flexibility
of choosing
between aand
better
good cross-correlation
code set and a larger
good
code
set.
polyphase
[28]
good
code
set
and
the
Deng’s
discrete
frequency
[20]
set
designs.
The
autocorrelation
Table 1 shows the average autocorrelation and average cross-correlation values for the Deng’s
and cross-correlation
sidelobe
peaks
of the codes
introduced
in [20]
this paper
are smaller,
as shown in
polyphase
[28] good code
set and
the Deng’s
discrete
frequency
set designs.
The autocorrelation
the plots in Figures 7 and 17 respectively. The proposed codes constructed via concatenating biphase
and cross-correlation sidelobe peaks of the codes introduced in this paper are smaller, as shown
Golay complementary code pairs and using discrete frequency chips results in a zero-sidelobe
in the plots in Figures 7 and 17 respectively. The proposed codes constructed via concatenating
domain and an adjacent region of very small cross-correlation sidelobes. In [28], it is shown that the
biphase Golay complementary code pairs and using discrete frequency chips results in a zero-sidelobe
average autocorrelation sidelobe and cross-correlation peaks decrease with increase in code length
domain and an adjacent region of very small cross-correlation sidelobes. In [28], it is shown that the
and approach ( ) for large . Their discrete frequency good code set [20] being wideband,
√
average autocorrelation
sidelobe and cross-correlation peaks decrease with increase in code length and
smaller
achieve much
sidelobe peaks. In the case of the good code set design proposed in this paper,
1
approach O √
for large N. Their discrete frequency good code set [20] being wideband, achieve
N length ( ), increasing the number of discrete frequencies ( ) in the chips results in
for a given code
much smaller sidelobe peaks. In the case of the good code set design proposed in this paper, for a given
a better code set.
code length (Ng ), increasing the number of discrete frequencies (N) in the chips results in a better
code set.
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.
Table 1. Table containing the average autocorrelation and cross-correlation peaks for the proposed set
Average
in comparison with Deng’s polyphase and discrete frequency Average
code sets.
Good Code Set
Good Code Set
Autocorrelation
Average Autocorrelation
Peak (dB)
Peak (dB)
Deng’s polyphase set
Deng’s
polyphase
set
(code
length
= 128 and
= 3 codes)
(code length = 128 and L = 3 codes)
Deng’s discrete frequency set
Deng’s discrete
frequency
setand = 3 codes)
(# of discrete
frequencies
= 128
(# of discrete frequencies = 128 and L = 3 codes)
= 16,
Proposed code set with
Proposed code=
set32
with
Ng ==
16,4
and
N = 32 and L = 4
−20.9606
−20.9606
−32.2641
−32.2641
−39.8280
−39.8280
Cross-correlation
Average Peak
Cross-Correlation
(dB)
Peak (dB)
−19.1525
−19.1525
−32.2522
−32.2522
−37.9926
−37.9926
Aerospace 2017, 4, 28
23 of 24
8. 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 DFCW chips based on LFM
and PLFM waveforms. It is also shown that mates of the complementary code pair could be used
to generate a second good code set using the same DFCW chips. These two sets are shown to be
quasi-orthogonal with a zero-sidelobe domain. Thus, resulting in a larger good code set with twice the
number of codes, while maintaining the same cross-correlation properties. Or a better good code set
could be constructed by choosing the best candidates from the two sets.
As a topic for further research, the proposed codes could be constructed using polyphase
versions (as shown in [29,30]) of the LFM/PLFM waveforms as chips with the complementary
code pairs and their mates. Exploring other waveforms or codes that satisfy the two conditions
of symmetry/anti-symmetry and quazi-orthogonality and improving the Doppler properties of these
codes are the other topics of future research.
Author Contributions: Adly T. Fam and Ravi Kadlimatti conceived and designed the proposed good code set.
Ravi Kadlimatti performed the simulations.
Conflicts of Interest: The authors declare no conflict of interest.
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