This paper considers the problem of waveform synthesis given a desired power spectrum. The properties of the designed waveforms are such that the overall system performance is increased. The metric used to evaluate the optimality of the synthesized time domain signals is the peak-to-average power ratio (PAPR). We discuss how to synthesize waveforms using the technique of partial transmit sequence (PTS). The key point is that the gradient can explicitly be derived from the objective function. Furthermore, the result is extended by allowing the power spectrum to deviate from its original shape, yielding a further reduction in the PAPR. The method is applied to derived power spectra for wideband multiple-input-multiple-output (MIMO) radar. It is shown that the proposed technique can achieve optimal or near optimal performance with PAPR below 0.5 dB

Ström, Marie

Viberg, Mats

Marie Strom

Mats Viberg

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Low PAPR waveform synthesis with application to wideband MIMO radar

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Low PAPR waveform synthesis with application to wideband MIMO radar
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Ström, M. ; Viberg, M. (2011) "Low PAPR waveform synthesis with application to
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Low PAPR Waveform Synthesis with Application to
Wideband MIMO Radar
Marie Stro¨m
Department of Signals and Systems
Chalmers University of Technology
Gothenburg, Sweden
Email: marie.strom@chalmers.se
Mats Viberg
Department of Signals and Systems
Chalmers University of Technology
Gothenburg, Sweden
Email: viberg@chalmers.se
Abstract—This paper considers the problem of waveform
synthesis given a desired power spectrum. The properties of the
designed waveforms are such that the overall system performance
is increased. The metric used to evaluate the optimality of the
synthesized time domain signals is the peak-to-average power
ratio (PAPR). We discuss how to synthesize waveforms using the
technique of partial transmit sequence (PTS). The key point is
that the gradient can explicitly be derived from the objective
function. Furthermore, the result is extended by allowing the
power spectrum to deviate from its original shape, yielding
a further reduction in the PAPR. The method is applied to
derived power spectra for wideband multiple-input-multiple-
output (MIMO) radar. It is shown that the proposed technique
can achieve optimal or near optimal performance with PAPR
below 0.5 dB.
I. INTRODUCTION
The design of time domain signal sequences that achieve
certain system requirements has been an important research
area for decades. The result can be used in several applications,
and here the focus is to design waveforms for wideband
multiple-input-multiple-output (MIMO) radar (for an overview
of MIMO radar, see e.g. [1], [2], and [3]).
The radar system performance is directly linked to the time
domain properties of the signal. As we are free to utilize
arbitrary transmit signals, we seek waveforms that coincide
with specific system requirements, for instance low peak-to-
average power ratio (PAPR) or even constant modulus. The
PAPR measures the largest power of a signal sample com-
pared to the average power. Signals with large PAPR require
higher dynamic range on the analog-to-digital converters as
well as power amplifiers with a large linear range. Thus,
the overall cost of the system is increased. The problem to
synthesize a waveform from a periodic signal with a given
power spectrum is studied already in [4]. Formulas to adjust
the phase angles of periodic signals that yield a low PAPR
are presented, and closed form solutions derived (for specific
power spectra). Continuing, the problem to construct multitone
signals with a low PAPR is addressed in [5], [6] and [7].
Furthermore, in [8], four different partial transmit sequence
(PTS) algorithms are discussed, and an extended version of
the time-frequency swapping algorithm [7] is proposed as
the preferred method. On the other hand, if the covariance
matrix of the waveforms is known, a cyclic algorithm that
uses the results from [9] gives the possibility to synthesize
waveforms with low PAPR or constant modulus, such that
the covariance matrix is approximately accomplished [10].
In [11] an algorithm to generate waveforms that match a
desired beampattern is proposed. Furthermore, algorithms that
use the expression of the signal-to-noise ratio (SNR) to design
waveforms that achieve low PAPR under the constraint of good
Doppler resolution are discussed in [12].
Herein, we seek to synthesize waveforms that have an
already derived desired power spectrum. The waveforms are
designed to achieve a low PAPR. Obviously, PAPR is not
the only concern in radar waveform design. In addition, the
waveform must have a good ambiguity function. Note that the
range resolution, i.e. the autocorrelation of the time domain
waveform is directly given by the power spectrum. Thus, it
is not of concern in this work. The Doppler resolution is of
interest for moving targets. However, it is not considered in
this paper.
A recently developed algorithm for MIMO communica-
tion [13] is exploited and reformulated to coincide with our
problem formulation. The key point is to formulate an analytic
expression of the objective function and to derive its gradient.
The result is compared with the preferred approach in [8],
where small modifications are necessary to fit our problem
statement. However, this is only a minor difference in the
application of the same principle. To further reduce the PAPR,
we introduce an extended version of the proposed algorithm,
where we allow the power spectrum to deviate from its
original shape. Finally, we investigate the trade off between
the reduction in PAPR and the loss in signal-to-noise-and-
interference ratio (SNIR) when the optimal power spectrum is
distorted.
Notation: In the sequel, time domain samples are denoted
as lowercase letters with indices n, vectors by boldface low-
ercase letters and matrices with boldface uppercase letters.
In comparison, frequency domain samples are denoted as
uppercase letters with indices k. The real and imaginary part
of a complex valued vector or matrix is denoted <(·) and =(·),
respectively.
II. PROBLEM FORMULATION
Consider a discrete time signal sequence y[n], n = 0 . . . N−
1, that has an already derived optimal power spectrum. Thus,
we seek to synthesize a waveform y = [y[0] . . . y[N − 1]]T
whose frequency behavior is agreeable with |Y [k]|2 = Pk,
where k = 0 . . . N − 1 and Y [k] is the discrete Fourier
transform (DFT) of y[n] at frequency index k. Moreover, the
designed signal sequence should possess a low PAPR yielding
both enhancement of the system performance and an overall
lower system cost. The PAPR is defined as
PAPR =
maxn |y[n]|2
1
N
∑N−1
n=0 |y[n]|2
. (1)
In the following section, we describe the waveform synthesis
while maintaining the power spectrum. The result is then
extended by allowing a small deviation of the desired power
spectrum.
III. WAVEFORM SYNTHESIS
Herein, we discuss the actual design of the waveforms. First,
we make use of Parseval’s theorem to write the energy of the
signal sample as
N−1∑
n=0
|y[n]|2 = 1
N
N−1∑
k=0
|Y [k]|2 = 1
N
N−1∑
k=0
Pk. (2)
Further, as we are interested in waveforms that coincide with a
desired power spectrum, the use of the inverse discrete Fourier
transform (IDFT) yields
y[n] =
1
N
N−1∑
k=0
√
Pk · ejφkej2pi knN , (3)
where the phases, φk ∈ [0 2pi), do not change the given
power spectrum. However, the PAPR changes dramatically
with the choice of the phases [4]. Hence, by tuning the
phases φk, we can synthesize a signal with desirable properties
while maintaining the power spectrum. Note that the PAPR is
reduced by minimizing the maximal |y[n]|2, since the energy
is fixed. Thus, we seek to find the phase angles by solving
φˆ = argmin
φk
max
n
|y[n]|2. (4)
Define the objective functions as fn(φ) = |y[n]|2, where
φ = [φ0 . . . φN−1]T . Herein, we make use of a sequential
quadratic programming (SQP) technique to solve the opti-
mization problem (see [13] for pseudo code and complexity
analysis). The optimization problem is formulated as
min
φ
κ (5)
subject to fn(φ) ≤ κ, n = 0 . . . N − 1
In (5), κ is the maximum of fn(φ). To avoid increasing the
power of any other signal sample, the functions fn(φ) ≤ κ are
set as constraints to the minimization problem. The cost func-
tion has several local minima, which have slightly different
levels [13]. Thus, by reaching a local minimum, we satisfy the
PAPR reduction aim. Moreover, this implies that the algorithm
is insensitive to the initialization vector. In order to derive the
gradients of fn(φ), rewrite (3) as
y[n] =
1
N
√
p
T
Wng(φ), (6)
where
√
p = [
√
P0 . . .
√
PN−1]T , Wn is a diagonal matrix
with the inverse discrete Fourier transform (IDFT) coeffi-
cients, (1, . . . , ej2pi
(N−1)n
N ), on the diagonal and g(φ) =
[ejφ0 , . . . ejφN−1 ]T . Rewriting, ejφ = cos(φ)+j sin(φ) yields
fn(φ) =
1
N2
|√pTWng(φ)|2 = (7)
=
1
N2
√
p
T
(<(Wn) cos(φ)−=(Wn) sin(φ))2︸ ︷︷ ︸
A
√
p+
+
1
N2
√
p
T
(<(Wn) sin(φ) + =(Wn) cos(φ))2︸ ︷︷ ︸
B
√
p.
The gradients of (7) are now calculated as
∂fn(φ)
∂φk
= − 2
N2
√
p
T
A
(
cos(2pi
nk
N
) sin(φk)+ (8)
+ sin(2pi
nk
N
) cos(φk)
)√
p+
2
N2
√
p
T
B·
·
(
cos(2pi
nk
N
) cos(φk)− sin(2pink
N
) sin(φk)
)√
p.
Next, by allowing the power spectrum to deviate from its
original shape with a constant, k, i.e.
√
P˜k =
√
Pk + k, we
investigate if further reduction of the PAPR is possible. Hence,
the optimization problem is reformulated as
min
k,φk
max
n
| 1
N
(
√
p+ )TWng(φ)|2 + λ||||2 (9)
subject to
√
Pk + k ≥ 0 and ||√p+ ||2 ≤ P tot,
Here, λ is a design parameter that controls the total amount
of allowed distortion for all entries,  = [0 . . . N−1]T and
P tot =
∑N−1
k=0 Pk. To solve the complete optimization prob-
lem, an alternating approach is used. Thus, first the optimal
phase angles, φ, are found for a fixed  by solving (5).
Second, (9) is minimized with respect to  with fixed phase
angles. The alternation continues until the reduction in the
PAPR is below a pre-determined convergence criterion. The
initial values of k are chosen as zero.
IV. RELATION TO PREVIOUS WORK
To evaluate the performance of the proposed waveform syn-
thesis, a comparison with equivalent algorithms is performed.
Herein, we compare our method to other PTS based tech-
niques, e.g. [4], [5], [6] and [8]. In particular, [8] discusses four
different algorithms to reduce the PAPR, and a preferred time-
frequency swapping algorithm is presented in detail. Thus, this
algorithm is compared with our approach. For the preferred
method to coincide with our proposed approach, the stated
complex multitone signal of [8] with N tones
m(t) =
N−1∑
n=0
ejφnejn∆ωt (10)
is reformulated into
m[n] =
1
N
N−1∑
k=0
√
Pke
jφkej2pi
kn
N . (11)
However, the same proposed algorithm can be used also in
this more general setting.
V. EXPERIMENTAL VALIDATION
To evaluate the proposed approach, the derived frequency
behavior of the signal is obtained from [14], where the optimal
waveform designs for wideband MIMO radar are investigated.
However, any system with prior knowledge of the power
spectrum can be used as a reference. In the sequel, we
present the case where three waveforms are synthesized, each
associated with a specific power spectrum. The power spectra
are visualized in Fig. 1.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Normalized frequency
|
Y
[k
] 
|2
P
1
P
2
P
3
Fig. 1. Desired power spectra for the three waveforms.
First, the waveforms are synthesized to maintain their de-
sired power spectra. It is assumed that convergence of the
algorithms are achieved when the reduction in PAPR ≤ 10−4
per iteration. Figure 2 illustrates the synthesized waveform
without the use of a PAPR reduction technique, when the
phases, φ, are drawn from a uniform distribution on the
interval φ ∈ [0 2pi). In comparison, Fig. 3 visualizes the
corresponding waveform, designed to achieve a low PAPR.
As seen, the maximum level of the signal is reduced.
The PAPR for the optimized signal sequences is com-
pared with the initial PAPR and the time-frequency swap-
ping method [8], see Table I. Unless stated otherwise, the
results are average over 200 Monte Carlo trials. As seen,
TABLE I
THE PAPR BEFORE (PAPR0) AND AFTER (PAPR1) OPTIMIZATION
COMPARED WITH THE TIME–FREQUENCY SWAPPING ALGORITHM [8].
PAPR0 [dB] PAPR1 [dB] PAPR [8] [dB]
Waveform 1 7.6 0.32 1.2
Waveform 2 6.6 0.23 1.0
Waveform 3 7.3 0.31 1.1
the proposed method performs significantly better than the
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Sample
|y[
n]|
2
Fig. 2. Synthesized time domain waveform associated with P1 without using
a PAPR reduction technique.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0.22
Sample
|y[
n]|
2
Fig. 3. Synthesized time domain waveform associated with P1 optimized to
achieve a low PAPR.
comparable approach. However, the time–frequency algorithm
is less complex than that of the proposed approach, and
is therefore less time consuming. In the sequel, the initial
guess for the proposed algorithm is the output from the time-
frequency swapping algorithm. Thus, faster convergence is
ensured as the initial guess is closer to a minimum.
Continuing, we investigate the optimal solution to the
minimization problem and to the time-frequency swapping
algorithm. To identify the variation of the achieved PAPR,
200 test trials are performed. The cumulative distribution
function (CDF) for the PAPR, illustrated in Fig. 4, shows
that 93% of the obtained PAPR is contained in the interval
PAPR ∈ [0.2 0.4] dB for the proposed approach.
Second, the waveforms are synthesized to approximately
coincide with their desired power spectra. The reduction in
the PAPR for the alternating algorithm with λ = 10−2 is
depicted in Fig. 5. On the x-axis, one iteration corresponds to
one step in the alternation method. As illustrated, the algorithm
converges fast and the PAPR is decreased in each iteration. To
investigate the degradation in performance, the SNIR defined
as
SNIR =
Ptarget
Pinterference + Pnoise
, (12)
where Ptarget, Pinterference and Pnoise are the power contribution
from the target, interference and receiver noise, respectively,
is evaluated for the distorted power spectra. The original
0 0.2 0.4 0.6 0.8 1 1.2 1.4
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
X [dB]
Pr
ob
( P
AP
R ≤
 
X 
)
 
 
Proposed
Friese [5]
Fig. 4. Prob(PAPR ≤ X) for 200 Monte Carlo trials
1 2 3 4 5 6 7 8
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Iteration
PA
PR
 [d
B]
 
 
Waveform 1
Waveform 2
Waveform 3
Fig. 5. The average reduction of the PAPR versus iteration count for the
alternating algorithm.
power spectra are optimal for a specific radar scenario,
consisting of: a target, deceptive jamming and receiver
noise (for further information see [14]). The results
are presented in Fig. 6, where the SNIR versus the
PAPR for a varying λ is depicted. Here, 2 complete
iterations of the alternating algorithm is performed and λ =
[1, 10−1, 50−2, 10−2, 50−3, 10−3, 50−4, 10−4, 50−5, 10−5].
Note that the illustrated PAPR is the average of the combined
waveforms, and that the design parameter, λ, decreases from
left to right. As seen, small variations of the power spectra
do not drastically affect the SNIR.
VI. CONCLUSION
In this paper, a waveform synthesis algorithm that reduces
the PAPR is presented. The designed time domain expression
has a specific frequency behavior, that is agreeable with a
desired power spectrum. The proposed algorithm is compared
with the time-frequency swapping method described in [8],
and shows a larger reduction in the PAPR. However, a draw-
back is that the proposed method is more time consuming.
The algorithm is extended by allowing a small deviation of
the power spectrum. Results show that further reduction of the
PAPR is possible and that the system performance, measured
as the SNIR, is not drastically degraded for a small distortion.
0.050.060.070.080.090.10.110.120.130.14
0
2
4
6
8
10
12
14
PAPR [dB]
SN
IR
 [d
B]
 
 
Fig. 6. The degradation in SNIR versus PAPR obtained for different values
of the design parameter λ.
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Low PAPR waveform synthesis with application to wideband MIMO radar

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