# Pilot design for a cellular wireless system based on Costas arrays.

**ABSTRACT** In this paper, we present a pilot design for an OFDM cellular system based on Costas arrays. The design provides pilot symbols for both common and dedicated pilots for MIMO transmission. The pilot symbols are designed so that pilots from different cells intersect minimally. The design results in a large number of available distinct pilot symbol patterns with minimally overlapping resource elements that may be assigned to different cells which simplifies cell planning. The design is described in the context of IEEE 802.16m, but is applicable to any OFDM system. Performance of the pilot design in noise and interference limited environments is presented and is shown to be on par with currently used pilot designs while providing a larger number of patterns.

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**ABSTRACT:**A conjecture that the number of Costas arrays is a monotonic increasing function of array size is disproved by extension of the known values to n =17. A probabilistic estimation formula is developed which predicts the peak at n =16 and tracks the known values to (typically) 5-6%Proceedings of the IEEE 08/1988; · 6.91 Impact Factor - SourceAvailable from: psu.edu
##### Article: A study of a class of detection waveforms having nearly ideal range—Doppler ambiguity properties

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**ABSTRACT:**A special class of permutation matrices is considered. It is shown that these matrices may be beneficially used to determine the frequency-time pattern of a uniform pulse train. Proper choice of burst waveform parameters is shown to result in a detection wave-form having range and Doppler resolution properties consistent with the overall signal duration and bandwidth. The range-Doppler sidelobe peaks are well-controlled so that the ideal "thumbtack" ambiguity function behavior is closely approximated by the synthesis procedure presented.Proceedings of the IEEE 09/1984; · 6.91 Impact Factor - SourceAvailable from: costasarrays.org[Show abstract] [Hide abstract]

**ABSTRACT:**A Costas array is an n × n array of dots and blanks with exactly one dot in each row and column, and with distinct vector differences between all pairs of dots. As a frequency-hop pattern for radar or sonar, a Costas array has an optimum ambiguity function, since any translation of the array parallel to the coordinate axes produces at most one out-of-phase coincidence. We conjecture that n × n Costas arrays exist for every positive integer n. Using various constructions due to L. Welch, A. Lempel, and the authors, Costas arrays are shown to exist when n = p - 1, n = q - 2, n = q - 3, and sometimes when n = q - 4 and n = q - 5, where p is a prime number, and q is any power of a prime number. All known Costas array constructions are listed for 271 values of n up to 360. The first eight gaps in this table occur at n = 32, 33, 43, 48, 49, 53, 54, 63. (The examples for n = 19 and n = 31 were obtained by augmenting Welch's construction.) Let C(n) denote the total number of n × n Costas arrays. Costas calculated C(n) for n ≤ 12. Recently, John Robbins found C(13) = 12828. We exhibit all the arrays for n ≤ 8. From Welch's construction, C(n) ≥ 2n for infinitely many n. Some Costas arrays can be sheared into "honeycomb arrays." All known honeycomb arrays are exhibited, corresponding to n = 1, 3, 7, 9, 15, 21, 27, 45. Ten unsolved problems are listed.Proceedings of the IEEE 10/1984; · 6.91 Impact Factor

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Pilot Design for a Cellular Wireless System Based on

Costas Arrays

Havish Koorapaty, Jiann-Ching Guey, Rajaram Ramesh, Kumar Balachandran

Ericsson Research

200 Holger Way

San Jose, CA 95134

{havish.koorapaty, jiann-ching.guey, rajaram.ramesh, kumar.balachandran}@ericsson.com

Abstract— In this paper, we present a pilot design for an OFDM

cellular system based on Costas arrays. The design provides pilot

symbols for both common and dedicated pilots for MIMO

transmission. The pilot symbols are designed so that pilots from

different cells intersect minimally. The design results in a large

number of available distinct pilot symbol patterns with minimally

overlapping resource elements that may be assigned to different

cells which simplifies cell planning. The design is described in the

context of IEEE 802.16m, but is applicable to any OFDM system.

Performance of the pilot design in noise and interference limited

environments is presented and is shown to be on par with

currently used pilot designs while providing a larger number of

patterns.

Keywords: Pilots, Costas Arrays, Reference Symbols

I.

INTRODUCTION

The design of reference signals for OFDM systems based

on a Costas Arrays [1,4,5,6] has been proposed in [2]. The

proposed reference signals in [2] are designed to facilitate

multiple tasks including device identification, initial time and

frequency synchronization and channel estimation. In this

paper, a pilot design using such reference signals for a practical

system is presented. The design illustrates the use of such

reference signals to enable MIMO transmission. Furthermore,

enhancements to the reference signals described in [2] are

presented in order to enable channel estimation for small

resource blocks while alleviating performance degradation due

to edge effects. The design is described in the context of the

downlink for IEEE 802.16m [3] and includes both common

and dedicated pilots. Performance of the pilot design is

evaluated and is compared to some of the other pilot designs

considered in IEEE 802.16m.

A central question to be addressed with pilot designs is

whether pilot symbols from a given cell should interfere only

with pilot symbols from other cells or also with data symbols

from other cells. Designs where pilots interfere with data from

other cells show some gain with pilot-boosting in an

interference-limited environment. This paper investigates the

performance of pilot designs where pilots interfere with data

from other cells in both noise-limited and interference-limited

environments and compares the performance to pilot designs

where pilots interfere only with pilots from other cells.

Section 2 briefly reviews the Costas Array [4]. Section 3

presents the pilot design presented in this paper in the context

of IEEE 802.16m. Section 4 presents the performance of the

proposed pilot structure obtained via link and hybrid link-

system simulations. Section 5 concludes the paper.

II.

COSTAS ARRAYS

The pilot design described in Section 3 is derived from a

base pattern over 6 symbols and 48 subcarriers that is repeated

in time and frequency. The base pattern is derived from a

Costas array [1]. Figure 1 shows a Costas array of length 6.

Figure 1: A 6x6 Costas array pattern

For the pilot design problem of interest, the horizontal

direction represents OFDM symbols in time and the vertical

direction represents frequency subcarriers. The Costas array in

figure 1 has the property that all cyclic shifts in time and

frequency have at most 2 coincidences with the original

pattern. Thus, the pattern has a uniqueness property that is not

present with regular patterns. Figure 2 illustrates the

coincidences between cyclic shifts of the Costas array [2]. The

6x6 base pattern is repeated once in time and frequency to form

a 12x12 overall pattern. This base pattern is represented by the

squares shaded in black in the figure. A time-frequency shifted

version of the pattern is super-imposed on the original pattern

and is represented in red with the shift being one subcarrier in

frequency and two symbols in time.

Inspection of the 6x6 array within the black border in the

figure shows that there are two collisions between the shifted

pattern represented in red and the base pattern represented in

black. The number of collisions for various time and frequency

shifts, referred to as the sidelobe distribution array (SDA), is

shown in the right hand side of the figure. The number of

collisions for the particular shift shown in the left hand side of

2010 IEEE 21st International Symposium on Personal Indoor and Mobile Radio Communications

978-1-4244-8016-6/10/$26.00 ©2010 IEEE218

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the figure is circled in the SDA. In the SDA shown on the right,

blank squares indicate zero collisions.

Figure 2: Periodic sidelobes distribution array (SDA) of a

Costas array

A step in the frequency dimension can represent multiple

subcarriers for the Costas array shown in Figures 1 and 2. That

is, a step in the frequency dimension may represent a step of M

subcarriers. The SDA shown in Figure 2 is therefore for

subcarrier shifts that are a multiple of M. The SDA is zero for

shifts in frequency that are not a multiple of M.

III.

PILOT DESIGN BASED ON COSTAS ARRAYS

The main goals of the design described here are to enable

the functions of cell identification and channel estimation for

MIMO transmission modes with very small time-frequency

block sizes while simplifying the task of cell planning by

providing a large set of minimally overlapping patterns. The

pilot design presented uses a basic resource block that is 18

subcarriers by 6 symbols in size which is specific to IEEE

802.16m. However, the principles of the design described are

more generally applicable and the use of IEEE 802.16m

resource block sizes is only for purposes of illustration.

The design presented here includes the option of a common

pilot signal that extends across the whole bandwidth but may

not extend over all OFDM symbols in a frame. This signal is

always present in certain parts of the frame and is not

associated with a particular data allocation to a mobile station

(MS). A separate set of patterns that is associated with a

particular resource block assigned to a MS is also defined. This

set of patterns could be used either as dedicated or common

pilots when transmitting data in the associated resource block.

The sets are designed in a manner so that they may co-exist

without overlaps in time-frequency positions between them.

The lack of overlap between sets is achieved by using a

different set of subcarriers for each set. This restricts the

number of patterns available within a set. However, if only one

of the set of patterns is used, the number of patterns available

in the set may be expanded. The design for both the allocation

independent and allocation dependent sets of patterns described

above is based on Costas arrays. This paper focuses on the

design for allocation dependent sets of patterns that may be

used as common or dedicated pilots. The design for allocation

independent sets is similar to the design for allocation

dependent sets of patterns.

When the allocation dependent pilot patterns are used as

dedicated pilots, the pilot symbols transmitted from the

antennas are precoded in the same manner as the data symbols

in the resource block. The number of streams included within a

resource block may vary from 1 to 4. When dedicated pilots are

used, the number of pilot patterns transmitted is the same as the

number of streams. Hence, if a particular transmission uses 3

streams, 3 pilot patterns are needed to enable channel

estimation for the 3 streams. On the other hand, when common

pilots are used, the number of patterns used must equal the

number of antennas used for the transmission irrespective of

the number of streams in the transmission. In addition, the pre-

coding weights used at the transmitter must be known to the

MS.

The basic resource block size in IEEE 802.16m is set to 18

subcarriers by 6 symbols. Within this block, 6 pilot symbols

(per stream) are allocated when one or two streams are used

and 4 pilot symbols per stream are allocated when more than 2

streams are used. The design presented uses the same pilot

overhead.

A. Pilot Design for 1 or 2 Streams

The pilot design when one or two streams are used is based

on a base pattern over a group of 36 subcarriers and 6 symbols

shown in Figure 3a. The base pattern shown in Figure 3a is

composed of two Costas Array patterns with the second pattern

being a shifted version of the first. The shifted pattern is shown

with shading in the figure. The second array is a cyclic shift of

the base array by 3 subcarriers in frequency and 3 symbols in

time. This enhancement ensures good channel estimation

performance with small block sizes. It is known that channel

estimation performance at the edges of a resource block suffers

in comparison to the performance in the middle of a resource

block. When the resource block includes just one base Costas

array pattern, every symbol carrying data is effectively very

close to an edge. Given the possibility of using resource blocks

that contain only one or a few 18x6 subcarrier-symbol blocks,

using two Costas Arrays of 6 symbols each spread over two

contiguous resource blocks in frequency is more robust to edge

effects than using one array of 6 symbols within a single

resource block.

Pilot patterns for different streams and base stations are

generated by cyclically shifting the base pattern in Figure 3a

within the 36x6 subcarrier-symbol group. Cyclic shifts in

frequency are limited to multiples of 6 sub-carriers. This

restriction is imposed for two reasons. The first reason is to

ensure that the pilot patterns used for streams 1 and 2 do not

interfere with those used for streams 3 and 4 and for the

common pilot patterns that are not associated with a data

allocation. Secondly, due to the design of the base pattern as a

combination of two Costas array patterns that are cyclic shifts

of one another, the number of patterns available is reduced by

half. Hence, shifts in frequency are restricted to multiples of six

subcarriers instead of three subcarriers. Therefore, possible

shifts in frequency are in the set [0, 6, …, 30] subcarriers.

Cyclic shifts in time are not restricted and therefore shifts of up

to 6 symbols in time are possible. The indices for a resource

element indexed as a result of a shift of M subcarriers in

frequency and N symbols in time of the resource element at

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index (i,j), i.e., at sub-carrier index j and OFDM symbol index i

(0 <= i <= 5, 0 <= j <= 35) may be represented as (mod(i+N,6),

mod(j+M,36)). For the dedicated pilot for streams 1 and 2, M

belongs to the set [0, 6, …, 30] and N to the set [0, 1, …, 5].

For two streams from the same base station, only shifts in

frequency and/or time of the base pattern that ensure that the

pilot symbols for the two streams don’t overlap are used.

Figure 3b shows an example pattern where common pilot

symbols for two antennas are transmitted from the same base

station. Pilots may be boosted in power over the data

subcarriers in the same symbol by up to 9 dB.

Figure 3a & 3b: Base pattern for streams 1 and 2 and

example of dedicated pilots for 2 streams

The total number of possible patterns is equal to the number

of possible cyclic time-frequency shifts. Possible frequency

shifts are in the set, [0, 6, …, 30] and possible time shifts are in

the set, [0, 1, 2, …, 5]. Hence, there are 6 frequency shifts

possible and 6 time shifts yielding a total of 36 possible

patterns. Note that the set of frequency shifts is restricted in

order to prevent overlaps with the common pilots referred to

earlier which are allocated independently of data allocations

and with the pilot patterns utilized when 4 streams are used. If

the two sets of pilot patterns are not used together in the same

parts of the frame in time and if interference between patterns

used for 3 or 4 streams and patterns used for 1 or 2 streams is

acceptable, then this restriction is not necessary. The number of

frequency shifts then increases three-fold and there are a total

of 108 pilot patterns possible. The number of distinct sectors is

then determined by the number of streams per sector for which

patterns from the set are used. If for example, two streams in

each sector are allocated distinct pilot patterns, the number of

distinct sectors that may be identified is 54. If the restricted set

of patterns defined to avoid interference with pilot patterns for

up to 4 streams and with common pilots is used, the total

number of distinct sectors is 36/2 = 18. This is a much larger

set of minimally overlapping patterns than is provided in

current standards such as IEEE 802.16m and LTE [7].

B. Pilot Design for 3 or 4 Streams

When 3 or 4 streams are used, it is assumed that the MS

must be experiencing a high SINR and is therefore probably not

moving very fast or experiencing high channel delay spread.

This motivates reducing the overhead further when 3 or 4

streams are used. Thus, the dedicated pilot structure when more

than 2 streams are used is based on a base pattern over a group

of 18 subcarriers and 6 symbols shown in Figure 4a. Pilot

sequences for different streams and base stations are generated

by cyclically shifting this base pattern within the 18x6

subcarrier-symbol group in the same manner as is done for

common pilots. Cyclic shifts in frequency are limited to

multiples of 3 sub-carriers. Therefore, possible shifts in

frequency are in the set [0, 3, 6, …, 15] subcarriers. Cyclic shifts

in time are performed over the last 5 symbols in the base pattern

so that no pilot symbols are transmitted in the first symbol.

Therefore, cyclic shifts in time of upto 5 symbols are allowed.

Figures 4a & 4b: Base pattern for streams 3 and 4

When three or four streams are used from a base station, only

shifts in frequency and/or time of the base pattern that ensure

that the pilot symbols for the four patterns don’t overlap are

used. Figure 4b shows an example pattern where pilot symbols

for four antennas are transmitted from the same base station.

Pilots are boosted in power over the data subcarriers in the same

symbol by up to 9 dB. Both dedicated and common pilots may

be transmitted in the same resource block without any overlap in

pilot symbols transmitted from the same base station.

The total number of possible patterns is equal to the number

of possible cyclic time-frequency shifts. Possible frequency

shifts are in the set, [0, 3, 6, …, 33] and possible time shifts are

in the set, [0, 1, 2, …, 4]. Hence, there are 12 frequency shifts

possible and 5 time shifts yielding a total of 60 possible

Time (6 symbols)

(3a)

Frequency (36 subcarriers)

Stream 1

Stream 2

(3b)

Time (6 symbols)

(4a)

Frequency (18 subcarriers)

Stream 1

Stream 2

Stream 3

Stream 4

(4b)

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patterns. Once again, note that the set of frequency shifts is

restricted in order to prevent overlaps with the common pilots

allocated independently of data allocations and with the pilot

patterns defined when 2 streams are used. If the two sets of pilot

patterns are not used together in the same parts of the frame in

time and if interference between patterns used for 3 or 4 streams

and patterns used for 1 or 2 streams is acceptable, then this

restriction is not necessary. The number of frequency shifts then

increases three-fold and there are a total of 180 pilot patterns

possible. The number of distinct sectors is then determined by

the number of streams per sector for which patterns from the set

are used. If for example, four streams in each sector are

allocated distinct pilot patterns, the number of distinct sectors

that may be identified is 45. If the restricted set of patterns

defined to avoid interference with pilot patterns for up to 2

streams and with common pilots is used, the total number of

distinct sectors is 60/4 = 15. Again, as in the case with up to 2

streams, this is a much larger set of patterns than is used in

current systems.

IV. PERFORMANCE

The previous sections presented a pilot design based on

Costas arrays that provides a much larger set of patterns to

choose from than the patterns used in IEEE 802.16m. It is

shown in this section that this added flexibility does not result

in a meaningful loss in performance. It is also shown that the

use of boosting with such patterns can yield gains in

interference limited environments.

Link simulations are used to assess channel estimation

performance of some of the various sets of pilot patterns that

were considered for IEEE 802.16m and the effect of channel

estimation performance on link performance. A single transmit

and receive antenna is used to transmit information over one

block of 18 subcarriers by 6 symbols. Six symbols in the block

are used to transmit a single pilot pattern from each of the sets

of pilot patterns considered. The pilot patterns vary in the

locations chosen for the pilots. A turbo encoder is used.

Channel estimation is performed using an MMSE estimator

with knowledge of the covariance matrix of the channel and the

noise.

A. Noise-Limited Environment

Figure 5 shows the throughput observed with QPSK

modulation, rate-1/2 coding on a Vehicular A channel [8] at

120 kmph. The impairment is assumed to be AWGN. Results

are shown for some of the pilot patterns proposed for IEEE

802.16m that have an overhead of 6 symbols over the 18x6

resource block. The pattern proposed in this paper is labeled

“Proposed Pattern” while other patterns considered for IEEE

802.16m are labeled “Alternate Pattern x”. It can be seen that

the differences between the different patterns are quite small.

Figure 6 shows results for some of the patterns considered

in IEEE 802.16m with 16QAM and with the vehicle speed

increased to 350 kmph. The proposed pattern is quite close in

performance to the best pattern while providing a much larger

set of patterns that may be used by cells. However, some of the

other patterns show significant degradation which shows the

importance of dealing with edge effects when using small

resource blocks and motivates the critical role played by the

enhancements proposed in Figure 3a for small blocks.

Figure 5: Link simulation with QPSK rate-1/2 in Veh. A

channel at 120 kmph

Figure 6: Link simulation with 16QAM rate-1/2 in Veh. A

channel at 350 kmph

B. Interference-Limited Environment

Figure 7 shows gains that may be obtained via pilot

boosting in an interference limited environment. The

interference limited environment is modeled by setting the

signal-to-noise ratio (SNR as opposed to SINR) to 50 dB which

is much higher than the range of SINR values considered in the

figure. The pilot design proposed in this paper is simulated

with different pilot boosting levels. Interference is modeled

using two equal strength interferers. Independent fading

channel realizations are applied to the signals from the desired

sector and the interfering sectors.

In Figure 7, the desired channel and the two interferers use

different pilot locations. This is achieved by using different

cyclic rotations of the base pattern. Figure 8 shows the

corresponding performance with different boosting levels when

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pilot locations of the desired and interfering sectors use the

same locations. It is notable that the gains achieved through

boosting are lower than those shown in Figure 7 where pilot to

pilot interference is not present.

Figure 7: Performance in interference limited environment

with various boosting levels. Pilots interfere with data between

sectors.

Figure 8: Performance in interference limited environment.

Pilots interfere only with pilots between sectors.

In Figure 9, the pattern proposed in this paper is compared

to the one adopted in IEEE 802.16m. The figure shows that the

differences in throughput obtained using the two patterns are

negligible for 16QAM modulations. The same is the case for

QPSK. This shows that the increased flexibility with the much

larger set of patterns available for cell planning does not come

at a meaningful performance cost.

V.

CONCLUSIONS

A set of pilot patterns using Costas arrays is presented in

the context of IEEE 802.16m. The set has a large number of

patterns that eases cell planning in a cellular system. Sets of

patterns are defined for the case when pilots are associated with

data allocations that may be used as dedicated or as common

pilots. One of the sets is used when up to two streams are used

in the transmission and the second set is used when more than

two streams are used. The channel estimation performance of

these patterns is compared to the performance obtained with

pilot patterns in the 802.16m standard. A key facet of the

presented design is that any two patterns in a set have very few

common locations. Thus, pilots from a given sector interfere

with data from a neighboring sector. This allows gains to be

realized when pilot boosting is used. Simulations in noise and

interference-limited environments are used to verify the

performance of the proposed pilot patterns and to verify the

gains obtained via pilot boosting in an interference limited

environment. The large number of distinct patterns available

with the proposed design provides flexibility in deployment

which is a distinct advantage while no meaningful performance

degradation is seen with respect to the current pilot design

being used in IEEE 802.16m.

Figure 9: Performance comparison with boosting in

interference limited environment with 16QAM modulation.

REFERENCES

[1] J.P. Costas, “Medium Constraints on Sonar Design and Performance,” in

EASCON Conv. Rec., 1975, pp. 68A-68L

[2] Jiann-Ching Guey, “Synchronization Signal Design for OFDM Based on

Time-Frequency Hopping Patterns”, Proceedings of IEEE International

Conference on Communications, 2007, pp. 4329 – 4334.

[3] IEEE 802.16m/D3, December 2009, “Draft Amendment to IEEE

Standard for Local and Metropolitan Area Networks, Part 16: Air

Interface for Fixed and Mobile Broadband Wireless Systems”.

[4] J.P. Costas, “A Study of a Class of Detection Waveforms Having Nearly

Ideal Range-Doppler Ambiguity Properties”, Proceedings of the IEEE,

Vol. 72, No. 8, August 1984.

[5] S.W. Golomb, “Constructions and Properties of Costas Arrays”,

Proceedings of the IEEE, Vol. 72, No. 9, September 1984.

[6] J. Silverman, V.E. Vickers and J.M. Mooney, “On the Number of Costas

Arrays as a Function of Array Size”, Proceedings of the IEEE, Vol. 76,

No. 7, July 1988.

[7] E. Dahlman, S. Parkvall, J. Skold and P. Beming, “3G Evolution: HSPA

and LTE for Mobile Broadband”.

[8] Recommendation ITU-R M. 1225, “Guidelines for evaluation of radio

transmission technologies for IMT-2000,” 1997.

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