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IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 31, NO. 3, MARCH 2013 1

Spectrum Sensing for OFDM Signals Using Pilot

Induced Auto-Correlations

Yonghong Zeng, Senior Member, IEEE, Ying-Chang Liang, Fellow, IEEE, and The-Hanh Pham, Member, IEEE

Abstract—Orthogonal frequency division multiplex (OFDM)

has been widely used in various wireless communications systems.

Thus the detection of OFDM signals is of significant importance

in cognitive radio and other spectrum sharing systems. A

common feature of OFDM in many popular standards is that

some pilot subcarriers repeat periodically after certain OFDM

blocks. In this paper, sensing methods for OFDM signals are

proposed by using such repetition structure of the pilots. Firstly,

special properties for the auto-correlation (AC) of the received

signals are identified, from which the optimal likelihood ratio test

(LRT) is derived. However, this method requires the knowledge of

channel information, carrier frequency offset (CFO) and noise

power. To make the LRT method practical, we then propose

an approximated LRT (ALRT) method that does not rely on

the channel information and noise power, thus the CFO is the

only remaining obstacle to the ALRT. To handle the problem, we

propose a method to estimate the composite CFO and compensate

its effect in the AC using multiple taps of ACs of the received

signals. Computer simulations have shown that the proposed

sensing methods are robust to frequency offset, noise power

uncertainty, time delay uncertainty, and frequency selectiveness

of the channel.

Index Terms—Cognitive radio, spectrum sensing, OFDM, covariance,

auto-correlation, cyclostationary, statistics, detection,

robust

I. INTRODUCTION

COGNITIVE radio is a promising technology to enable

reusing under-utilized radio spectrum, particularly in

television (TV) broadcasting bands [1, 2]. To do so, a cognitive

radio needs to sense the environment and identify the temporally

unused bands of the primary users before transmitting [2–

4]. On the other hand, orthogonal frequency division multiplex

(OFDM) has been widely adopted in many TV broadcasting

standards, such as digital TV standard (DVB-T) and the China

Mobile Multimedia Broadcasting (CMMB) standard. How to

reliably sense the OFDM signals is thus crucial to provide

sufficient protections to the primary users [1].

The methods for sensing OFDM signals can be categorized

into two broad groups: blind detection method and featurebased

detection method. The major problem of the blind

detections is the vulnerability to noise uncertainty, particularly

for energy detection and its variations [5], and non-robustness

to interference uncertainty [3] for, e.g., eigenvalue-based detection

[6]. The feature-based detections use special features

Manuscript received 13 April 2012; revised 30 August 2012. Part of the

work was presented at IEEE Intern. Conference on Communication Systems

(ICCS), Singapore, Nov. 2012.

Y. H. Zeng, Y.-C. Liang and T. H. Pham are with the Institute for Infocomm

Research, A ∗ STAR, Singapore (e-mails: {yhzeng, ycliang, thpham}@i2r.astar.edu.sg).

Y.-C. Liang is also with University of Electronic Science &

Technology of China (UESTC), China.

Digital Object Identifier 10.1109/JSAC.2013.1303xx.

0733-8716/13/$31.00 c○ 2013 IEEE

of the OFDM signals, thus they are in general more robust to

noise and interference uncertainty. The most useful features

of OFDM signals that can be used for sensing are the cyclic

prefix (CP) property and the pilot/preamble structure. The CP

induces periodicity property in the received signals, which

can be used to detect the presence of the signal by checking

the known periodicity property. For example, detectors that

directly utilize this property have been derived in [7–10].

Meanwhile, detectors in [11–14] use the cyclostationarity

induced by the CP to detect the signal. In general, CP-based

methods have relatively poor performances as the CP length

is short and the periodicity induced by CP is affected by

frequency selective channel [15].

In many OFDM based standards, pilot subcarriers are inserted

into the OFDM symbols to assist the dedicated receivers

to estimate channels and to do synchronization for information

recovery purpose. This property can also be exploited

by the cognitive radio in sensing algorithm design. There

are four types of sensing schemes which make use of the

pilot subcarriers: (a) matched filtering (MF) [16–18], which

correlates the received signal with the known pilots/preamble;

(b) pilot-induced auto-correlation (AC) detection, which uses

the periodicity property of the AC caused by the periodically

repeated pilots [15]; (c) the power of pilot subcarriers can be

used for detection if the positions of the pilot subcarriers are

available [19]; and (d) pilot-aided cyclostationary detection,

which uses the cyclostationarity induced by the pilots [20–

26].

While MF is the optimal detector under ideal conditions [3,

4, 27], it is vulnerable to time dispersive channel, frequency

offset and timing error [3, 28]. The pilot subcarrier power

detection [19] is also very sensitive to frequency offset as

it usually uses time average of the received signal to improve

the performance. On the other hand, the pilot-aided

cyclostationary detections in general need higher computational

complexity and are vulnerable to frequency offset and

synchronization errors as well. In [15], a pilot-induced autocorrelation

detection is proposed, which has been shown to

have good performance, low complexity and at the same time

be robust to synchronization error, time dispersive channel and

frequency offset.

In this paper, we propose new detection methods that are

derived from the generalized likelihood ratio tests (GLRT)

using the pilot-induced auto-correlations. The major differences

from the approaches in [15] are in the following. (1)

We provide a rigorous derivation for the statistical property

of the AC coefficients at any channel conditions including

frequency selective fading, timing error and frequency offset.


2 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 31, NO. 3, MARCH 2013

(2) We derive an LRT to optimally combine multiple AC

coefficients. In [15], the LRT is only derived for single AC

coefficient and a maximum ratio combining (MRC) is used

to combine multiple AC coefficients. (3) New sensing and

CFO estimation methods are found that are not in [15]. The

major contributions of the the paper are as follows. (1) Special

properties are found for the AC of the received signal induced

by the pilots. (2) Based on the special properties of the AC, the

likelihood ratio tests (LRT) is derived. (3) An approximated

LRT (ALRT) is proposed for low signal-to-noise ratio (SNR)

detection that does not rely on the channel and noise power

knowledge. (4) An efficient method is proposed to estimate

the composite carrier frequency offset (CFO) based on the

special properties of the AC. (5) Based on the estimated CFO,

a method is presented to enhance the sensing performance by

compensating and searching the composite CFO in a limited

interval. The proposed methods are robust to frequency offset,

noise power uncertainty, frequency selective channel, and time

delay uncertainty. Simulations show that the proposed methods

have better performance than the method in [15] and are truly

robust to dispersive channel and frequency offset.

The rest of the paper is organized as follows. The OFDM

signal model and its properties are briefly discussed in Section

II. Derivations of the detection methods are described in detail

in Section III. A frequency offset estimation based on the ACs

of the received signal is investigated in Section IV. Some suboptimal

detections are given in Section V. The performances of

the methods are verified by simulations in Section VI. Finally

conclusions are drawn in Section VII.

II. OFDM SIGNAL PROPERTIES AND SYSTEM MODEL

Let S i (n) be the frequency domain OFDM signal of the

primary user at block i and subcarrier n, wherei =0, 1, ··· ,

and n =0, 1, ··· ,N f − 1 with N f being the block size. Each

block is transformed by the inverse discrete Fourier transform

(IDFT), giving the time domain signal as

s i (k) =

N f −1


n=0

S i (n)e j2πnk/N f

, k =0, 1, ··· ,N f − 1. (1)

To combat frequency selective fading, guard intervals are

usually inserted in the time domain signal. There are usually

two ways to insert guard intervals. The first way is to add a

cyclic prefix (CP). CP is a block of N g data that is the copy

of the last N g data of the OFDM block. Hence, each block

s i (k) is turned to a i (k) as

a i (k) =s i (N f − N g + k), k =0, 1, ··· ,N g − 1 (2)

a i (k) =s i (k − N g ), k = N g , ··· ,N − 1, (3)

where N = N f + N g is the block length after adding the CP.

OFDM system using the CP is called CP-OFDM. The second

way is to do zero-padding (ZP), for which N g zeros are added

to the original block. So each block s i (k) is turned to a i (k)

as

a i (k) =s i (k), k =0, 1, ··· ,N f − 1 (4)

a i (k) =0, k = N f , ··· ,N − 1. (5)

OFDM system using the ZP is called ZP-OFDM. In both ways,

the time domain transmitted signal can be written as

x(n) =a ⌊n/N⌋ (n − N⌊n/N⌋), n =0, 1, ··· . (6)

where ⌊v⌋ denotes the largest integer not greater than v. The

time domain signal x(n) is shaped by a pulse shaping filter

and transmitted through a wireless channel.

For channel estimation and synchronization purpose, pilots

are usually inserted into the frequency domain signals. Usually

there are two types of pilots: continual pilots and scattered

pilots. A particular communication system may use only one

type of pilots or both types of the pilots. For example, in

DVB-T, two types of pilots are used at the same time.

Continual pilots are inserted to every OFDM block and their

locations are fixed. Let Ω c be the set of subcarrier indices

for the continual pilots. Then S i (n) =S (p) (n), forn ∈ Ω c

and all i, whereS (p) (n) is the pilot symbol at subcarrier n.

Since continual pilots appear in every OFDM block, we have

S i (n) =S j (n) for any i, j and n ∈ Ω c .

On the other hand, scattered pilots are inserted to every

OFDM block but their locations are different at different

blocks. For example, in DVB-T standard, the locations of

scattered pilots at different blocks are periodical: they are

at the same locations after four blocks. In general, let L

be the repetition period of the scattered pilots and Ω s be

the set of subcarrier indices for the scattered pilots. Then

S i (n) =S i+mL (n) for any i, m and n ∈ Ω s .

Let H 0 be the hypothesis of “signal absent” and H 1 be

the hypothesis of “signal present”. The objective of spectrum

sensing is to choose one of the two hypotheses (H 0 or H 1 )

based on the received signals. The probability of detection,

P d , and probability of false alarm, P fa ,aredefined as follows:

P d = P (H 1 |H 1 ), P fa = P (H 1 |H 0 ), respectively.

At the sensing receiver, the received signal is downconverted

to the baseband and sampled at the same rate of

the transmitted signal. Let y(n) (n =0, 1, ··· ,N s − 1) bethe

baseband received discrete signal with sample size N s . Thus

we have

H 0 : y(n) =w(n)

H 1 : y(n) =e j(2πnɛ+ϕ)

∑L h

m=0

h(m)x(n − m − τ)+w(n),

n =0, 1, ··· ,N s − 1 (7)

where ɛ is the normalized CFO, ϕ is a random phase, h(m)

is the channel response, τ is the time delay and w(n) is the

noise.

The CP/ZP structure, continual and scattered pilots are

unique properties of OFDM signals, which can be used

for feature detection. We claim that a sensing algorithm is

“optimal” if it achieves the highest P d for a given P fa with a

fixed number of samples, though there could be other criteria

to evaluate the performance of a sensing algorithm.

III. PILOT INDUCED AUTO-CORRELATION DETECTION

The sample AC of the received signal is defined as

r(l) = 1

N s−l−1


y(n + l)y ∗ (n). (8)

N s − l

n=0


ZENG et al.: SPECTRUM SENSING FOR OFDM SIGNALS USING PILOT INDUCED AUTO-CORRELATIONS 3

Due to the special structure of the OFDM signal, the AC

has some special properties that can be used for detection. In

the following we propose new methods that use the features

induced by the pilot structure.

A. Detection based on the continual pilots

The transmitted signal contains both pilots and data. Thus

the IDFT of each block can be written into two parts:

where

s i (k) =s (p)

i (k)+s (d)

i (k), k =0, 1, ··· ,N f − 1, (9)

s (p)

i (k) = ∑

S i (n)e j2πnk/N f

, (10)

n∈Ω c

s (d)

i (k) = ∑

S i (n)e j2πnk/N f

. (11)

n/∈Ω c

Accordingly, the transmitted time domain signal can be decomposed

into two parts:

x(n) =x (p) (n)+x (d) (n), n =0, 1, ··· ,N s − 1, (12)

where the first and second terms are due to the pilots and data,

respectively. As the same continual pilots are inserted into

every OFDM block, the signal part due to the pilots contains

identical contents at different time blocks. That is,

x (p) (n) =x (p) (n + kN), for any n, k. (13)

The identical contents at every block cause the AC r(kN) to

have a fixed positive term. However, this fixed positive term

is also corrupted by the CFO. In fact, we have the following

theorem.

Theorem 1: The AC at lag kN of the received signal can

be written as

H 0 : r(kN) =η(k)

H 1 : r(kN) =e j2πkNɛ μ + ζ(k), (14)

k =1,...,K,

where μ is a fixed positive term induced by the continual

pilots, ζ(k) is contributed by the interference from other nonpilot

symbols and noise, η(k) is the AC of pure noise at lag

kN, andK = ⌊N s /N ⌋. Assume that noise and transmitted

data samples are independent and identically distributed (iid),

and have zero means. Then ζ(k) and η(k) will have Gaussian

distributions with

E(ζ(k)) = E(η(k)) = 0, (15)

1

Var(ζ(k)) =

N s − kN σ4 1 , (16)

1

Var(η(k)) =

N s − kN σ4 η , (17)

where ση 2 is the expected noise power and σ1 2 is the expected

(average) power of combined data signal (including fading)

and noise.

Proof. We consider hypothesis H 1 first. The AC of the

received signal can be decomposed into two parts:

r(kN) =r 1 (kN)+r 2 (kN), (18)

where r 1 (kN) is the AC of the pure signal and r 2 (kN) is the

cross term of signal and noise. In fact,

r 1 (kN) = ej2πkNɛ

N s − kN


N s−kN−1

·

n=0

∑L h

∑L h

m 1=0 m 2=0

h(m 1 )h ∗ (m 2 )

x(n + kN − m 1 − τ)x ∗ (n − m 2 − τ) (19)

r 2 (kN)

=

+

∑L h

m=0

∑L h

m=0

Furthermore we have

where


N s−kN−1

·

n=0

N

h(m) s−kN−1


N s − kN

n=0

·x(n + kN − m − τ)w ∗ (n)

h ∗ N

(m) s−kN−1


N s − kN

n=0

e j[2π(n+kN)ɛ+φ]

e −j(2πnɛ+φ)

·x ∗ (n − m − τ)w(n + kN)

N

1

s−kN−1


+

w(n + kN)w ∗ (n) (20)

N s − kN

n=0

r 1 (kN) =r 11 (kN)+r 12 (kN), (21)

r 11 (kN) = ej2πkNɛ

N s − kN

∑L h

∑L h

m 1=0 m 2=0

h(m 1 )h ∗ (m 2 )

(


x (p) (n + kN − m 1 − τ) x (p) (n − m 2 − τ))

(22)

r 12 (kN) = ej2πkNɛ

N s − kN


N s−kN−1

·

n=0

∑L h

∑L h

m 1=0 m 2=0

x (p) (n + kN − m 1 − τ)

+ ej2πkNɛ

N s − kN


N s−kN−1

·

n=0

∑L h

∑L h

m 1=0 m 2=0

x (d) (n + kN − m 1 − τ)

+ ej2πkNɛ

N s − kN


N s−kN−1

·

n=0

∑L h

∑L h

m 1=0 m 2=0

x (d) (n + kN − m 1 − τ)

h(m 1 )h ∗ (m 2 )

h(m 1 )h ∗ (m 2 )

h(m 1 )h ∗ (m 2 )

(

) ∗

x (d) (n − m 2 − τ)

(

) ∗

x (p) (n − m 2 − τ)

(

x (d) (n−m 2 − τ)) ∗.

(23)


4 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 31, NO. 3, MARCH 2013

Note that x (p) (n + kN) =x (p) (n) for all n and k. Thus

we have

r 11 (kN) = ej2πkNɛ

N s − kN


N s−kN−1

·

n=0

+ ej2πkNɛ

N s − kN


N s−kN−1

·

n=0

∑L h

m 1=0

|x (p) (n − m 1 − τ)| 2

∑L h

m 1=0


x (p) (n − m 1 − τ)

|h(m 1 )| 2

m 2≠m 1

h(m 1 )h ∗ (m 2 )

(

x (p) (n − m 2 − τ)) ∗

.

(24)

Without loss of generality, we assume that the sample size N s

is an integer multiples of N. Then

N

1

s−kN−1


|x (p) (n − m 1 − τ)| 2

N s − kN

= 1 N

n=0

N−1


|x (p) (n − m 1 − τ)| 2 , (25)

n=0

which is not related to k. Let

μ =

∑L h

m 1=0

Thus we have

|h(m 1 )| 2 1 N

N−1


n=0

|x (p) (n − m 1 − τ)| 2 . (26)

r 11 (kN) =e j2πkNɛ μ + v(k), (27)

where v(k) represents the second term. Finally we have

r(kN) =e j2πkNɛ μ + v(k)+r 12 (kN)+r 2 (kN), (28)

where μ is the received power of the pilots; v(k) is the

cross-correlation of the pilots; r 12 (kN) includes the crosscorrelation

of the pilots and data, and the AC of data; r 2 (kN)

includes the cross-correlation of the signal and noise, and the

AC of noise. To simplify the notations, we use ζ(k) to denote

v(k)+r 12 (kN)+r 2 (kN).

Assume that noise and transmitted data samples are iid,

and both have zero means. Furthermore, it is assumed that the

cross-correlations of the continual pilots are very small 1 .Then

it is obvious that

E(v(k)) ≈ E(r 12 (kN)) = E(r 2 (kN)) = 0. (29)

Thus, E(ζ(k)) = 0. The variance of it can be computed

directly from the expressions. In fact

Var(ζ(k)) ≈ Var(r 12 (kN)) + Var(r 2 (kN)). (30)

From the expressions we have

Var(r 2 (kN)) =

2

N s − kN

∑L h

m=0

|h(m)| 2 σ 2 xσ 2 η

1

+

N s − kN σ4 η , (31)

1 This is one of the design criteria for the pilots in practical systems

Var(r 12 (kN))

=

2

N s − kN

1

+

N s − kN

∑L h

∑L h

m 1=0 m 2=0

∑L h

∑L h

m 1=0 m 2=0

|h(m 1 )| 2 |h(m 2 )| 2 σ 2 x,p σ2 x,d

|h(m 1 )| 2 |h(m 2 )| 2 σ 4 x,d , (32)

where σ 2 x = E(|x(n)|2 ), σ 2 x,p = E(|x(p) (n)| 2 ), σ 2 x,d =

E(|x (d) (n)| 2 ),andσ 2 η =E(|w(n)| 2 ).Let

σ 2 1 = (σ 4 η +2σ2 x σ2 η

·

∑L h

m=0

∑L h

|h(m)| 2 +(σ 4 x,d +2σ2 x,p σ2 x,d )

∑L h

m 1=0 m 2=0

|h(m 1 )| 2 |h(m 2 )| 2 ) 1/2

, (33)

which is related to transmitted signal power, channel and noise

1

power. Then Var(ζ(k)) =

N s−kN σ4 1 .

Based on the central limit theorem, we can assume that

ζ(k) has Gaussian distribution with zero mean and variance

σ 4 1

N .

s−kN

At hypothesis H 0 , the AC is simplified as

N

1

s−kN−1


r(kN) =η(k) =

w(n + kN)w ∗ (n). (34)

N s − kN

n=0

Based on the central limit theorem, r(kN) will approach to

Gaussian distribution. It is proved [27, 29] that η(k) has zero

1

mean and variance

N s−kN σ4 η .


We can use the model (14) to derive the optimal detection

based on the AC. We know that the optimal detection is the

likelihood ratio test (LRT) [27]. Based on the model (14), it

is easy to derive that the test statistic of the LRT is

K∑

T LRT −CP = (N s − kN)

k=1

·

(− |r(kN) − ej2πkNɛ μ| 2

σ 4 1

)

+ |r(kN)|2

ση

4 . (35)

We call this method the LRT based on continual pilots (LRT-

CP). Although it is optimal, it cannot be used in practice as

ɛ, σ 1 , μ and σ η are unknown. Thus we need to approximate

these parameters by their estimations.

A major challenge for cognitive radio is reliable sensing at

very low signal-to-noise ratio (SNR). At very low SNR, the

data signal contribution to the AC is much smaller than that of

noise. Thus the noise power approaches to the received signal

power. In fact, as proved in [30], the maximum likelihood

(ML) estimation for the noise power is

σ 2 η ≈ r(0) =

1 N∑

s−1

N s

n=0

|y(n)| 2 . (36)

On the other hand, σ1 2 is the combined signal (excluding pilots)

and noise power. Thus it can also be approximated by r(0).

The noise power can also be estimated by using other

methods. For example, we can use the average power of the

received signal at the null subcarrier locations of the OFDM


ZENG et al.: SPECTRUM SENSING FOR OFDM SIGNALS USING PILOT INDUCED AUTO-CORRELATIONS 5

as the estimation, if the OFDM signal has null subcarriers and

they are known to the receiver.

So, replacing σ 4 η and σ 4 1 by r 2 (0) in the LRT, we have the

approximated LRT (ALRT)

T ALRT −CP =

K∑

(N s − kN)

k=1

·

(− |r(kN) − ej2πkNɛ μ| 2

)

r 2 + |r(kN)|2

(0)

r 2 (0)

= 1 K∑

r 2 (N s − kN) ( 2μRe(e −j2πkNɛ r(kN)) − μ 2)

(0)

k=1

(

)

= 2μ

r(0) Re 1

K∑

(N s − kN)e −j2πkNɛ r(kN)

r(0)

k=1

− (N s − kN)μ 2

r 2 , (37)

(0)

where Re means taking the real part. Note that μ is related

to the primary user’s channel and thus it is usually unknown.

μ

r(0)

However, is almost a constant for large N s and fixed

channel. We know that any constants will not affect the

detection performance. Thus we ignore the constant terms and

obtain an equivalent test as

T ALRT −CP

(

)

1

K∑

=Re (N s − kN)e −j2πkNɛ r(kN) . (38)

r(0)

k=1

B. Detection based on the scattered pilots

In many systems like DVB-T, the scattered pilots repeats the

structure after a few OFDM symbols. Let LN be the repetition

period of the scattered pilots. For example, in DVB-T standard,

L =4. Similar to the continual pilots, scattered pilots also

cause the received signal to contain some identical contents at

some blocks, which causes the AC to have peaks at dedicated

lags. In general, the AC should have a peak at lag kLN,

k =1, 2, ···. Similar to the derivations for the continual pilots,

we have the following theorem.

Theorem 2: The AC at lag kLN of the received signal can

be written as

H 0 : r(kLN) =η(k)

H 1 : r(kLN) =e j2πkLNɛ μ + ζ(k), (39)

k =1,...,K.

where μ is a fixed positive term induced by the scattered pilots,

ζ(k) is contributed by the interference from other non-pilot

symbols and noise, η(k) is the AC of pure noise at lag kLN,

and K = ⌊N s /(LN)⌋. Assume that noise and transmitted data

samples are iid, and have zero means. Then ζ(k) and η(k) will

have Gaussian distributions with

E(ζ(k)) = E(η(k)) = 0, (40)

1

Var(ζ(k)) =

N s − kLN σ4 1, (41)

1

Var(η(k)) =

N s − kLN σ4 η, (42)

where ση 2 is the noise power and σ2 1 is the combined data

signal (including fading) and noise power.

Similar to the derivation for continual pilots, we can get the

approximated LRT based on scattered pilot (ALRT-SP) at low

SNR as

T ALRT −SP



=Re⎝ 1

K/L


(N s − kLN)e −j2πkLNɛ r(kLN) ⎠ (43)

r(0)

k=1

C. Detection using both continual and scattered pilots

If there exist both continual and scattered pilots, the value

of μ in model (14) may change with k. Especially, the values

at k and kL are different. In fact, r(kLN) also includes contributions

from the scattered pilots. Thus the detection based

on continual pilots has already partially used the scattered

pilots as well. Similarly the detection based on scattered pilots

also includes contributions from the continual pilots. However,

both methods have not fully used the two types of pilots.

It is hard to derive the optimal detection to fully use both

types of pilots. Here we simply consider the combining of

the two methods. Borrowing the well-known MRC technique,

we propose a combining method as follows. Let M c and

M s be the number of continual pilots and scattered pilots,

respectively. Let p c and p s be the power of each continual

pilot and each scattered pilot, respectively. Define a constant

as

λ =(p s M s )/(p c M c ) (44)

λ is approximately the ratio of the scattered pilots power to

the continual pilots power. A heuristic combining method is:

T ALRT −CSP



=Re⎝ λ

K/L


(N s − kLN)e −j2πkLNɛ r(kLN) ⎠

r(0)

k=1

(

)

1

K∑

+Re (N s − kN)e −j2πkNɛ r(kN) . (45)

r(0)

k=1

IV. FREQUENCY OFFSET ESTIMATION AND SEARCHING

Now the major obstacle is the unknown frequency offset in

the tests. How to estimate the frequency offset under very

low SNR is a very challenging problem. There have been

quite a few CFO estimation methods. These methods can be

in general grouped into two classes: (1) non-data aided (blind)

estimations [31–34]; and (2) data-aided (preamble/pilot-based)

methods [35–41]. The non-data aided (blind) estimations

usually are based on the OFDM property like CP and null

subcarriers. The data-aided methods usually use specially designed

preambles. Most of them require the preamble OFDM

symbol to have periodical property in time domain within one

OFDM block, which can be realized by placing dedicated null

subcarriers (NSCs). The major problem of known methods is

that they do not work well at very low SNR.

As described in (14) and (39), a special feature here is that

we do not need to estimate the original CFO ɛ but just Nɛ


6 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 31, NO. 3, MARCH 2013

(for continual pilots) or LNɛ (for scattered pilots). This will

gives us special advantage to design the algorithm. If we use

known methods to estimate ɛ first and then get Nɛ or LNɛ,

the estimation error is effectively multiplied by N or LN

times. Thus in the following we will propose a new method

that directly estimates Nɛ (for continual pilots) or LNɛ (for

scattered pilots) using the ACs of the received signal.

A. Frequency offset estimation

We consider continual pilot case of model (14). Let ϑ = Nɛ

mod 1. We further define δ = ϑ if 0 ≤ ϑ ≤ 0.5 and δ = ϑ−1

if 0.5


ZENG et al.: SPECTRUM SENSING FOR OFDM SIGNALS USING PILOT INDUCED AUTO-CORRELATIONS 7

where f k is a weight and ∑ K

k=1 f k =1.

Theorem 4: Assume that the sensing time is long and the

initial CFO estimation is sufficiently accurate. The optimal

linear estimator based on (61) should have the coefficients

f k = k 2 (N s − kN)/c, (62)

where c = ∑ K

k=1 k2 (N s − kN) is a constant.

Proof: To obtain the optimal estimator, we need the best

combining coefficients f k for BLUE. As discussed above, we

need the variance of θ1(k)

2πk .Letˆζ(k) =χ 1 (k)+jχ 2 (k). Then

(

)

μ sin(2πˆδk)+χ 2 (k)

φ(kN) = arctan

μ cos(2πˆδk)+χ 1 (k)


= arctan⎝ tan(2πˆδk)+


χ2(k)

μ cos(2πˆδk)

⎠ . (63)

1+ χ1(k)

μ cos(2πˆδk)

Assume that μ>>|ˆζ(k)| (valid at long sensing time) and the

remaining error ˆδ is very small. Using the first order Taylor

approximation, we have

Thus

φ(kN) ≈ 2πˆδk +

χ 2 (k)

μ cos(2πˆδk) ≈ 2πˆδk + χ 2(k)

μ . (64)

φ(kN)

2πk

≈ ˆδ + χ 2(k)

2πkμ . (65)

That is, at H 1 , we approximately have

θ 1 (k)

2πk ≈ χ 2(k)

2πkμ . (66)

Note that χ 2 (k) is the imagine part of ˆζ(k). Thus

1

Var(χ 2 (k)) = Var(ˆζ(k))/2 =Var(ζ(k))/2 =

2(N s−kN) σ4 1 .

Hence, finally we have

( θ1 (k)

)

1

Var =

2πk 8π 2 μ 2 k 2 (N s − kN) σ4 1. (67)

Thus the BLUE should have combining coefficients f k =

k 2 (N s − kN)/c.


The final estimation of the CFO δ is: δ 1 = δ 0 + ˆδ 0 .

If there are scattered pilots, we can use them to estimate

the composite CFO (LNɛ mod 1) based on model (39). The

derivation and the algorithm are similar. We can simply replace

N by LN in all the equations above. However, we can only

estimate LNɛ using the scattered pilots. It may not be possible

to obtain Nɛ mod 1 from LNɛ mod 1. On the other hand,

if we have estimated Nɛ mod 1 based on the continual pilots,

it is easy to get the estimation for LNɛ mod 1. Itmaybe

possible to use both the continual and scattered pilot for the

estimation of Nɛ mod 1.

B. Search the frequency offset

After the estimation, we first subtract the normalized CFO

in the test statistics. The remaining (residue) of the CFO is

much smaller than before. Let the remaining (residue) of the

normalized CFO is in interval (−Δ, Δ], whereΔ ≪ 0.5. We

then search the CFO within this interval. For each CFO in the

interval, we obtain a value for the test statistic. The maximum

of all the values are then used for the final decision. We use

continual pilot case as an example to describe the details of

the method in the following.

First the CFO is compensated, that is, we multiply r(kN)

by e −j2πkδ1 to obtain ¯r(kN) =r(kN)e −j2πkδ1 .Let¯ζ(k) =

ζ(k)e −j2πkδ1 ,and¯η(k) =η(k)e −j2πkδ1 .Wehave

H 0 :¯r(kN) =¯η(k)

H 1 :¯r(kN) =e j2πk¯δμ + ¯ζ(k), (68)

k =1,...,K,

where ¯δ = δ − δ 1 is the remaining CFO (the estimation error).

The ALRT test based on the compensated system is therefore

¯T ALRT −CP

(

)

1

K∑

=Re (N s − kN)e −j2πk¯δ ¯r(kN) . (69)

r(0)

k=1

Although ¯δ is still unknown, but it is assumed to be much

small than δ = Nɛ mod 1. So we can search ¯δ in the

interval (−Δ, Δ] with limited searching steps. Let M be the

number of searching steps and ¯δ m = −Δ+2mΔ/M be the

searching points, m =1, 2, ··· ,M. For each searching point,

we compute the test

¯T (m)

ALRT −CP

(

1

K∑

=Re (N s − kN)e −j2πk¯δm ¯r(kN)

r(0)

k=1

Let the maximum of all the tests be

)

. (70)

¯T (max)

ALRT −CP = max ¯T (m)

ALRT −CP . (71)

1≤m≤M

The searching complexity is only O(KM)=O(MN s /N ).

This complexity is usually much lower than the complexity

for computing the AC r(kN), which is O(Ns 2/N ), asN s

is usually much larger than M. Hence, the searching only

increases the complexity marginally.

V. SUB-OPTIMAL APPROACHES

At very low SNR, the frequency offset estimation may

not be very accurate. We should also consider alternative

approaches without frequency offset estimation.

One such approach is simply taking absolute value in the

ALRT. We get three methods for continual pilot, scattered

pilots and their combination as follows.

T ALRT −CP−A = 1

r(0)

K∑

(N s − kN)|r(kN)|. (72)

k=1

T ALRT −SP−A = 1

K/L


(N s − kLN)|r(kLN)|. (73)

r(0)

T ALRT −CSP−A =

k=1

K/L

λ ∑

(N s − kLN)|r(kLN)|

r(0)

+ 1

r(0)

k=1

K∑

(N s − kN)|r(kN)|. (74)

k=1


8 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 31, NO. 3, MARCH 2013

Continual pilots

Scattered pilots

Continual pilots

Scattered pilots

Absolute error

Absolute error

10 −1

10 −2

Sensing time

10 −2

−15 −14 −13 −12 −11 −10 −9 −8 −7 −6 −5

10 −1 SNR

4 6 8 10 12 14 16

Fig. 1. Absolute error of the CFO estimation at different SNR (dB) (sensing

time: 5ms)

Fig. 3. Absolute error of the CFO estimation at different sensing time

(SNR=-12dB)

Continual pilots

Scattered pilots

Continual pilots

Scattered pilots

Absolute error

10 −2

Absolute error

10 −2

10 −3

10 −3

10 −1 SNR

−20 −19 −18 −17 −16 −15 −14 −13 −12 −11 −10

10 −1 Sensing time

4 6 8 10 12 14 16

Fig. 2. Absolute error of the CFO estimation at different SNR (dB) (sensing

time: 40ms)

Fig. 4. Absolute error of the CFO estimation at different sensing time

(SNR=-8dB)

VI. SIMULATIONS

In the computer simulations, the OFDM signals are generated

based on the DVB-T standard. As an example, we

consider 2K mode: there are N f = 2048 subcarriers, including

45 continual pilot subcarriers, 141 scattered pilot subcarriers,

and 343 null subcarriers. The power of a pilot subcarrier is

16/9 times that of a data subcarrier, and the CP length is

chosen as N g = N f /8. We consider a 8MHz bandwidth

channel. The signal is down-converted to baseband and the

sampling rate is 64/7 MHz.

Frequency selective channels are generated that have 20

taps with exponential power delay profile. At each Monte-

Carlo run, the tap coefficients of the channel are generated as

complex random numbers. The proposed sensing algorithm

does not require time synchronization, thus we assume the

received signal has an unknown random time delay.

It is assumed that the oscillator mismatch is up to 10ppm

and the carrier frequency is 1GHz. Thus the maximum CFO

is 10kHz, which makes the maximum normalized CFO to be

0.07/64 = 0.0011. For each Monte-Carlo test, we choose the

normalized CFO ɛ randomly (with even distribution) within

interval (−0.0011, 0.0011]. For ALRT methods, the composite

CFO Nɛis estimated and compensated first and then searching

is performed in the interval (−0.08, 0.08] with 30 searching

points. For the sub-optimal approaches, however, no composite

CFO estimation is required.

The thresholds are determined by Monte-Carlo tests. As

all the proposed methods are normalized by the instantaneous

received signal power (r(0)), the probability of false alarm

for any of them will be a constant at different noise powers

(known as constant false alarm (CFA) detection). Thus the

threshold can be set based on any fixed noise power. As an


ZENG et al.: SPECTRUM SENSING FOR OFDM SIGNALS USING PILOT INDUCED AUTO-CORRELATIONS 9

1

0.95

0.9

0.9

0.8

Probability of detection

0.85

0.8

0.75

0.7

0.65

0.6

ALRT−CP

ALRT−SP

ALRT−CP−A

ALRT−SP−A

ALRT−CSP

ALRT−CSP−A

Chen−Gao−Daut

Probability of detection

0.7

0.6

0.5

0.4

0.3

0.2

ALRT−CP

ALRT−SP

ALRT−CP−A

ALRT−SP−A

ALRT−CSP

ALRT−CSP−A

Chen−Gao−Daut

0.55

10 −3 10 −2 10 −1

Probability of false alarm

0.1

−14 −13 −12 −11 −10 −9 −8

SNR

Fig. 5. Detection probability versus false alarm rate at SNR=-12dB (sensing

time: 5ms)

Fig. 6. Detection probability versus SNR at fixed false alarm rate 0.001

(sensing time: 5ms)

example, for a target false alarm rate P fa , we run the detection

methods I times using the complex Gaussian noises of unit

power as input (without primary signal). We then keep the

test statistics of all runs and choose the ⌊I(1 − P fa )⌋ largest

value as the threshold. As long as I is large, the threshold can

be quite accurate. This approach is widely used for any CFA

detections to set the threshold when very accurate closed-form

expression is not available. Note that the threshold is set once

for all at any fixed P fa . Thus it can be done off-line.

Detailed comparisons for various known methods have been

given in [15], which show that the method in [15] is in general

better than other methods. So, in this paper we just show the

comparisons of the proposed methods with that in [15], which

is called Chen-Gao-Daut method in the following. The method

uses scattered pilots for the detection.

Figure 1 and 2 show the absolute errors (|δ − δ 1 |)ofthe

CFO estimation at different SNRs, where the sensing time

is fixed at 5ms (milli second) and 40ms, respectively. Here

we see that the estimation performance improves as the SNR

increases, and the algorithms works well at very low SNR (say,

-18dB) when the sensing time is long. Figure 3 and 4 show

the performances of the CFO estimation at different sensing

times with unit of ms, where the SNR is fixed at -12 dB and -8

dB, respectively. We see that longer sensing time gives better

estimation, but the improvement will be saturated as the SNR

is fixed. Obviously using scattered pilots always gives better

estimation as the number of scattered pilots is much larger than

that of continual pilots in the DVB-T standard. Simulations

also show that the proposed CFO estimation method is valid

for any frequency offset (full range estimation).

Figure 5 shows the probability of detection versus probability

of false alarm at SNR=-12dB. Figure 6 and 7 show

the probability of detection versus SNR at fixed probability of

false alarm 0.001 and 0.1, respectively. For these figures the

sensing time is 5ms. To show the performances at very long

sensing time, we give a result in Figure 8, where the sensing

time is 20ms with sample size of 182857 and the probability of

Probability of detection

1

0.95

0.9

0.85

0.8

0.75

0.7

0.65

0.6

ALRT−CP

ALRT−SP

ALRT−CP−A

ALRT−SP−A

ALRT−CSP

ALRT−CSP−A

Chen−Gao−Daut

0.55

−14 −13 −12 −11 −10 −9 −8

SNR

Fig. 7. Detection probability versus SNR at fixed false alarm rate 0.1 (sensing

time: 5ms)

false alarm is 0.1. Fromthefigures, we can see the following

conclusions. (1) Just using the continual pilots is not a good

approach as the the number of continual pilots in the DVB-T

standard is very small and the composite CFO estimation is

not accurate (shown in Figure 1 to 4). (2) For detections based

on the scattered pilots, the proposed ALRT-SP combined with

CFO estimation is much better than Chen-Gao-Daut method.

(3) The method using the absolute value approximation and

scattered pilots has similar performance to the Chen-Gao-

Daut method. (4) The heuristic combining of the continual

and scattered pilot methods (ALRT-CSP and ALRT-CSP-A)

does not give better performance, as the detection based

on continual pilots reduces the overall performance. Another

reason is that the detection based on scattered pilots has

already partially used the continual pilots as explained in

Section III.C. (5) The proposed methods are robust to channel,


10 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 31, NO. 3, MARCH 2013

Probability of detection

1

0.9

0.8

0.7

0.6

0.5

ALRT−CP

ALRT−SP

ALRT−CP−A

ALRT−SP−A

ALRT−CSP

ALRT−CSP−A

Chen−Gao−Daut

−20 −19 −18 −17 −16 −15 −14

SNR

Fig. 8. Detection probability versus SNR at fixed false alarm rate 0.1 (sensing

time: 20ms)

time delay and noise power uncertainties. In fact, in the

simulations the frequency selective channel, time delay, and

frequency offset are randomly set, and the noise power is not

used in the detection.

VII. CONCLUSIONS

In this paper, we have proposed sensing methods using

the repetition structure of the pilots in OFDM-based systems.

The derived approximated LRT (ALRT) methods only use

the sample auto-correlations of the received signal. We have

also proposed a method to estimate the CFO based on the

auto-correlations and compensate its effect in sensing design.

Extensive simulations with the DVB-T signals have shown that

using the ACs induced by the scattered pilots achieves the

best performance. Compared to the methods using matched

filtering, the proposed methods are more robust to frequency

offset, frequency selective channel and time delay uncertainty.

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Yonghong Zeng received the B.S. degree from Peking University, Beijing,

China, and the M.S. degree and the Ph.D. degree from National University

of Defense Technology, China. Since Nov. 2004, he has been working

in the Institute for Infocomm Research, A ∗ STAR, Singapore, as a senior

research fellow, research scientist, and senior scientist. His current research

interests include signal processing and wireless communications, especially

on cognitive radio and software defined radio, cooperative communications,

MIMO, channel estimation, equalization, detection, and synchronization.

He has published/invented six books, a number of papers and patents.

He received the first IEEE Communication Society Asia Pacific Outstanding

Paper Award in 2012. He received the Certificate of Appreciation for

outstanding contributions to the IEEE 802.22 standard in 2011. He was

the recipient of the IES (Institute of Engineers Singapore) prestigious

engineering achievement awards in 2007 and 2009. He received the ministrylevel

Scientific and Technological Development Awards in China four times.

He was an EXCO member of IEEE Singapore Section in 2009. Currently

he serves as editor for IEEE Trans. Wireless Communications and Journal

of Engineering. He served as an organizing committee member or Track

chair or session chair or TPC member for many prestigious international

conferences such as IEEE ICC, IEEE GlobeCom, IEEE DySPAN, APSIPA,

IEEE WCNC, IEEE VTC, IEEE ICIEA and CrownCom in recent years. See

more details here: http://www1.i2r.a-star.edu.sg/ yhzeng/

Ying-Chang Liang (SM’00-F’11) is now a Principal Scientist with the

Institute for Infocomm Research (I2R), Agency for Science, Technology

and Research (A*STAR), Singapore, and holds an adjunct professorship

position with University of Electronic Science & Technology of China

(UESTC), China. He was a visiting scholar with the Department of Electrical

Engineering, Stanford University, from Dec 2002 to Dec 2003, and taught

graduate courses in National University of Singapore from 2004 - 2009.

His research interest includes cognitive radio, dynamic spectrum access,

reconfigurable signal processing for broadband communications, information

theory and statistical signal processing.

Dr Liang was elected a Fellow of the IEEE in 2011, and has received

five Best Paper Awards, including IEEE ComSoc APB outstanding paper

award in 2012, and EURASIP Journal of Wireless Communications and

Networking best paper award in 2010. He also received the Institute of

Engineers Singapore (IES)’s Prestigious Engineering Achievement Award in

2007, I2R’s Achiever of the Year Award in 2008, and the IEEE Standards

Association’s Certificate of Appreciation Award in 2011, for his contributions

to the development of IEEE 802.22, the first worldwide standard based on

cognitive radio technology.

Dr Liang currently serves as Editor-in-Chief of IEEE Journal on Selected

Areas in Communications - Cognitive Radio Series. He was an Associate

Editor of IEEE Transactions on Wireless Communications from 2002 to

2005, and served as a Guest Editor of five special issues on emerging topics

published in IEEE, EURASIP and Elsevier journals in the past five years.

He is also a Distinguished Lecturer of the IEEE Vehicular Technology

Society, an Associate Editor of IEEE Transactions on Vehicular Technology,

and a member of the Board of Governors of the IEEE Asia-Pacific

Wireless Communications Symposium. He served as technical program

committee (TPC) Co-Chair of 2010 IEEE Symposium on New Frontiers

in Dynamic Spectrum Access Networks (DySPAN’10), General Co-Chair

of 2010 IEEE International Conference on Communications Systems

(ICCS’10), and Symposium Chair of 2012 IEEE International Conference

on Communications (ICC’12).

The-Hanh Pham received the B.E. degree in Electronics and Telecommunications

Engineering from Hanoi University of Technology, Vietnam, in 2001.

At National University of Singapore (NUS), he earned the Ph.D. in Electrical

Engineering in 2008. He was a Research Fellow with NUS and a Scientist with

the Institute for Infocomm Research (I2R), Singapore. His research interests

include cognitive radio and relay networks.

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