ULTRASONIC OFDM PULSE DETECTION FOR TIME OF FLIGHT

MEASUREMENT OVER WHITE GAUSSIAN NOISE CHANNEL

Daniel F. Albuquerque, Jos

´

e M. N. Vieira, Carlos A. C. Bastos and Paulo J. S. G. Ferreira

Signal Processing Lab – IEETA/DETI, University of Aveiro, 3810-193 Aveiro, Portugal

Keywords:

Pulse Detection, OFDM, Ultrasonic Location System.

Abstract:

In this paper we evaluate the probability of detection and the probability of false alarm for an OFDM pulse

over AWGN channel. This type of pulse is useful for indoor location systems using ultrasounds due to the

ability to accurately measure the time of ﬂight by pulse detection while transmitting some data. Moreover, we

can avoid the RF auxiliary channel by using an OFDM-based sync. The probability of OFDM pulse detection

over white Gaussian noise will be presented as function of the probability of false alarm when a threshold

technique is employed at the receiver. Furthermore, the pulse detection probability will be compared with a

chirp pulse.

1 INTRODUCTION

Indoor location is an active area of research in the sig-

nal processing community with a large potential from

the point of view of applications (Sayed et al., 2005;

Liu et al., 2007). To perform indoor location, there are

2 main types of solutions: Ultrasonic (US) and Radio

Frequency (RF) based systems. RF power strength

based systems are inexpensive but require the proﬁl-

ing of the entire location scenario to get a RF ﬁnger-

print resulting in an accuracy from 1 to 5 meters ap-

proximately (Stuntebeck et al., 2008; Bahl and Pad-

manabhan, 2000). On the other hand, the ultrasound

technology is the best suited to achieve the accuracy

required on indoor environments, that can be less than

1 cm in some cases (Gonzalez and Bleakley, 2009;

Prieto et al., 2007). Our location system, LocUS, aim

to be entirely based on ultrasound signals. Most of

the known ultrasonic location systems use an auxil-

iary RF channel for synchronization. An RF pulse is

used as a time reference for measuring the propaga-

tion delay between the source and the receiver (Hazas

and Hopper, 2006). Although this auxiliary RF chan-

nel allows very simple clock synchronization and de-

lay measurement solutions, it also gives away two im-

portant advantages that US-based systems bear in ref-

erence to RF-based ones: the immunity to RF interfer-

ence, and the ability to safely operate in the presence

of critical electronic instrumentation such as medical

or life-support systems. Therefore, one way to avoid

the use of an auxiliary RF signal to measure the time-

of-ﬂight (TOF) is to synchronize the clocks of the

nodes (Skeie et al., 2001). To achieve this, the nodes

should be able to send to each other the clock infor-

mation using the ultrasonic channel. Due to the reﬂec-

tion of the ultrasonic signals on the walls the acoustic

communication channel presents a strong multipath

effect causing inter-symbolic interference.

LocUS solved this problem by using OFDM (Or-

thogonal Frequency Division Multiplexing) to per-

form data transmission. This technique has al-

ready been used in ultrasonic underwater communi-

cations (Mason et al., 2007; Nakashima et al., 2006).

OFDM is a very ﬂexible modulation technique, robust

to multipath and that simpliﬁes the channel equal-

ization. It is also very sensitive to synchronization,

which may be an advantage when the application re-

quires the measurement of the TOF (Levanon and

Mozeson, 2004). As we intend to send just one

OFDM data pulse with the carriers phase modulated,

to measure the phases we need to send another pulse

to measure the phases difference. That way, the ﬁrst

pulse has a double purpose, it should have a high en-

ergy to be effectively detected by a matched ﬁlter and

it will be used as a phase reference by the second

pulse to decode the transmitted data.

In this paper it will be presented an analytical ex-

pression for the probability of OFDM pulses detec-

tion in the presence of white Gaussian noise. For

that propose it will be presented, in section 2, the

345

F. Albuquerque D., M. N. Vieira J., A. C. Bastos C. and J. S. G. Ferreira P..

ULTRASONIC OFDM PULSE DETECTION FOR TIME OF FLIGHT MEASUREMENT OVER WHITE GAUSSIAN NOISE CHANNEL.

DOI: 10.5220/0003375103450350

In Proceedings of the 1st International Conference on Pervasive and Embedded Computing and Communication Systems (PECCS-2011), pages

345-350

ISBN: 978-989-8425-48-5

Copyright

c

2011 SCITEPRESS (Science and Technology Publications, Lda.)

asynchronous data transmission with OFDM pulses.

In section 3 it is presented the probability of detec-

tion and the the probability of false alarm for OFDM

pulses over a white Gaussian noise channel when a

threshold technique is used in the receiver. Some sim-

ulation results will be presented in section 4. At the

end it will be presented a brief conclusion.

2 ASYNCHRONOUS DATA

TRANSMISSION WITH OFDM

An architecture for asynchronous data transmission

using OFDM pulses is proposed. We use two con-

catenated OFDM pulses, one for time synchroniza-

tion (e.g. time-of-ﬂight measurement) and another for

some data information transmission (e.g. source iden-

tiﬁcation). The group of the synchronization pulse

and the data information pulse is called frame.

2.1 Frame Prototype

Figure 1 presents the proposed frame prototype, as

mention before, there are two different main pulses in

the frame: The OFDM Sync. and the OFDM Data.

The ﬁrst pulse is detected by the receiver by matched

ﬁlter and used to synchronization. It will also be used

to demodulate the second pulse, the OFDM Data, by

a differential demodulation scheme (Haykin, 2001).

This method was chosen manly because the OFDM

pulse is robust to environments with multipath but it

does not produce a very high resolution in time syn-

chronization, as will be seen later on. Therefore, the

differential demodulation will be robust to this small

jitter.

Guard

FFT

OFDM Sync.

Guard

Time

OFDM Data

Figure 1: Asynchronous data transmission with OFDM,

frame prototype.

The Guard FFT, presented in Figure 1 is a cyclic

extension to protect the demodulation process due to

the time synchronization jitter. The Guard Time is

a cyclic extension of the OFDM data pulse to avoid

inter-symbolic interference caused by the room im-

pulse response (Schulze and Luders, 2005).

3 OFDM PULSE DETECTION

Two different techniques to perform pulse detec-

tion can be used, the conventional time domain MF

(Matched Filter) or the FFT (Fast Fourier Transform)

followed by a small scalar product. Figure 2 presents

the FFT technique, where the incoming signal, x

(pulse to detect plus noise) enters in a Buffer Delay

that has the same size of the pulse s. Therefore, the

information in the buffer is converted to the frequency

domain. Moreover, due to the OFDM properties S(k)

(Fourier transform of s) is only different from zero in

the information carriers so the system only needs to

perform the dot product in that carriers.

Delay

Buffer

FFT

...

S*(k)

a¢b

a

x(n)

y(n)

...

b

... ...

... ...

Figure 2: Pulse detection with FFT.

In the following sections is considered that the

FFT and IFFT are normalize to 1/

√

N:

X(k) =

1

√

N

N−1

∑

n=0

x(n)e

−j2π

nk

N

(1)

x(n) =

1

√

N

N−1

∑

k=0

X(k)e

j2π

nk

N

. (2)

Therefore, the output y

FFT

(n) of FFT detector can

be written as:

y

FFT

(n) =

1

√

N

N−1

∑

k=0

S

∗

(k)

N−1

∑

m=0

x(m + n −N + 1)W

−mk

.

(3)

where W = e

j

2π

N

and ()

∗

is the conjugate.

On the other hand, the output of the MF detector

y

MF

(n) is given by (Levanon and Mozeson, 2004):

y

MF

(n) =

N−1

∑

m=0

s

∗

(N −1 −m)x(n −m)

=

N−1

∑

m=0

x(m + n −N + 1)s

∗

(m)

(4)

where s(m) is the inverse Fourier transform of S(k):

y

MF

(n) =

N−1

∑

m=0

x(m+n−N +1)

1

√

N

N−1

∑

k=0

S(k)W

mk

!

∗

(5)

reorganizing the terms in the equation:

y

MF

(n) =

1

√

N

N−1

∑

k=0

S

∗

(k)

N−1

∑

m=0

x(m + n −N + 1)W

−mk

.

(6)

PECCS 2011 - International Conference on Pervasive and Embedded Computing and Communication Systems

346

As can be seen, equation (6) is equal to equation (3).

In the FFT technique the scalar product has the

size of the number of carriers. On the other hand, the

MF technique needs to have a ﬁlter with the size of the

pulse. So the choice between one of these techniques

depends on the pulse size and the number of carri-

ers. If the pulse is small and there is a large number

of carriers, the MF is better than the FFT technique,

otherwise, if the pulse is big and there are only a few

carriers the FFT is better than the MF technique.

In order to simplify equation (6) and without loss

of generalization in the next section we will consider

the detector output with a time advance of (N −1)

y(n) =

1

√

N

N−1

∑

k=0

S

∗

(k)

N−1

∑

m=0

x(m + n)W

−mk

. (7)

3.1 Probability of Detection

In order the detect the presence of a pulse the output

y(n) must be compared to a given threshold. There-

fore, we consider that a pulse is detected if the output

y(n) is greater than the given threshold. Therefore,

the threshold must be chosen in a way that a very

weak pulse must be detected and the noise does not

produce false alarms. In this way, it is very impor-

tant to keep the probability of false alarm, which is

the probability of detecting the pulse when in the in-

put there is only noise, very small and the probability

of detection very high. To evaluate these probabilities

we must consider two competing hypotheses:

H

0

: x(n) = w(n)

H

1

: x(n) = w(n) + As(n)

(8)

which is equivalent to:

H

0

: x(n) = w(n) + As(n) , A = 0

H

1

: x(n) = w(n) + As(n) , A 6= 0

(9)

where H

0

is called the null hypothesis, H

1

the alter-

native hypothesis and w(n) is white Gaussian noise.

The system choose the hypothesis by the magnitude

of the signal y(n). Therefore, if the magnitude of y(n)

is greater than a given threshold γ the system decides

H

1

otherwise the system decides H

0

. This procedure

is shown in Figure 3.

> °

< °

jy(n)j

2

H

1

H

0

j j

2

y(n)

Figure 3: Detection decision block.

The output of the ﬁrst block, after the compara-

tors, is given by:

|y(n)|

2

=

1

N

N−1

∑

k=0

S

∗

(k)

N−1

∑

m=0

x(m + n)W

−mk

2

(10)

Using (9):

|y(n)|

2

=

1

N

N−1

∑

k=0

S

∗

(k)

N−1

∑

m=0

w(m + n)W

−mk

+

+A

N−1

∑

m=0

s(m + n)W

−mk

!

2

(11)

The signal S(k) is only different from zero in a set

of carriers of the OFDM pulse, k ∈ (k

0

,k

1

,...,k

N

p

−1

)

(where N

p

is the number of OFDM pulse carriers),

they will be real due to the differential demodulation

and without loss of generalization we can consider

that |S(k)| = 1 for k ∈ (k

0

,k

1

,...,k

N

p

−1

). So we can

simplify the equation 11 to:

|y(n)|

2

=

1

N

k

N

p

−1

∑

k=k

0

S(k)

U

k

N

+ jV

k

N

!

+

A

k

N

p

−1

∑

k=k

0

S(k)

N−1

∑

m=0

s(m + n)W

−mk

2

(12)

where U(k/N) and V (k/N) for k = 1, 2,..., N/2 −

1 are random independent and Gaussian vari-

ables (Steven, 1993). Therefore, if w(m) is white

noise with zero mean and variance σ

2

, the U(k/N)

and V (k/N) also has zero mean and variance Nσ

2

/2.

Normalizing U and V to have unit variance, |y(n)|

2

becomes:

|y(n)|

2

=

N

p

σ

2

2

U(k/N)

p

Nσ

2

/2

+ j

V (k/N)

p

Nσ

2

/2

+ µ(n)

2

(13)

where µ(n) can be seen as the mean of the sum of this

two variables, moreover, it is dependent of the pulse

and it is only different from zero when the pulse is

present, µ(n)can be written as:

µ(n) =

A

σ

k

N

p

−1

∑

k=k

0

k

N

p

−1

∑

l=k

0

S(k)S(l)W

l(n−n

i

)

P(k −l, n)

N

p

N

p

/2

(14)

where n

i

is the ﬁrst sample of the pulse and P(k,n) is

the Fourier transform of the boxcar function:

P(k, n) = W

−

(N−1+n

i

−n)k

2

ψ(k, n) (15)

ULTRASONIC OFDM PULSE DETECTION FOR TIME OF FLIGHT MEASUREMENT OVER WHITE GAUSSIAN

NOISE CHANNEL

347

where, for k 6= 0 ψ(k,n) is deﬁned as:

ψ(k, n) =

sin

(N+n

i

−n)πk

N

sin

(

πk

N

)

n

i

≤ n < n

i

+ N

sin

(N+n−n

i

)πk

N

sin

(

πk

N

)

n

i

−N < n < n

i

0 otherwise.

(16)

Rewriting this equation, ψ(k, n) becomes:

ψ(k, n) =

sin

(n−n

i

)πk

N

sin

πk

N

(−1)

k

υ(n) (17)

were υ(n) is:

υ(n) = u(n −n

i

−1 + N) −2u(n −n

i

) +u(n −ni −N)

(18)

and u(n) is a unitary step (it is zero for n < 0 and one

otherwise). For k = 0, ψ(k, n) is deﬁned as:

ψ(0,n) = Nυ(n)

2

+ (n −n

i

)υ(n) (19)

So the equation 14 can be simpliﬁed to:

µ(n) =

A

σ

k

N

p

−1

∑

k=k

0

k

N

p

−1

∑

l=k

0

S(k)S(l)W

(k+l)(n−n

i

)

2

ψ(k −l, n)

N

p

N

p

/2

(20)

Separating µ(n) in real and complex part it can be

written as:

µ(n) = µ

U

(n)+ jµ

V

(n) =

√

η

h

µ

0

U

(n)+ jµ

0

V

(n)

i

(21)

where η is:

η =

A

2

σ

2

. (22)

As S is real, µ

0

U

(n) and µ

0

V

(n) can be written sepa-

rately:

µ

0

U

(n) =

k

N

p

−1

∑

k=k

0

k

N

p

−1

∑

l=k

0

S(k)S(l)ψ

U

(k, l; n)

N

p

N

p

/2

(23)

µ

0

V

(n) =

k

N

p

−1

∑

k=k

0

k

N

p

−1

∑

l=k

0

S(k)S(l)ψ

V

(k, l; n)

N

p

N

p

/2

(24)

where ψ

U

(k, l; n) and ψ

V

(k, l; n) for k 6= l are deﬁned

as:

ψ

U

(k, l; n) =

sin

2πl(n−n

i

)

N

−sin

2πk(n−n

i

)

N

2sin

π(k−l)

N

υ(n)

(25)

ψ

V

(k, l; n) =

cos

2πl(n−n

i

)

N

−cos

2πk(n−n

i

)

N

2sin

π(k−l)

N

υ(n)

(26)

for k = l ψ

U

(k, l; n) and ψ

V

(k, l; n) are deﬁned as:

ψ

U

(l, l;n) = cos

2πl(n −n

i

)

N

ψ(0,n) (27)

ψ

V

(l, l;n) = −sin

2πl(n −n

i

)

N

ψ(0,n) (28)

Rewriting the equation 13, |y(n)|

2

becames:

|y(n)|

2

=

N

p

σ

2

2

"

U(k/N)

p

Nσ

2

/2

+ µ

U

(n)

!

2

+

+

V (k/N)

p

Nσ

2

/2

+ µ

V

(n)

!

2

#

(29)

The two quadratic terms in equation 29 can be

seen as a Chi-Squared distribution with two Gaussian

white independents variables (χ

0

2

2

(λ(n))) (Steven,

1993), one with µ

U

(n) mean and another with µ

V

(n)

mean. Where λ(n) is given by (Steven, 1993):

λ(n) = µ

U

(n)

2

+ µ

V

(n)

2

= ηλ(n)

0

(30)

where λ(n)

0

is given by:

λ(n)

0

= µ

0

U

(n)

2

+ µ

0

V

(n)

2

(31)

From equations 23, 24, 30 and 31 we can get two

important conclusions: First, the value of N does not

change the magnitude of λ

0

, second, λ is proportional

to η consequently λ is proportional to the signal to

noise ratio.

For the particular case when n = n

i

, (ﬁrst sample

of the pulse), λ

0

takes the value 2N

p

so λ becomes:

λ(n

i

) = η2N

p

(32)

The Signal to Noise Ratio (SNR) is equal to

ηN

p

/N, therefore, λ can also be written in function

of the SNR and the size of the pulse:

λ(n

i

) = 2N ×SNR (33)

With these results it would be easy to compute the

probability of false alarm (P

f a

) and the probability of

detection (P

d

(n)) for any instant n.

The probability of false alarm can be seen as the

probability of choose H

1

when the system is in the

presence of H

0

i.e. the system believes to be in the

presence of the pulse but it only has noise in the input

(µ

U

(n) and µ

V

(n) are equal to zero). The probability

of false alarm can be deﬁned as:

P

f a

= Pr

n

|y(n)|

2

> γ;H

0

o

(34)

Replacing |y(n)|

2

we obtain:

P

f a

= Pr

U(k/N)

√

Nσ/2

2

+

V (k/N)

√

Nσ/2

2

>

>

2γ

N

p

σ

2

;H

0

(35)

PECCS 2011 - International Conference on Pervasive and Embedded Computing and Communication Systems

348

Therefore the probability of false alarm is:

P

f a

= Q

χ

0

2

2

(0)

2γ

N

p

σ

2

= Q

χ

2

2

2γ

N

p

σ

2

= e

−

γ

N

p

σ

2

(36)

where χ

2

2

represent a Chi-Squared distribution with

two Gaussian white independents variables with unit

variance and zero mean (Steven, 1993).

The probability of detection can be seen as the

probability of choosing H

1

when the system is in the

presence of the pulse i.e. the system considers that a

pulse was received and and one is present. The prob-

ability of detection can be deﬁned as:

P

d

(n) = Pr

n

|y(n)|

2

> γ;H

1

o

(37)

Doing the same for P

d

that it was done for P

f a

P

d

(n) = Q

χ

0

2

2

λ(n)

2γ

N

p

σ

2

. (38)

Using (36) P

d

can be written as function of P

f a

P

d

(n) = Q

χ

0

2

2

λ(n)

ln(P

f a

−2

)

(39)

or

P

d

(n) = Q

χ

0

2

2

ηλ

0

(n)

ln(P

f a

−2

)

. (40)

4 RESULTS

In order to validate the theoretical detection probabil-

ity a test for a probability of false alarm of 10

−16

was

performed. In this test, it was considered the follow-

ing practical situation:

• An OFDM pulse with 17 carriers sampled at

100 kHz with the following amplitudes

S(k) = {1, 1,-1, -1,-1,-1,1,1, -1,1, 1,...

...-1,1,-1,1,-1,1}, k = k

0

.. .k

N

p

−1

;

• the lowest carrier frequency was set to 40 KHz

(k

0

= 400);

• the pulse has 1000 samples;

• the signal to noise ratio was set to 0 dB;

• for each sample 1000 simulations were per-

formed;

Figures 4 and 5 present the probability of detection

for a probability of false alarm of 1 ×10

−16

and 1 ×

10

−8

respectively. The horizontal axis represents the

incoming instant sample of the pulse with n

i

as the

instant when the last sample of the pulse was received.

Figure 4 shows that there is a great probability to

detect the pulse before the system receive the whole

ni−1000 ni ni+1000

0

0.5

1

P

d

Samp l e

expression

simulation

Figure 4: Probability of detection for a probability of false

alarm of the 1 ×10

−16

.

ni−1000 ni ni+1000

0

0.5

1

P

d

Samp l e

expression

simulation

Figure 5: Probability of detection for a probability of false

alarm of the 1 ×10

−8

.

pulse. Therefore, to improve the detection capabilities

we can use a very simple algorithm: When the system

detects a pulse it can wait for a period of time equal

to the length of the pulse and choose the maximum

value at the match ﬁlter output.

4.1 Comparison with other Usual Pulse

It was chosen a linear chirp also known as a linear

frequency-modulated pulse (Levanon and Mozeson,

2004) to compare with the proposed pulse. A lin-

ear chirp is a signal where its instantaneous frequency

change linearly with time. Moreover, this type of sig-

nals are usually used in radar and sonar systems (Lev-

anon and Mozeson, 2004).

For this test we chosen a chirp pulse with similar

characteristics (bandwidth, length, maximum ampli-

tude) to the OFDM pulse. The autocorrelation func-

tions of the OFDM and chirp pulses are shown in Fig-

ure 6.

−N 0 N

0

0.5

1

Sample

Amplitude

OFDM

Chirp

Figure 6: Comparison of the autocorrelation functions of a

chirp pulse and an OFDM pulse.

From Figure 6 it is possible to say that the Chirp

pulse is better for detection over noise environments

than the proposed OFDM pulse. Manly because it

ULTRASONIC OFDM PULSE DETECTION FOR TIME OF FLIGHT MEASUREMENT OVER WHITE GAUSSIAN

NOISE CHANNEL

349

produces a bigger amplitude in the matched ﬁlter out-

put. To prove this statement we evaluated the proba-

bility of detection as a function of the probability of

false alarm for 0 dB ratio of signal amplitude to noise

standard deviation. The results are presented in Fig-

ure 7.

10

−200

10

−150

10

−100

10

−50

10

0

0

0.2

0.4

0.6

0.8

1

P

fa

P

d

OFDM

Chirp

Figure 7: Probability of detection in function of probability

of false alarm.

5 CONCLUSIONS

In this paper we evaluated the probability of detection

of an OFDM pulse over white Gaussian noise chan-

nels for a given probability of false alarm. Therefore,

the resultant expression was validated with computer

simulations. The results demonstrated that the proba-

bility of detection of an OFDM pulse is lower than a

chirp with similar characteristics. This lower perfor-

mance comes from the reduced energy of the OFDM

compared to the chirp, this energy difference is due

to the amplitude concerns. However, this worst per-

formance does not compromise the potential for us-

ing OFDM pulses for asynchronous communication.

Simulation results have sown that if the probability of

false alarm is set to 10

−50

the pulse has a 90% prob-

ability of detection. Moreover, the proposed OFDM

pulse will be used not only for TOF measurements but

also for some data communication.

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