110B, to ease practical aspects concerning transceiver implementation.

In some embodiments, the transmitting device 110B may also utilize non-sinusoidal wavelets or may utilize a plurality of wavelets where each signal element utilizes a different wavelet for further scalability and security. The transmitting device 110B may transmit the pilot sequence 130 over T_n∈ [Image Omitted] baud intervals, and the pilot symbol modulating the sinusoid transmitted from the m∈th antenna during the t∈th baud interval is given in the mth row and tth column of the pilot matrix X_n∈ and denoted by X_n(m, t) in the following formula 3 wherein X_n(m, t)˜(−π, π):

❘
"[LeftBracketingBar]"

X
n

(

m
,
t

)

❘
"[RightBracketingBar]"

=

{

~

𝒰
⁡
(

0
,
1

)

,

v

n
,
m
,
t

≥

v

n

0
,

o
.

w
.

(
3
)

In the formula 3, v_n,m,t^˜ [Image Omitted] (0,1) is a random variable that determines whether the transmitting device's 110B mth antenna is activated during the tth baud interval and Un is the transmitting device's 110B activation threshold that determines the antenna activation probability; furthermore ƒ_p(v_n,m₀_,t₀, v_n,m₁_,t₁)=ƒ_p(v_n,m₀_,t₀) ƒ_p(v_n,m₁_,t₁)∀m₀, m₁∈ [Image Omitted] ; t₀, t₁∈. A zero entry in X_nimplies that no transmission occurs from that antenna during that baud interval. Therefore, X_n, describes the particular antenna activation sequence used by the transmitting device 110B in its columns, and the particular pilot sequence 130 transmitted by the transmitting device 110B and its elements. Together the variables v_n,m,tand v_ndescribe the activation sequence 140 for the transmitting device.

In some embodiments, the receiving device 110A is equipped with N_s∈ [Image Omitted] antennas that are formed in a nonmodified 2D ULAA and have the default spatial signature 125 thereof. The difference in the unit spatial signature 125 in the transmit direction of Ω term denoted by e_t(Ω) may be replaced with the term e_n(Ω) derived in formula 2. The signal received at the receiving device's 110A n<N_sth antenna at the end of the t<T_nth baud interval is given on the nth row and tth column of y_s∈ [Image Omitted] in the following formula 4:

y
s

=

H
n

⁢

X
n

+
w

(
4
)

In the formula 4, w∈ [Image Omitted] is the additive white Gaussian noise (AWGN) matrix comprising independent elements identically distributed with ˜(0, (σ_h/γ_n)²). γ_nis the transmitting device's 110B SNR.

III. Detection Algorithm

Since the receiving device 110A relies on random deviations of spatial signatures 125, a detection algorithm for the transmitting device's 110B spatial signature deviation can be implemented to decide if the received signal is coming from the transmitting device 110B or not. This algorithm is illustrated in FIG. 5 as the method 500. The detection algorithm may be derived by correlating the transmitting device's 110B expected received signal over y_s. The correlation is calculated using the following formula 5:

ρ
=

tr
⁡
(

X
n
H

⁢

H
n
H

⁢

y
s

)

(
5
)

As the noise floor is sensitive to temperature and possible signal emission from imposters, the noise variance,

σ
^

n
2

∈ [Image Omitted] , may be estimated by similarly correlating y_swith any signature orthogonal to that of the transmitting device 110B and all other possible authorized users. The detection metric β∈ is then given by the following formula 6:

β
=

ρ

σ
^

n
2

(
6
)

The detection metric is then compared to a threshold value (ψ). To minimize the error, one threshold can be selected as half of the distance between two states according to the following formula 7:

ψ
e

=

tr
⁢

(

X
n
H

⁢

H
n
H

⁢

H
n

⁢

X
n

)

2
⁢

σ
^

n
2

(
7
)

However, ψ_eis not a good threshold for low SNR scenarios, which results in high false alarm (FA) rates. To prevent that, a threshold (ψ_FA) can be precalculated to fix the FA probability to a designated value. FA probability can be represented using the following formula 8:

Pr
⁡
(
FA
)

=

Pr
⁡
(

β
>

ψ

F
⁢
A

|

y
s

)

=
w

(
8
)

The formula 8 may be ˜ [Image Omitted] (0, tr(X_n^HH_n^HH_nX_n)). The final threshold is found as the combination of both threshold as the following formula 9:

ψ
=

max
⁡
(

ψ
e

,

ψ
FA

)

(
9
)

Performance analysis for a variety of false alarm thresholds is presented in Sec. IV.

IV. Performance Analysis

To evaluate the proposed authentication method, link level simulations have been performed under highly scattering Rayleigh channel. The receiving device 110A is assumed to have 512 antennas at all times, while the transmitting device 110B may have different number of active antennas depending on v_n.

FIG. 3 is an illustration of a graph 300 of the missed detection rate of a transmitting device's 110B signature against SNR. The proposed method fails to authenticate a transmitting device 110B with less than 1% probability at most at 13 dB SNR if M_n=16 antennas are activated while Pr(FA)=0.001. It is also seen that 1% misdetection probability can be achieved when 8 dB SNR for both M_n=16 and M_n=128 active antennas with relaxed FA requirement of Pr(FA)=0.01. Lower rates are possible as the number of active antennas is increased, or the false alarm probability requirement is relaxed.

FIG. 4 illustrates a graph 400 of false detection rates of a transmitting device's 110B signature 125 when a receiving device 110A is only receiving noise. As it is seen from FIG. 4, the desired FA rates of Pr(FA)=0.001 and Pr(FA)=0.01 are closely achieved for SNR values up to 12 dB and 10 dB, respectively. After those SNR values the secondary threshold of ψ_ebecomes effective and improves the false alarm performance for a transmitting device's 110B signature 125 detection algorithm.

V. Example Embodiments