Spectral and temporal stealthy fiber-optic communication using sampling and phase encoding

Tomer Yeminy; Dan Sadot; Zeev Zalevsky

doi:10.1364/OE.19.020182

1. Introduction

The rapid spread of optical communication systems has increased the need for proper security solutions in order to prevent eavesdropping and jamming. Spread spectrum encryption techniques seem to be appropriate for optical communication systems due to their large bandwidth which enables to achieve high processing gain [1].

Methods for covert transmission using coherent optical code-division multiple-access (OCDMA) are proposed in [2–4]. In these methods, the signal is encrypted in the time domain using a dispersive element which encodes the spectral phase of the signal. Implementing the approaches presented in [5,6], the temporal phase of the dispersed signal is encoded, hence assigning different phase to each spectral component. However, using these encryption methods when transmitting outside the bandwidth of a public channel, the spectral concealment of the signal is not promised since the signal is not necessarily spectrally hidden under the noise level. If the signal is not concealed in the frequency domain, an adversary that coherently detects and samples the signal can perform Discrete Fourier Transform, therefore disclosing the spectral amplitude of the signal. Hence, the transmitted signal is not spectrally stealthy.

In the encryption method which is proposed in this work, the spectral amplitude of the signal is deliberately spread wide, essentially enabling to transmit a signal with low power spectral density (PSD), keeping the signal below the noise level in the frequency domain. The spectral spreading is achieved by sampling the signal. At the receiver, all the spectral replicas of the signal are folded to the baseband, therefore the PSD of the signal is reconstructed and in turn, the signal to noise ratio (SNR) is improved. This is achieved by coherently adding all the signal's spectral replicas at the baseband (hence the signal is reinforced) whereas the spectral replicas of the noise are added incoherently (consequently they are averaged to a low value).

The proposed covert communication system performances are analytically derived and shown to be supported by those accomplished by simulating the communication system. Finally, the encryption strength is estimated.

2. System description

Suppose an information source generating a bit sequence which modulates an optical carrier with frequency $f_{c}$ . The modulation results in the pulse sequence $s (t)$ which we would like to encrypt in both time and frequency domains. Each pulse $p (t)$ in the sequence has pulse width $T$ and double sided bandwidth $Δ f$ . The gap between two sequential pulses in the time domain is $Δ t$ , satisfying $Δ t \geq T$ . A digital implementation of the proposed communication system is illustrated in Fig. 1 . For the simplicity of the mathematical analysis we discuss an equivalent analog system presented in Fig. 2 .

Fig. 1 Digital covert communication system. TPE-Temporal Phase Encoder, FFT- Fast Fourier Transform, SPE-Spectral Phase Encoder, IFFT-Inverse Fast Fourier Transform, D/A-Digital to Analog converter, Mod.-Modulator, EDFA – Erbium Doped Fiber Amplifier, AWGN-Additive White Gaussian Noise, SPD-Spectral Phase Decoder, A/D-Analog to Digital converter, MF-Matched Filter, TPD-Temporal Phase Decoder.

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Fig. 2 Proposed covert communication system. Mod.-Modulator, TPE-Temporal Phase Encoder, SPE-Spectral Phase Encoder, EDFA-Erbium Doped Fiber Amplifier, AWGN-Additive White Gaussian Noise, SPD-Spectral Phase Decoder, MF-Matched Filter, TPD-Temporal Phase Decoder.

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First, each pulse is multiplied by a different temporal phase term. Accordingly, the addition of the Fourier Transform (FT) of the various pulses in the signal bandwidth is incoherent, thus reducing the PSD of $s (t)$ . Then, a sampler with sampling frequency $Δ f$ is applied to have $M = B W / Δ f$ spectral replicas of the signal in the frequency domain, $B W$ being the communication system bandwidth. An analog sampler can be implemented by a modulator transmitting a burst of the optical signal each $1 / Δ f$ seconds. In addition, the spectral phase of the signal is optically encoded, spreading the signal in the time domain. Finally, the signal's amplitude is amplified by $\sqrt{M}$ in order to compensate for the attenuation of the spectral amplitude of the signal stemming from the spectral spreading of the signal due to the sampling. White Gaussian noise $n (t)$ is added due to amplification and channel noises. The signal spreading in the time and frequency domains keeps it below the noise level in both domains.

The decryption process begins with optical spectral phase decoding followed by coherent detection. The signal is subsequently sampled at sampling frequency $Δ f$ in order to fold all its spectral replicas to the baseband where they are coherently added. Then, a filter matched to $p (t)$ is applied. Finally, the temporal phase of the signal is decoded and the signal is sampled with sampling interval $Δ t$ . The original bit sequence is recovered by a decision circuit.

2.1 Encoder configuration

Suppose a bit sequence modulating an optical carrier, generating a pulse sequence represented by the analytical signal:

s (t) = \sum_{n = 1}^{N} a_{n} p (t - n Δ t) e^{j 2 π f_{c} t} = b (t) e^{j 2 π f_{c} t}

where

{a_{n}}_{n = 1}^{N}

are statistically independent Bernoulli distributed random variables with equally probable values 0 and 1,

p (t)

is real and:

b (t) = \sum_{n = 1}^{N} a_{n} p (t - n Δ t)

is the baseband pulse sequence carried by the optical carrier.

The temporal phase of the signal is encoded with the following phase:

ρ (t) = \sum_{n = 1}^{N} Φ_{n} r e c t (\frac{t - n Δ t}{Δ t})

Hence, the analytical signal at point 1 in Fig. 2 resulting from the temporal phase encoding is:

s_{1} (t) = s (t) e^{j ρ (t)} = \sum_{n = 1}^{N} a_{n} e^{j Φ_{n}} p (t - n Δ t) e^{j 2 π f_{c} t} = b_{1} (t) e^{j 2 π f_{c} t}

where the phases

{Φ_{n}}_{n = 1}^{N}

encoding the temporal phase of the pulse sequence can get Q_t possible values equally spaced between 0 and

2 π

, and:

b_{1} (t) = \sum_{n = 1}^{N} a_{n} e^{j Φ_{n}} p (t - n Δ t)

is the temporal phase encoded baseband pulse sequence having FT

B_{1} (f)

. The FT of

s_{1} (t)

is

S_{1} (f) = B_{1} (f - f_{c})

.

The signal is subsequently sampled, yielding the analytical signal $s_{2} (t)$ at point 2. Supposing the bandwidth of the communication system is BW, the sampled signal has the following FT:

S_{2} (f) = \frac{1}{\sqrt{M}} \sum_{m = - (M - 1) / 2}^{(M - 1) / 2} S_{1} (f - m Δ f) = \frac{1}{\sqrt{M}} \sum_{m = - (M - 1) / 2}^{(M - 1) / 2} B_{1} (f - m Δ f - f_{c})

where

M = B W / Δ f

is the number of spectral replicas of

S_{1} (f) / \sqrt{M}

in the frequency domain, generated due to the sampling.

Then, the spectral phase of the signal is encoded. The phase encoding has two goals. The first is signal spreading in the time domain in order to conceal it under the noise level. The second goal is to prevent reconstruction of the signal spectral amplitude by an eavesdropper that samples the signal in order to coherently add its spectral replicas at the baseband while the spectral replicas of the noise are added incoherently. The spectral phase encryption turns the addition of the spectral replicas to incoherent addition, thus the signal is kept bellow the noise level in the frequency domain.

The phase encrypting the spectral frequency of the signal has the following form in the positive frequency domain:

ψ (f) = \sum_{k = - (K - 1) / 2}^{(K - 1) / 2} φ_{k} r e c t (\frac{f - k δ f - f_{c}}{δ f})

Where

K = B W / δ f

is the number of phase bins in the communication system bandwidth and

{φ_{k}}_{k = - (K - 1) / 2}^{(K - 1) / 2}

getting Q_f possible values equally spaced between 0 and

2 π

. In addition:

r e c t (f) = {\begin{cases} 1 | f | \leq 1 / 2 \\ 0 ​ | f | > 1 / 2 \end{cases}

Various methods for spectral phase encoding are available [7–11]. Sub-GHz spectral resolution is expected to be available as commercial tunable wavelength-division multiplexing (WDM) lasers already reach 1GHz resolution using Bragg gratings for optical filtering. Using sub-GHz optical filters enables spectral phase encoding at this resolution. The phase encoding can be either spatial, using a spatial light modulator (SLM) as performed in [7,8], or in-fiber by using in-fiber phase modulators.

Consequently, the spectral phase encoded analytical signal at point 3, $s_{3} (t)$ has FT:

S_{3} (f) = S_{2} (f) e^{j ψ (f)} = \frac{e^{j ψ (f)}}{\sqrt{M}} \sum_{m = - (M - 1) / 2}^{(M - 1) / 2} S_{1} (f - m Δ f)

An Erbium doped fiber amplifier is then used to amplify the signal by

\sqrt{M}

in order to compensate for the attenuation of the signal's spectral amplitude stemming from the spectral spreading of the signal due to the sampling.

After transmission, white Gaussian noise $n (t)$ with autocorrelation $R_{n} (t) = σ^{2} δ (t)$ and FT $N (f)$ is added to the signal due to amplifier and channel noises. It should be noted that $N (f)$ is a white Gaussian noise with autocorrelation $R_{N} (f) = σ^{2} δ (f)$ . The noise is represented by the analytical signal $n_{A} (t)$ having FT $2 u (f) N (f)$ where $u (f)$ is the step function:

u (f) = {\begin{matrix} 1 & f \geq 0 \\ 0 & f < 0 \end{matrix}

Thus, the analytical signal

s_{4} (t)

at point 4 is:

s_{4} (t) = \sqrt{M} s_{3} (t) + n_{A} (t)

with FT:

S_{4} (f) = \sqrt{M} S_{3} (f) + 2 u (f) N (f) = e^{j ψ (f)} \sum_{m = - (M - 1) / 2}^{(M - 1) / 2} S_{1} (f - m Δ f) + 2 u (f) N (f)

The signal spreading in the time and frequency domains keeps the signal below the noise level, resulting in a stealthy signal in both domains.

2.2 Decoder configuration

The decryption process begins with dispersion compensation followed by spectral phase decoding. Chromatic dispersion can be either compensated by dispersion compensating fibers or avoided by using dispersion shifted fibers for transmission. Polarization mode dispersion (PMD) can be compensated using the coherent method offered in [12]. It should be mentioned that the PMD compensation dynamics is in the sub-MHz rate, thus much slower than the on-line decryption process. Spectral phase decoding can be optically implemented by a spectral phase decoder (SPD) multiplying the FT of the analytical signal with $e^{- j ψ (f)}$ . The bandwidth of the SPD is BW, giving rise to the analytical signal $s_{5} (t)$ at point 5, with FT:

S_{5} (f) = S_{4} (f) r e c t (\frac{f - f_{c}}{B W}) e^{- j ψ (f)} = \sum_{m = - (M - 1) / 2}^{(M - 1) / 2} S_{1} (f - m Δ f) + 2 N (f) r e c t (\frac{f - f_{c}}{B W}) e^{- j ψ (f)}

The signal is then coherently detected and passed through a low pass filter with bandwidth BW, yielding the electrical analytical signal $s_{6} (t)$ at point 6 which has FT:

S_{6} (f) = \frac{1}{2} S_{5} (f + f_{c}) = \frac{1}{2} [\sum_{m = - (M - 1) / 2}^{(M - 1) / 2} B_{1} (f - m Δ f) + 2 N (f + f_{c}) r e c t (\frac{f}{B W}) e^{- j ψ (f + f_{c})}]

Using the following notations for the detected noise and spectral phase encoding:

N_{d} (f) = N (f + f_{c})

and:

ψ_{d} (f) = ψ (f + f_{c})

Equation (14) can be written as:

S_{6} (f) = S_{5} (f + f_{c}) = \frac{1}{2} [\sum_{m = - (M - 1) / 2}^{(M - 1) / 2} B_{1} (f - m Δ f) + 2 N_{d} (f) r e c t (\frac{f}{B W}) e^{- j ψ_{d} (f)}]

A sampler having a transfer function with bandwidth BW and sampling frequency

Δ f

is subsequently applied to fold the spectral replicas of the signal to the baseband, resulting in the analytical signal

s_{7} (t)

at point 7. The FT of the signal is:

\begin{array}{l} S_{7} (f) = S_{6} (f) * \frac{1}{\sqrt{M}} \sum_{l = - (M - 1) / 2}^{(M - 1) / 2} δ (f - l Δ f) = \\ = \frac{1}{2 \sqrt{M}} \sum_{l = - (M - 1) / 2}^{(M - 1) / 2} \sum_{m = - (M - 1) / 2}^{(M - 1) / 2} B_{1} (f - m Δ f - l Δ f) + \\ + \frac{1}{2 \sqrt{M}} \sum_{l = - (M - 1) / 2}^{(M - 1) / 2} 2 N_{d} (f - l Δ f) r e c t (\frac{f - l Δ f}{B W}) e^{- j ψ_{d} (f - l Δ f)} \end{array}

where * denotes the convolution operator.

Then, a filter matched to $p (t)$ filters $s_{7} (t)$ . Consequently, the signal $s_{8} (t)$ is achieved at point 8 having FT:

\begin{array}{l} S_{8} (f) = S_{7} (f) P^{*} (f) = \\ = \frac{P^{*} (f)}{2 \sqrt{M}} \sum_{l = - (M - 1) / 2}^{(M - 1) / 2} \sum_{m = - (M - 1) / 2}^{(M - 1) / 2} B_{1} (f - m Δ f - l Δ f) + \\ + \frac{P^{*} (f)}{2 \sqrt{M}} \sum_{l = - (M - 1) / 2}^{(M - 1) / 2} 2 N_{d} (f - l Δ f) r e c t (\frac{f - l Δ f}{B W}) e^{- j ψ_{d} (f - l Δ f)} \end{array}

Since the bandwidth of

P^{*} (f)

is limited to the baseband, all the cross terms in Eq. (19) zero, therefore, only the terms with

l = m

are left. Thus, Eq. (19) turns to:

\begin{array}{l} S_{8} (f) = \frac{P^{*} (f)}{2 \sqrt{M}} \sum_{m = - (M - 1) / 2}^{(M - 1) / 2} B_{1} (f) + \\ + \frac{P^{*} (f)}{2 \sqrt{M}} \sum_{m = - (M - 1) / 2}^{(M - 1) / 2} 2 N_{d} (f - m Δ f) r e c t (\frac{f - m Δ f}{B W}) e^{- j ψ_{d} (f - m Δ f)} \end{array}

Hence:

\begin{array}{l} S_{8} (f) = \frac{P^{*} (f)}{2 \sqrt{M}} M B_{1} (f) + \\ + \frac{P^{*} (f)}{2 \sqrt{M}} \sum_{m = - (M - 1) / 2}^{(M - 1) / 2} 2 N_{d} (f - m Δ f) r e c t (\frac{f - m Δ f}{B W}) e^{- j ψ_{d} (f - m Δ f)} \end{array}

The first term at the right wing of Eq. (21) denotes the signal whereas the right term denotes the noise. Multiplying these terms by

\sqrt{M}

, the first term linearly increases with M since it comprises M spectral replicas of the signal which are coherently added at the baseband and filtered by the matched filter. The noise term does not linearly increase with M since the spectral noise shifts are added incoherently. This distinction between the decoded signal and the noise will be further expressed later when the SNR of the decoded signal is derived.

The temporal phase of the signal is then decoded. Hence, the analytical signal at point 9 is:

\begin{array}{l} s_{9} (t) = e^{- j ρ (t)} \int_{- \infty}^{\infty} S_{8} (f) e^{j 2 π f t} d f = \\ = e^{- j ρ (t)} \int_{- \infty}^{\infty} \frac{P^{*} (f)}{2} \sqrt{M} B_{1} (f) e^{j 2 π f t} d f + e^{- j ρ (t)} \int_{- \infty}^{\infty} \frac{P^{*} (f)}{2 \sqrt{M}} \cdot \\ \cdot \sum_{m = - (M - 1) / 2}^{(M - 1) / 2} 2 N_{d} (f - m Δ f) r e c t (\frac{f - m Δ f}{B W}) e^{- j ψ_{d} (f - m Δ f)} e^{j 2 π f t} d f \end{array}

We use the following notations:

{\tilde{s}}_{A} (t) = e^{- j ρ (t)} \int_{- \infty}^{\infty} \frac{P^{*} (f)}{2} \sqrt{M} B_{1} (f) e^{j 2 π f t} d f

and:

{\tilde{n}}_{A} (t) = e^{- j ρ (t)} \int_{- \infty}^{\infty} \frac{P^{*} (f)}{2 \sqrt{M}} \cdot \sum_{m = - (M - 1) / 2}^{(M - 1) / 2} 2 N_{d} (f - m Δ f) r e c t (\frac{f - m Δ f}{B W}) e^{- j ψ_{d} (f - m Δ f)} e^{j 2 π f t} d f

where

{\tilde{s}}_{A} (t)

and

{\tilde{n}}_{A} (t)

are the analytical signal and noise terms at point 9 respectively.

Equation (23) can be further developed to yield:

{\tilde{s}}_{A} (t) = e^{- j ρ (t)} \int_{- \infty}^{\infty} \frac{P^{*} (f)}{2} \sqrt{M} B_{1} (f) e^{j 2 π f t} d f = \frac{\sqrt{M}}{2} e^{- j ρ (t)} \int_{- \infty}^{\infty} p^{*} (τ - t) \sum_{n = 1}^{N} a_{n} e^{j Φ_{n}} p (τ - n Δ t) d τ

It should be noted that:

\begin{array}{l} {\tilde{s}}_{A} (n Δ t) = \frac{\sqrt{M}}{2} e^{- j ρ (t)} \int_{- \infty}^{\infty} p^{*} (τ - n Δ t) \sum_{k = 1}^{N} a_{k} e^{j Φ_{k}} p (τ - k Δ t) d τ = \\ = \frac{\sqrt{M}}{2} \sum_{l = 1}^{N} e^{- j Φ_{l}} r e c t (\frac{n Δ t - l Δ t}{Δ t}) \int_{- \infty}^{\infty} p^{*} (τ - n Δ t) \sum_{k = 1}^{N} a_{k} e^{j Φ_{k}} p (τ - k Δ t) d τ = \\ = \frac{\sqrt{M}}{2} a_{n} E_{p} \end{array}

where:

E_{p} \equiv \int_{- \infty}^{\infty} {| p (t) |}^{2} d t

The signal

s_{9} (t)

is sampled each temporal interval

Δ t

, generating the sampled signal

s_{10} (t)

at point 10:

s_{10} (t) = {\begin{matrix} s_{9} (n Δ t) = {\tilde{s}}_{A} (n Δ t) + {\tilde{n}}_{A} (n Δ t) & t = n Δ t \\ 0 & o t h e r w i s e \end{matrix}

where n is an integer. A decision circuit is then applied to recover the original bit sequence.

3. System expected performances

Having the signal and noise expressed in Eq. (26) and Eq. (24) respectively, the SNR and the bit error rate (BER) measured by an authorized user after signal decryption are subsequently derived.

3.1 SNR after decryption

The SNR of the decrypted signal is given by:

S N R = \frac{{(\tilde{s} (n Δ t))}^{2}}{E [{(\tilde{n} (n Δ t))}^{2}]}

where the following notations are used:

\tilde{s} (t) = Re {{\tilde{s}}_{A} (t)}

and:

\tilde{n} (t) = Re {{\tilde{n}}_{A} (t)}

From Eq. (26) it can be seen that:

\tilde{s} (n Δ t) = Re [{\tilde{s}}_{A} (n Δ t)] = \frac{\sqrt{M}}{2} a_{n} E_{p}

Assuming the SNR is measured at time

t = n Δ t

for which

a_{n} = 1

, Eq. (32) gives:

\tilde{s} (n Δ t) = \frac{\sqrt{M}}{2} E_{p}

Therefore, the signal power is:

{(\tilde{s} (n Δ t))}^{2} = {(\frac{\sqrt{M}}{2} E_{p})}^{2} = \frac{M}{4} E_{p}^{2}

In addition:

\begin{array}{l} E [{(\tilde{n} (t))}^{2}] = E [{(Re {{\tilde{n}}_{A} (t)})}^{2}] = \\ = E [{(\frac{1}{2} {{\tilde{n}}_{A} (t) + {\tilde{n}}_{A}^{*} (t)})}^{2}] = \\ = \frac{1}{4} E [{({\tilde{n}}_{A} (t))}^{2}] + \frac{1}{2} E [{\tilde{n}}_{A} (t) {\tilde{n}}_{A}^{*} (t)] + \frac{1}{4} E [{({\tilde{n}}_{A}^{*} (t))}^{2}] \end{array}

Using Eq. (24), it can be shown that

E [{({\tilde{n}}_{A} (t))}^{2}] = E [{({\tilde{n}}_{A}^{*} (t))}^{2}] = 0

. Hence:

E [{(\tilde{n} (t))}^{2}] = \frac{1}{2} E [{\tilde{n}}_{A} (t) {\tilde{n}}_{A}^{*} (t)] = \frac{σ^{2}}{2 M} \int_{- \infty}^{\infty} \sum_{m = - (M - 1) / 2}^{(M - 1) / 2} {| P (f) |}^{2} d f = \frac{σ^{2}}{2} E_{p}

Therefore, the SNR measured after decryption is:

S N R = \frac{{| \tilde{s} (n Δ t) |}^{2}}{E [{(\tilde{n} (n Δ t))}^{2}]} = \frac{\frac{M}{4} E_{p}^{2}}{\frac{σ^{2}}{2} E_{p}} = M \frac{E_{p}}{2 σ^{2}}

The term

E_{p} / σ^{2}

is the SNR for the case of a baseband pulse sequence transmitted without going through the encoder, and simply detected by a matched filter followed by a sampler and a decision circuit. This term is multiplied by factor M since the M spectral replicas of the signal are added coherently while the addition of the M spectral shifts of the noise is incoherent. The factor 2 at the denominator is due to the modulation of the baseband signal with the optical carrier having frequency

f_{c}

which doubles the transmission bandwidth, therefore doubling the noise power.

There are two effects enabling to conceal the signal below the noise level in the time domain. The first is the processing gain which is shown in Eq. (37) to increase the SNR by factor $M = B W / Δ f$ where BW is the bandwidth of the encrypted signal and $Δ f$ is the bandwidth of the unencrypted signal. This gain stems from the coherent addition of the signal's M spectral replicas at the decryption process. Hence, a low power signal can be transmitted, relying on the processing gain.

The second effect is the signal spreading in the time domain resulting from the spectral phase encryption. The unencrypted signal has bandwidth $Δ f$ , which is widened to bandwidth BW at point 2 in Fig. 2 due to the sampling. The spectral phase encoder (SPE) generates $B W / δ f$ chips in the frequency domain at point 3, $δ f$ being the spectral resolution of the spectral phase code. Hence, each of the signal's samples is spread by factor $B W / δ f$ . Since the unencrypted signal is wider than its samples in the time domain by $M = B W / Δ f$ , the encrypted signal is temporally spread by factor $B W / (M \cdot δ f)$ relatively to the unencrypted signal. Assuming BW = 80GHz, $Δ f = 5 G H z$ and $δ f = 100 M H z$ , the encrypted signal contains $M = 80 G H z / 5 G H z = 16$ spectral replicas and it is spread by factor $80 G H z / (16 \cdot 100 M H z) = 50$ in the time domain.

3.2 BER after decryption

Since the decoder is a linear system the noise after the decryption process remains Gaussian. The average of the sampled noise at point 10 in Fig. 2 is:

E [\tilde{n} (t)] = \frac{1}{2} E [{\tilde{n}}_{A} (t) + {\tilde{n}}_{A}^{*} (t)] = 0

The variance of the noise is given by Eq. (36).

The BER for the case of a Gaussian noise is given by:

B E R = \frac{1}{2} e r f c (\frac{\sqrt{S N R}}{2 \sqrt{2}})

Using Eq. (37), the BER is:

B E R = \frac{1}{2} e r f c (\frac{\sqrt{S N R}}{2 \sqrt{2}}) = \frac{1}{2} e r f c (\frac{1}{2 \sqrt{2}} \sqrt{M \frac{E_{p}}{2 σ^{2}}}) = \frac{1}{2} e r f c (\frac{1}{4} \sqrt{M \frac{E_{p}}{σ^{2}}})

4. Signal hiding in the frequency domain

Suppose an optical signal processed by a spectrum analyzer. Coherent detection yields the electrical signal $v (t)$ , measured in Volts. The FT of $v (t)$ is $V (f)$ . The PSD of $v (t)$ is:

G_{v} (f) = \frac{1}{\tilde{T}} {| V (f) |}^{2}

The motivation for this definition is that

d f = 1 / \tilde{T}

is the spectral resolution of the signal

v (t)

. Hence, the voltage content of the signal in frequency

f

can be found by locating a BPF with bandwidth

1 / \tilde{T}

at frequency

f

. The amplitude of the filtered signal passed through an envelope detector will be

V (f) d f = V (f) / \tilde{T}

, measured in Volts. Its power will be

P_{v} (f) = {| V (f) d f |}^{2} = {| V (f) / \tilde{T} |}^{2}

, measured in Watts (assuming that the load over which the voltage of the signal was measured has

1 Ω

resistance).

P_{v} (f)

is the power content of

v (t)

within bandwidth

d f = 1 / \tilde{T}

in frequency

f

which is measured by the spectrum analyzer. The PSD in frequency

f

is therefore

G_{v} (f) = P_{v} (f) / d f = \frac{1}{\tilde{T}} {| V (f) |}^{2}

.

{| V (f) |}^{2}

is the energy spectral density (ESD) of

v (t)

, with units of J/Hz. When

v (t)

is a random process, its ESD is defined as the ensemble average of

{| V (f) |}^{2}

.

We would like to estimate the PSD of the signal and the noise measured by an adversary trying to reveal the cloaked signal in the frequency domain by means of coherently detecting and sampling the signal in order to fold its spectral replicas to the baseband. Ensuring a stealthy transmission in the frequency domain is an advantage of the proposed method over the methods described in [2–6]. According to Eq. (1), the duration of the pulse sequence is $\tilde{T} = N Δ t$ .

The encrypted signal was shown in Eq. (12) to be:

S_{4} (f) = e^{j ψ (f)} \sum_{m = - (M - 1) / 2}^{(M - 1) / 2} S_{1} (f - m Δ f) + 2 u (f) N (f)

Applying a band pass filter (BPF) with bandwidth BW at the carrier frequency, the analytical signal becomes:

S_{4} (f) = e^{j ψ (f)} \sum_{m = - (M - 1) / 2}^{(M - 1) / 2} S_{1} (f - m Δ f) + 2 r e c t (\frac{f - f_{c}}{B W}) N (f)

Considering the finite duration of the noise in the time domain, Eq. (42) becomes:

\begin{array}{l} S_{4} (f) = e^{j ψ (f)} \sum_{m = - (M - 1) / 2}^{(M - 1) / 2} S_{1} (f - m Δ f) + \\ + 2 r e c t (\frac{f - f_{c}}{B W}) {N (f) * [N Δ t \sin c (f \cdot N Δ t) \exp (- j 2 π f \cdot N Δ t / 2)]} \end{array}

Coherently detecting

s_{4} (t)

and applying a low pass filter with bandwidth BW without spectral phase decryption yields the signal

{\overset{⌢}{s}}_{5} (t)

with FT:

\begin{array}{l} {\overset{⌢}{S}}_{5} (f) = \frac{1}{2} e^{j ψ (f + f_{c})} \sum_{m = - (M - 1) / 2}^{(M - 1) / 2} B_{1} (f - m Δ f) + \\ + {r e c t (\frac{f - f_{c}}{B W}) [N (f) * (N Δ t \sin c (f \cdot N Δ t) e^{- j π f \cdot N Δ t})]} * δ (f + f_{c}) \end{array}

Sampling the detected signal yields the signal

{\overset{⌢}{s}}_{6} (t)

with FT:

\begin{array}{l} {\overset{⌢}{S}}_{6} (f) = {\overset{⌢}{S}}_{5} (f) * \frac{1}{\sqrt{M}} \sum_{l = - (M - 1) / 2}^{(M - 1) / 2} δ (f - l Δ f) = \\ = \frac{1}{2 \sqrt{M}} \sum_{m = - (M - 1) / 2}^{(M - 1) / 2} \sum_{l = - (M - 1) / 2}^{(M - 1) / 2} B_{1} (f - m Δ f - l Δ f) e^{j ψ (f + f_{c} - l Δ f)} + \frac{N Δ t}{\sqrt{M}} \sum_{l = - (M - 1) / 2}^{(M - 1) / 2} r e c t (\frac{f - l Δ f}{B W}) \cdot \\ \cdot \int_{- \infty}^{\infty} N (f_{1}) \sin c ((f + f_{c} - l Δ f - f_{1}) \cdot N Δ t) e^{- j π (f + f_{c} - l Δ f - f_{1}) \cdot N Δ t} d f_{1} \end{array}

Finally, applying a matched filter results in the signal

{\overset{⌢}{s}}_{7} (t)

having FT:

\begin{array}{l} {\overset{⌢}{S}}_{7} (f) = {\overset{⌢}{S}}_{6} (f) P^{*} (f) = \\ = \frac{P^{*} (f)}{2 \sqrt{M}} \sum_{m = - (M - 1) / 2}^{(M - 1) / 2} B_{1} (f) e^{j ψ (f + f_{c} - m Δ f)} + \\ + \frac{N Δ t P^{*} (f)}{\sqrt{M}} \sum_{l = - (M - 1) / 2}^{(M - 1) / 2} \int_{- \infty}^{\infty} N (f_{1}) \sin c ((f + f_{c} - l Δ f - f_{1}) \cdot N Δ t) e^{- j π (f + f_{c} - l Δ f - f_{1}) \cdot N Δ t} d f_{1} \end{array}

The PSD of the signal is therefore:

\begin{array}{l} G_{s} (f) = \frac{1}{N Δ t} E [{| \frac{P^{*} (f)}{2 \sqrt{M}} B_{1} (f) \sum_{m = - (M - 1) / 2}^{(M - 1) / 2} e^{j ψ (f + f_{c} - m Δ f)} |}^{2}] = \\ = \frac{{| P (f) |}^{2}}{4 M N Δ t} {| \sum_{m = - (M - 1) / 2}^{(M - 1) / 2} e^{j ψ (f + f_{c} - m Δ f)} |}^{2} \sum_{n = 1}^{N} \sum_{l = 1}^{N} E [a_{n} a_{l}] e^{j (Φ_{n} - Φ_{l})} e^{j 2 π f (n - l) Δ t} = \\ = \frac{{| P (f) |}^{4}}{8 M N Δ t} [\sum_{m = - (M - 1) / 2}^{(M - 1) / 2} \sum_{l \neq m}^{} e^{- j [ψ (f + f_{c} - m Δ f) - ψ (f + f_{c} - l Δ f)]} + M] [\sum_{n = 1}^{N} \sum_{l \neq n}^{} \frac{1}{2} e^{j (Φ_{n} - Φ_{l})} e^{j 2 π f (n - l) Δ t} + N] \end{array}

When

{φ_{k}}_{k = - (K - 1) / 2}^{(K - 1) / 2}

and

{Φ_{n}}_{n = 1}^{N}

are statistically independent and uniformly distributed between 0 and

2 π

, the following approximation is done:

G_{s} (f) \approx \frac{1}{8 M N Δ t} {| P (f) |}^{4} M N = \frac{1}{8 Δ t} {| P (f) |}^{4}

The PSD of the noise is:

\begin{array}{l} G_{n} (f) = \frac{1}{N Δ t} E ({| {r e c t (\frac{f - f_{c}}{B W}) [N (f) * (N Δ t \sin c (f \cdot N Δ t) e^{- j π f \cdot N Δ t})]} * δ (f + f_{c}) |}^{2}) = \\ = \frac{1}{N Δ t} E [| \frac{N Δ t P^{*} (f)}{\sqrt{M}} \cdot \\ \cdot {\sum_{l = - (M - 1) / 2}^{(M - 1) / 2} \int_{- \infty}^{\infty} N (f_{1}) \sin c ((f + f_{c} - l Δ f - f_{1}) \cdot N Δ t) e^{- j π (f + f_{c} - l Δ f - f_{1}) \cdot N Δ t} d f_{1} |}^{2}] = \\ = \frac{{(N Δ t)}^{2} {| P (f) |}^{2} σ^{2}}{M N Δ t} \sum_{l = - (M - 1) / 2}^{(M - 1) / 2} \sum_{m = - (M - 1) / 2}^{(M - 1) / 2} \int_{- \infty}^{\infty} \sin c ((f + f_{c} - l Δ f - f_{1}) \cdot N Δ t) \cdot \\ \cdot \sin c ((f + f_{c} - m Δ f - f_{1}) \cdot N Δ t) e^{j π (l - m) Δ f \cdot N Δ t} d f_{1} \end{array}

In addition,

Δ f > > {(N Δ t)}^{- 1}

. Hence, noting

\tilde{f} \equiv f + f_{c}

, the following approximation is used:

\sin c ((\tilde{f} - m Δ f - f_{1}) \cdot N Δ t) \sin c ((\tilde{f} - l Δ f - f_{1}) \cdot N Δ t) \approx \sin c^{2} [(\tilde{f} - m Δ f - f_{1}) \cdot N Δ t] δ_{m, l}

Thus, Eq. (49) becomes:

G_{n} (f) \approx \frac{N Δ t {| P (f) |}^{2} σ^{2}}{M} \sum_{m = - (M - 1) / 2}^{(M - 1) / 2} \frac{1}{N Δ t} = σ^{2} {| P (f) |}^{2}

Consequently, the spectral SNR measured by the eavesdropper is:

S N R_{E} (f) = \frac{G_{s} (f)}{G_{n} (f)} \approx \frac{\frac{1}{8 Δ t} {| P (f) |}^{4}}{σ^{2} {| P (f) |}^{2}} = \frac{{| P (f) |}^{2}}{8 σ^{2} Δ t}

The spectral SNR experienced by the adversary increases with ${| P (f) |}^{2}$ . Having $M$ spectral replicas of the signal, this term can be attenuated by factor $M$ due to the processing gain resulting from the coherent addition of the spectral replicas. Hence, taking a large number of spectral replicas enables to reduce the power of the pulses in the transmitted pulse sequence. In addition, the spectral SNR decreases with $σ^{2}$ which is the white Gaussian noise double sided PSD and with $Δ t$ , which is the temporal interval between two sequential pulses. Enlarging this interval immerses the signal more deeply in the spectral noise.

Equation (52) implies that the spectral SNR measured by the adversary depends neither on the number of pulses in the pulse sequence, nor in the number of the signal spectral replicas. The independence in the number of pulses is achieved due to the temporal phase encryption, which prevents the coherent addition of the various pulses in the pulse sequence at the baseband. In addition, the independence in the number of spectral replicas results from the spectral phase encryption which prevents the coherent addition of the signal's spectral replicas at the baseband.

5. Simulation and results

5.1 Simulated system configuration

The generated simulation realizes the system described in Fig. 2 with the following parameters:

• Number of pulses in a single analyzed pulse sequence - $N = 500$ .
• Pulse sequence duration - $1 μ \sec$ .
• Temporal gap between two sequential pulses - $Δ t = 2 n \sec$ .
• Temporal pulse width - $T = 0.5 n \sec$ . Taking $T = Δ t$ results in degradation of the SNR and BER after decoding due to inter-symbol interference.
• Pulse bandwidth - $Δ f = 5 G H z$ .
• Transmission bandwidth - $B W = 80 GHz$ (unless noted otherwise). Current analog to digital converters used for coherent detection are limited to bandwidth of about 50GHz.
• Additive white Gaussian noise double sided PSD - $σ^{2} = 0.2 m w / H z$ .
• Quantization level of the temporal phase encoding - Q_t = 64.
• Quantization level of the spectral phase encoding - Q_f = 64.
• Spectral resolution of spectral phase encoding - $δ f = 200 M H z$ .

The modulating format of the pulse sequence was on-off-keying. Each pulse in the sequence has a raised cosine shape in the time domain with rolloff factor $α = 0.25$ [13]. The temporal and spectral phases encoding the pulse sequence are uniformly distributed between 0 and $2 π$ .

A sequence of one million pulses (which is composed of 2000 pulse sequences, each having $N = 500$ pulses) was run in order to evaluate the BER after decryption.

5.2 Results

The SNR used for the original baseband pulse passing through a matched filter was $E_{p} / σ^{2} = 10$ . The SNR and BER of the decoded pulse sequence were examined for various values of the transmission bandwidth BW. Their expected values are given in Eq. (37) and Eq. (40) respectively.

Figure 3 presents the SNR after decoding. The theoretical SNR matches the one measured by the simulation. In addition, an opponent trying to recover the signal with randomly chosen temporal and spectral encoding phases, having the same quantization, temporal and spectral resolution as the authorized user was simulated. It is shown that the eavesdropper experiences very low SNR since signal is spread below the noise level. The SNR measured by the adversary varies with the transmission bandwidth since various spectral slices of the spectral phase encoder (SPE) are used, therefore influencing the matching between the SPD of the authorized user and the eavesdropper.

Fig. 3 SNR after decoding.

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The BER after decoding is illustrated in Fig. 4 . Three graphs are used for the authorized user. The first denotes the theoretical BER. The second graph exhibits the BER calculated by substituting the measured SNR from Fig. 3 in Eq. (40). The third one is the BER measured by the simulation. It should be noted that the deviation of the measured BER from the two other BER graphs (the theoretical graph and the one calculated from the SNR) for the case of an authorized user with transmission bandwidth of 80GHz stems from the limited statistics of the simulation. While the expected BER is about 10^-5.5, only 10⁶ pulses are run by the simulation, hence the number of measured erroneous bits is lower than that expected by the two other graphs. The BER measured by the adversary is very high (about 0.5). The threshold chosen for the eavesdropper is half the maximal power of its received noiseless signal.

Fig. 4 BER after decoding.

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Figure 5 shows the decoded pulse sequence in time and frequency domains for the authorized user and the adversary. The transmission bandwidth is BW = 80GHz. Figure 5(a) and Fig. 5(b) present a 5nsec interval of the noiseless original pulse sequence and the noisy decoded pulse sequence passed through a matched filter for the authorized user and the eavesdropper respectively. The eavesdropper cannot observe the signal while the authorized user can easily detect it. Figure 5(c) and Fig. 5(d) exhibit the PSD normalized by $1 / (N Δ t)$ , measured by the authorized user and the adversary respectively for a single pulse sequence consisting of 500 pulses. In the first case, the signal is well raised above the noise level while being significantly lower than the noise PSD for the latter due to the incoherent addition of the spectral replicas and incorrect temporal phase decoding, which prevents the coherent addition of the various temporal pulses in the frequency domain.

Fig. 5 Decoded signal, authorized user and eavesdropper. (a) Original noiseless pulse sequence and authorized user noisy decoded pulse sequence. (b) Original noiseless pulse sequence and eavesdropper noisy decoded pulse sequence. (c) Authorized user pulse sequence and noise power spectral density. (d) Eavesdropper pulse sequence and noise power spectral density.

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5.3 Encryption strength estimation

In order to evaluate the encryption strength the following assumptions are made:

• The SNR used for the original baseband pulse passing through a matched filter was $E_{p} / σ^{2} = 4$ (corresponding to SNR = 18dB and BER = 10^-4.5 after authorized decryption).
• The goal of the eavesdropper is to raise the signal above the noise level in either time or frequency domain.
• The eavesdropper has a-priory knowledge about the encryption and decryption method. The only parameters that should be guessed are the temporal and spectral phase codes.
• The eavesdropper tries to deduce the spectral and temporal phase codes by working with roughly quantized temporal and spectral phase codes.
• The eavesdropper needs to properly guess at least $N_{t}$ and $N_{f}$ temporal and spectral phases in order to reveal the signal in either time or frequency domain.
• When observing the signal, the adversary tries to refine the phase quantization of the guessed phases in order to improve the measured SNR and BER.
• The encryption strength is defined as the time interval needed to accomplish the goal of the eavesdropping process.
• The eavesdropper has the same spectral and temporal phase decoders as the authorized user. The maximal SPD's SLM update frequency of the eavesdropper is: $f_{S L M} = 1 M H z$ .

First, we assume that the adversary begins with binary quantized temporal and spectral phases and that the temporal phase code is already known. Tries are made by the adversary to get

S N R > 0 d B

in the time domain. Second, temporal phase quantization Q_t = 64 is assumed.

Using the simulation it was observed that the spectral resolution of the spectral phase code has major influence over the number of spectral phases needed for the eavesdropper to properly guess in order to reveal the signal in the time domain. Larger spectral resolution results in a higher $N_{f}$ . For example, using 100 different randomly chosen spectral phase codes for the authorized user and the eavesdropper with Q_f = 256 (which are already commercial) and $δ f = 100 M H z$ , the mean value observed for $N_{f}$ was 158 with standard deviation of 33.33. Reducing the spectral phase code resolution to $δ f = 200 M H z$ , the average $N_{f}$ became 76 with standard deviation of 23.14.

$N_{f}$ is also affected by the number of spectral phase code quantization levels Q_f. A larger Q_f results in a larger $N_{f}$ . For instance, when working with $δ f = 200 M H z$ , Q_f = 2, the average required $N_{f}$ reduced to 51.1 with standard deviation of 21.07.

The time interval required for the adversary to get $S N R > 0 d B$ in the time domain is give by:

T_{e a v} = \frac{{(Q_{f, e a v})}^{N_{f}}}{f_{S L M}}

where

Q_{f, e a v}

is the spectral phase quantization level of the spectral phase code used by the eavesdropper. The assumed value is

Q_{f, e a v} = 2

(having no prior knowledge about the spectral phase code of the authorized user, it was observed that eavesdropping with larger

Q_{f, e a v}

only increases

T_{e a v}

, hence the eavesdropper is assumed to start with binary quantization).

For the case of $δ f = 100 M H z$ and Q_f = 256, substituting $N_{f} = 125$ , Eq. (53) gives:

T_{E} = 2^{N_{f}} / f_{S L M} = 2^{125} / 10^{6} H z = 1 .35 \cdot 10^{24} y e a r s

While for

δ f = 200 M H z

and Q_f = 256, substituting

N_{f} = 53

, yields:

T_{E} = 2^{N_{f}} / f_{S L M} = 2^{53} / 10^{6} H z = 285 .6 y e a r s

And

δ f = 200 M H z

,Q_f = 2, resulting in

N_{f} = 30

gives:

T_{E} = 2^{N_{f}} / f_{S L M} = 2^{30} / 10^{6} H z = 3 .4 \cdot 10^{- 5} y e a r s

Thus, the spectral resolution and quantization level of the phase code strongly affect the encryption strength. It should be noted that the above estimated

N_{f}

is actually a lower bound for the number of spectral phases needed to be properly guessed in order to accomplish positive SNR in the time domain since it was assumed that the temporal phase code is completely known to the adversary.

$N_{t}$ can be found by estimating the number of temporal phases needed to be properly guessed by an adversary using a binary quantized temporal phase decoder in order to raise the signal above the noise level in the frequency domain after properly guessing $N_{f}$ spectral phases with the binary quantized SPD. For example, a spectral phase code and an adversary SPD yielding $N_{f} = 210$ were chosen ( $δ f = 100 M H z$ ). The peak of the spectral SNR became positive at $N_{t} = 320$ .

Hence, using high resolution and highly quantized phase decoding, the encryption system can cause the eavesdropping process involving a “brute force” attack in which the temporal and spectral phase codes are randomly guessed to last a very long time interval in order to raise the signal above the noise level in time and frequency domains. This time interval is irrelevant for the eavesdropper.

A second kind of possible attack is such that the eavesdropper is assumed to determine the original unencrypted signal known as “Chosen-plaintext attack” [14]. The encryption methods proposed in [2–4] enable the spectral phase code disclosure when the eavesdropper sends a known pulse sequence from the encoder of the authorized user when the signal is not concealed under the transmission of a public WDM user. The spectral phase of the sent pulse sequence is known to the eavesdropper, hence, the spectral phase code is revealed. The temporal phase encrypting the signal in the methods described in [5,6] can be disclosed by sending a narrowband pulse for which the effect of the dispersion is insignificant. The resulting encoded signal is the original signal having its temporal phase encoded. This way, the temporal phase code can be measured by the adversary. The spectral phase code can then be disclosed similarly to methods [2–4]. However, using the encryption method proposed in this work, the pulse sequence is well spread below the noise level in the frequency domain as well as in the time domain, therefore making the recovery of the spectral phase code much more difficult.

6. Conclusion

A method for spectral and temporal covert communication is presented. The encryption is based on encrypting the temporal phase of the pulse sequence, therefore reducing its power spectral density. The pulse sequence is subsequently spread in the frequency domain by means of sampling, therefore enabling to transmit a signal with low power spectral density. Spectral phase encryption is subsequently applied to spread the signal in the time domain and prevent the eavesdropper from observing the signal in the frequency domain by coherently adding its spectral replicas at the baseband. Hence, at the end of the encryption process the pulse sequence is spread below the noise level in the time and frequency domains. Mathematical modeling as well as numerical investigations demonstrate the applicability of the proposed approach.

References and links

1. A. J. Viterbi, “Spread spectrum communications – myths and realities,” IEEE Commun. Mag. 17(3), 11–18 (1979). [CrossRef]

2. B. B. Wu and E. E. Narimanov, “A method for secure communications over a public fiber-optical network,” Opt. Express 14(9), 3738–3751 (2006). [CrossRef] [PubMed]

3. K. Kravtsov, B. Wu, I. Glysk, P. R. Prucnal, and E. Narimanov, “Stealth transmission over a WDM network with detection based on an all-optical thresholder,” in Proceedings of IEEE Conference on Lasers and Electro-Optics (IEEE, 2007), pp. 480–481.

4. B. Wu, A. Agarwal, I. Glesk, E. Narimanov, S. Etemad, and P. R. Prucnal, “Steganographic fiber-optic transmission using coherent spectral-phase-encoded optical CDMA,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies, OSA Technical Digest (CD) (Optical Society of America, 2008), paper CFF5, http://www.opticsinfobase.org/abstract.cfm?URI=CLEO-2008-CFF5.

5. Z. Wang, M. P. Fok, L. Xu, J. Chang, and P. R. Prucnal, “Improving the privacy of optical steganography with temporal phase masks,” Opt. Express 18(6), 6079–6088 (2010). [CrossRef] [PubMed]

6. Z. Gao, X. Wang, N. Kataoka, and N. Wada, “Stealth transmission of time-domain spectral phase encoded OCDMA signal over WDM network,” IEEE Photon. Technol. Lett. 22(13), 993–995 (2010). [CrossRef]

7. D. Sinefeld, C. R. Doerr, and D. M. Marom, “Photonic spectral processor employing two-dimensional WDM channel separation and a phase LCoS modulator,” in Optical Fiber Communication Conference, OSA Technical Digest (CD) (Optical Society of America, 2010), paper OMP5, http://www.opticsinfobase.org/abstract.cfm?URI=OFC-2010-OMP5.

8. D. Sinefeld and D. M. Marom, “Hybrid guided-wave/free-space optics photonic spectral processor based on LCoS phase only modulator,” IEEE Photon. Technol. Lett. 22(7), 510–512 (2010). [CrossRef]

9. X. Wang, “Novel time domain spectral phase encoding/decoding technique for OCDMA application,” in International Conference on Transparent Optical Networks (IEEE, S. Miguel (Portugal), 2009), paper Th.A3.4.

10. X. Wang and N. Wada, “Spectral phase encoding of ultra-short optical pulse in time domain for OCDMA application,” Opt. Express 15(12), 7319–7326 (2007). [CrossRef] [PubMed]

11. D. Miyamoto and H. Tsuda, “Spectral phase encoder employing an arrayed-waveguide grating and phase-shifting structure,” IEEE Photon. Technol. Lett. 19(17), 1289–1291 (2007). [CrossRef]

12. E. Ip and J. M. Kahn, “Digital equalization of chromatic dispersion and polarization mode dispersion,” J. Lightwave Technol. 25(8), 2033–2043 (2007). [CrossRef]

13. J. G. Proakis and M. Salehi, Communication Systems Engineering (Prentice Hall, 1994), Chap. 8.

14. Y. Frauel, A. Castro, T. J. Naughton, and B. Javidi, “Resistance of the double random phase encryption against various attacks,” Opt. Express 15(16), 10253–10265 (2007). [CrossRef] [PubMed]

Spectral and temporal stealthy fiber-optic communication using sampling and phase encoding

Abstract

1. Introduction

2. System description

2.1 Encoder configuration

2.2 Decoder configuration

3. System expected performances

3.1 SNR after decryption

3.2 BER after decryption

4. Signal hiding in the frequency domain

5. Simulation and results

5.1 Simulated system configuration

5.2 Results

5.3 Encryption strength estimation

6. Conclusion

References and links

Cited By

Figures (5)

Equations (57)

Optics Express