# Analysis, optimization, and implementation of a hybrid DS/FFH spread-spectrum technique for smart grid communications

- Mohammed M Olama
^{1}Email author, - Xiao Ma
^{2}, - Stephen M Killough
^{1}, - Teja Kuruganti
^{1}, - Stephen F Smith
^{1}and - Seddik M Djouadi
^{2, 3}

**2015**:25

https://doi.org/10.1186/s13634-015-0208-z

© Olama et al.; licensee Springer. 2015

**Received: **30 October 2014

**Accepted: **17 February 2015

**Published: **12 March 2015

## Abstract

In recent years, there has been great interest in using hybrid spread-spectrum (HSS) techniques for commercial applications, particularly in the Smart Grid, in addition to their inherent uses in military communications. This is because HSS can accommodate high data rates with high link integrity, even in the presence of significant multipath effects and interfering signals. A highly useful form of this transmission technique for many types of command, control, and sensing applications is the specific code-related combination of standard direct sequence modulation with ‘fast’ frequency hopping, denoted hybrid DS/FFH, wherein multiple frequency hops occur within a single data-bit time. In this paper, error-probability analyses are performed for a hybrid DS/FFH system over standard Gaussian and fading-type channels, progressively including the effects from wide- and partial-band jamming, multi-user interference, and varying degrees of Rayleigh and Rician fading. In addition, an optimization approach is formulated that minimizes the bit-error performance of a hybrid DS/FFH communication system and solves for the resulting system design parameters. The optimization objective function is non-convex and can be solved by applying the Karush-Kuhn-Tucker conditions. We also present our efforts toward exploring the design, implementation, and evaluation of a hybrid DS/FFH radio transceiver using a single field-programmable gate array (FPGA). Numerical and experimental results are presented under widely varying design parameters to demonstrate the adaptability of the waveform for varied harsh smart grid RF signal environments.

## Keywords

## 1 Introduction

Hybrid spread-spectrum (HSS) systems, which combine direct-sequence (DS) and frequency-hopping (FH) spread-spectrum (SS) techniques, are attractive for their strong multiple-access capabilities, resistance to multipath fading and intentional/unintentional jamming, and the security they provide against eavesdroppers [1-6]. In recent years, there has been great interest in using HSS systems for commercial applications, particularly in the Smart Grid.

User requirements for the next generation wireless communication system have been specified for the Smart Grid advanced metering infrastructure (AMI) and distribution automation systems [7]. These requirements demonstrate the need for high capacity and highly secure networks for Smart Grid applications. There is a significant gap between commercially available communications systems and those needed to satisfy the demanding requirements associated with electric utility industry. HSS systems are a promising candidate for Smart Grid applications since they provide high data rates with excellent signal security.

Spreading the signal over a relatively wide bandwidth allows transmission with relatively low power density, leading to low probabilities of detection and interception. HSS systems also provide an inherent security against eavesdroppers because knowledge of the spreading codes is required. The choice of appropriate pseudo-noise (PN) codes and dynamic altering of signal parameters provides the opportunity for a strong security scheme in the physical (PHY) layer of the network [5]; details of these techniques will be addressed in future works. This specific paper will focus on implementation, exploration, and optimization of the parameter space of the HSS system for adapting the technique for application-level requirements in Smart Grid.

Based on the hopping rate, an HSS system is classified into a hybrid direct-sequence/slow frequency hopping (DS/SFH) system or a hybrid direct-sequence/fast frequency hopping (DS/FFH) version. In hybrid DS/FFH systems, multiple frequency hops occur within a single data-bit time. Specifically, each bit is represented by chip transmissions at multiple frequencies. If one or more chips are corrupted by multipath or interference in the RF link, statistically a majority should still be correct. Standard or slow frequency hopping, in contrast, transmits at least one (and usually several) data bits in each hopping interval. DS/FFH systems have not been previously widely implemented in many commercial or industrial applications since fast frequency-hopping rates were limited by the technology of frequency synthesizers. Today’s extremely fast hopping speed direct-digital synthesizers (DDSs) [8] are rapidly becoming an alternative to the traditional frequency-agile analog-based phase-locked loop (PLL) synthesizers. Output frequencies with micro-Hertz resolution and sub-degree phase tuning capabilities can thus be readily achieved using a single integrated circuit (IC).

Most of the works related to HSS in the literature have addressed evaluating its performance under different modulation techniques [2], channel conditions [1,3], multi-user interference [2,3], and jamming [4]. However, little research has yet evaluated the performance of a hybrid DS/FFH system under all combinations of the aforementioned cases. Moreover, few efforts have to date attempted to address the design and selection of the HSS system parameters that achieve optimal performance. The work in this paper extends the one in [9] and [10] from a DS system to a hybrid DS/FFH system, in addition to taking jamming impacts into consideration. In [11], the performance of a SFH system was considered. In [2] and [12], the performance of a DS/SFH system over an AWGN channel and with multi-user interference was considered. The performance of an FFH system over fading channels was examined in [13] and extended in [3] to include the effects of partial-band noise jamming. Although [4] and [14] computed the error probability of DS/SFH under jamming tones in both AWGN and Rician fading channels, only a single user was considered. In [15], the optimal spreading sequences for chip-synchronous CDMA are derived by minimizing the average bit error rate under the standard-Gaussian-approximation condition. The work in [16] presents a simulation-based study for evaluating the performance of a hybrid DS/FFH scheme. Some preliminary performance analysis and hardware designs for the hybrid DS/FFH scheme were initially presented in [17-19].

In this paper, error-probability analyses are performed for a hybrid DS/FFH system over standard Gaussian and fading-type channels, progressively including the effects from wide- and partial-band jamming, multi-user interference, and varying degrees of Rayleigh and Rician multipath fading. We present analytical derivations for evaluating the performance in terms of probability of bit error. In addition, an optimization approach is formulated that minimizes the average bit-error probability of a hybrid DS/FFH communication system and solves for the system design parameters that achieve an optimal performance level. The optimization objective function is non-convex and can be solved by applying the Karush-Kuhn-Tucker (KKT) conditions [20]. We also present our efforts toward exploring the design, implementation, and evaluation of a hybrid DS/FFH radio transceiver using a single field-programmable gate array (FPGA). Numerical and experimental results are presented under widely varying design parameters to demonstrate the adaptability of the waveform for varied harsh smart grid RF signal environments.

## 2 System model

*K*nodes that represent smart meters or data aggregation points in the Smart Grid wireless network. For the

*k*th node, the transmitted signal is given as

where *P* is the common transmitted signal power, *f*
_{c} is the carrier frequency, \( \left\{{f}_{\mathrm{h}}^{\mathrm{k}}(t)\right\} \) denotes the hopping frequency of the *k*th node, the data signal *b*
_{k}(*t*) is a sequence of statistically independent, unit-amplitude positive, and negative rectangular pulses of duration *T*
_{b}, and *a*
_{k}(*t*) is the PN-code waveform for the *k*th node in DSSS and is given as \( {a}_{\mathrm{k}}(t)={\displaystyle {\sum}_{n=-\infty}^{\infty }}{a}_{\mathrm{n}}^{\mathrm{k}}{P}_{{\mathrm{T}}_{\mathrm{c}}}\left(t-n{T}_{\mathrm{c}}\right), \) where \( \left\{{a}_{\mathrm{n}}^{\mathrm{k}}\right\} \) is the discrete periodic signature sequence assigned to the *k*th node and \( {P}_{{\mathrm{T}}_{\mathrm{c}}}(t) \) is a rectangular pulse that starts at *t* = 0 and ends at *t* = *T*
_{c}.

Consider *M* frequency hopping channels with *L* (assume *L* is odd) hops per bit. Let *T* = *T*
_{b}/*L* denote the duration of each hop and *T*
_{c} = *T*
_{b}/*NL* denote the chip duration for the PN-code sequence, where *N* is the period of the PN-sequence and is also assumed to be odd. The wide-band jamming fully corrupts *W* hopping channels and another single channel partially (let \( {W}_{\mathrm{J}}^{\mathrm{P}} \) be the part of the channel affected by the partial jamming).

*J*(

*t*) and

*n*(

*t*) represent the jamming term and AWGN term that have two-sided spectral densities

*N*

_{J}/2 and

*N*

_{0}/2,, respectively, and

*γ*

_{k}is the Rician channel coefficient for the

*k*th node;

*β*

_{k}(

*τ*,

*t*) is a zero-mean complex Gaussian random process that represents the equivalent low-pass time-varying impulse response for the fading channel [10]. The covariance function for the fading process in a WSSUS channel is [22,23].

*ρ*

_{k}(

*τ*,

*t*−

*s*) =

*ρ*

_{k}(0,

*t*−

*s*)

*δ*(

*τ*) [23], where the covariance function

*ρ*

_{k}(0,

*t*−

*s*) is defined as

where *λ* = (*n* + *β*) < *N*, *n* is a positive integer less than *N*, 0 ≤ *β* < 1 and *v* = (*λT*
_{c})^{−1}
*T* [10].

Similar to [24], the time delays and data symbols for the *k*th node are modeled as mutually independent random variables which are uniformly distributed on [0, *T*] and {−1, + 1}, respectively. We also assume *τ*
_{i} = 0 when considering the output of the *k*th (*k* ≠ *i*) correlation receiver.

## 3 Error probability analysis

In this section, we first investigate the average error probability for one hop, and then we employ a majority voting scheme to compute the overall error probability for one bit.

*k*, the other

*K-*1 users are considered as interference. Three different situations may occur in one hop:

*j*out of

*K-*1 users interfere with the same hopping channel of user

*k*and (1) no jamming corrupts the channel, (2) jamming fully corrupts the channel, or (3) jamming partially corrupts the channel. Thus, the total average error probability \( {P}_{\upvarepsilon}^{\mathrm{k}} \) of one hop for user

*k*can be computed as:

*j*interfering users. Expression (6) is equivalent to

*P*(

*a*,

*b*) is the joint probability of events

*a*and

*b*, and

*P*

^{k}(

*ε*|

*a*,

*b*) is the conditional probability of error, given events

*a*and

*b*have occurred. From the problem formulation, we can obtain:

- A.
*Case 1: No Jamming*

*k*’s channel,

*NSR*=

*N*

_{0}/2

*PT*is the noise-to-signal ratio, and

*Q*(•) is the complementary error function. Following the arguments in [10] and [24], \( {I}_{\mathrm{j}}^{\mathrm{k}} \) is computed as:

where *β*
_{
l
} = 1 for *l* < *n* and *β*
_{
n
} = *β*, *Δ*
_{i}(*l*) = *R*
_{i}(*l* + 1) − *R*
_{i}(*l*), *ζ*
_{i,k}(*l*) = *Δ*
_{i}(*l*)*R*
_{k}(*l*), *Γ*
_{l}(a) = *N* − *v*(*l* + *aβ*
_{
l
}), \( {m}_{\mathrm{i},\mathrm{k}}=2{\displaystyle {\sum}_{l=1-N}^{N-1}}{R}_{\mathrm{i}}(l){R}_{\mathrm{k}}(l)+{\displaystyle {\sum}_{l=1-N}^{N-1}}{R}_{\mathrm{i}}(l){R}_{\mathrm{k}}\left(l+1\right), \) and *R*
_{i}(*l*) is the usual aperiodic autocorrelation function for the PN-sequence.

*N*of the sequence. In this work, we employ a maximal-length sequence (MLS) as the signature sequence. However, by using an MLS code, there does not exist a closed-form expression of the aperiodic autocorrelation function,

*R*

_{i}(

*l*), for the general MLS code, which prevents us from finding a closed-form expression for \( {I}_{\mathrm{j}}^{\mathrm{k}} \). However, we can compute a closed-form expression if we know exactly which MLS code is used. Actually, two different MLS codes with the same length will have different aperiodic autocorrelation functions. Therefore, we consider an upper bound on an MLS’s aperiodic autocorrelation function derived in [26] to compute an upper bound on the error probability of the HSS system. From [26], we have

*R*

_{i}(0) =

*N*and \( {R}_{\mathrm{i}}(l)<{R}_{\mathrm{u}}=1+\frac{2}{\pi }{\left(N+1\right)}^{\frac{1}{2}}\mathrm{In}\left(\frac{4N}{\pi}\right),\;l\ne 0 \). Plugging them back into \( {I}_{\mathrm{j}}^{\mathrm{k}} \) in (10) and assuming

*γ*

_{k}=

*γ*as a constant for simplicity, we get an upper bound on \( {I}_{\mathrm{j}}^{\mathrm{k}} \) as:

- B.
*Case 2: Full Jamming*

*k*’s channel, the error probability for BPSK is given as:

*JSR*=

*N*

_{J}/2

*PT*is the jamming-to-signal ratio.

- C.
*Case 3: Partial Jamming*

*k*’s channel, the error probability includes two portions: one is the part of the channel corrupted and the other is the uncorrupted part. Let \( q={W}_{\mathrm{J}}^{\mathrm{p}}/\left(N{W}_{\mathrm{b}}\right) \) denote the fraction of the channel jammed, where

*W*

_{b}= 1/

*T*

_{b}. Then the error probability for the BPSK case is given as:

*γ*

_{k}=

*γ*, then, for simplicity, \( {P}_{\varepsilon}^{\mathrm{k}} \) can be represented as

*P*

_{ ε }. To compute the error probability for

*one bit*, denoted

*P*

_{E}, we employ a majority voting decision scheme given as:

Due to the monotonicity of *Q*(•), using (11) provides an upper bound on \( {P}_{\varepsilon}^{\mathrm{k}} \) and thus an upper bound on *P*
_{E}. The problem of determining the HSS system parameters for an optimal performance is now discussed in the next section.

## 4 Optimization problem formulation

In realistic HSS systems, the overall system performance always suffers from practical parameter constraints. Thus, we formulate the problem of minimizing the bit-error performance subject to some representative parameter constraints.

*M*, the length of the PN-sequence

*N*, the number of channels fully corrupted by jamming

*W*, and the number of hops per bit

*L*. Assume that these parameters satisfy the following constraints

together with integer constraints on the parameters (i.e., *M*, *N*, *W*, *L* are positive integers).

The physical meaning of these constraints can be explained as follows: (15) represents that the total bandwidth of the system (*MNW*
_{b}) is limited by *K*
_{1}, where *K*
_{1} > 0; (16) means that the number of frequency channels fully corrupted by the jamming are a portion of the total number of channels, where 0 ≤ *K*
_{2} ≤ 1; (17) provides a lower bound on the time duration of each hop \( \left(\frac{1}{L{W}_{\mathrm{b}}}\right) \) due to implementation limitations; and (18) restricts all the parameters to be positive.

The integer constraints are removed in the problem statement and the following analysis because they can be imposed after the solutions of (19) are found. This will be discussed in more detail in the following section.

## 5 Necessary conditions of the optimization problem

*P*

_{E}, we can further relax the constraint (17). Note

*P*

_{E}is a monotonically decreasing function with respect to

*L*, so constraint (17) can be written as:

which means that there is an upper bound on *L*. Moreover, as *P*
_{
ε
} does not depend on *L*, the error probability *P*
_{E} reaches its minimum when \( L=\frac{1}{K_3{W}_{\mathrm{b}}} \) with the other parameters fixed.

*P*

_{E}is a monotonically increasing function with respect to

*P*

_{ ε }on the interval [0, 1], and

*M*,

*N*,

*W*are all contained only in

*P*

_{ ε }; thus, the optimization problem in (19) can be further simplified as:

*P*

_{n}=

*P*(

*jusers*,

*no jam*),

*P*

_{f}=

*P*(

*jusers*,

*full jam*),

*P*

_{p}=

*P*(

*jusers*,

*partial jam*),

*P*

_{nj}=

*P*

^{k}(

*ε*|

*j*users, no jam),

*P*

_{fj}=

*P*

^{k}(

*ε*|

*j*user, full jam),

*P*

_{pj}=

*P*

^{k}(

*ε*|

*j*users, partial jam). Expression (22) can be further simplified by representing

*P*

_{pj}in terms of

*P*

_{nj}and

*P*

_{fj}as follows:

and also let *x* = (*M*, *N*, *W*).

From Section 3, we observe that *P*
_{n} and *P*
_{f} are functions of both *M* and *W*, while *P*
_{p} is only a function of *M*. Moreover, *P*
_{nj} and *P*
_{fj} are functions of *N*. We can also observe that the error probability to be minimized has a complex structure and is a non-convex function. Thus, to compute the optimal solution, we apply the Karush-Kuhn-Tucker (KKT) [20] conditions to problem (21).

*(Karush-Kuhn-Tucker Conditions) Let y**

*be a local minimum of the following problem*

*where f*and

*g*

_{ i }

*are continuously differentiable functions with appropriate dimensions. Then there exists an unique Lagrange multiplier vector μ*= (

*μ*

_{1}, …,

*μ*

_{m}),

*such that*

*where*
\( R\left(y,\mu \right)=f(y)+{\displaystyle {\sum}_{i=1}^{\mathrm{m}}}{\mu}_{\mathrm{i}}{g}_{\mathrm{i}}(y) \)
*is the Lagrangian function and A*(*y**) *is the set of active constraints at y** *defined as: For any feasible vector y (the vector that satisfies all constraints), the set of active inequality constraints is given as A*(*y*) = {*i*|*g*
_{
i
}(*y*) = 0} *and if j* ∉ *A*(*y*), *it is said that the jth constraint is inactive at y*.

*In addition, if f* and *g*
_{
i
}
*are twice continuously differentiable, then there holds*
\( {z}^T{\nabla}_{\mathrm{yy}}^2R\left({y}^{*},\mu \right)z\ge 0 \), *for all z in proper dimensions, such that* ∇*g*
_{
i
} (*y* *)^{
T
}
*z* = 0, ∀ *i* ∈ *A*(*y* *).

Now, the necessary conditions for a local minimum of problem (21) can be derived by applying the KKT conditions as follows:

*Let x** = (

*M**,

*N**,

*W**)

*be a local minimum of the problem*(21)

*, then there exists unique μ*

_{1}≥ 0,

*μ*

_{2}≥ 0,

*such that*

*where*

*In addition, the following inequality holds:*

Proof:

In order to apply the KKT conditions, we first need to check the types of inequality constraints, to determine whether they are active or inactive inequality constraints.

It is obvious that (18) is inactive at *x**. To check for (15), first assume that (15) is also inactive at *x**, which infers *M***N***W*
_{
b
} − *K*
_{1} < 0. However, it should be noted that *P*
_{
ε
} is a monotonically decreasing function with respect to both *M** and *N**; thus, *M***N***W*
_{b} − *K*
_{1} < 0 means there is still an ‘increasing space’ for either *M** or *N**, such that *P*
_{
ε
} can still be reduced by increasing *M** or *N** to *M***N***W*
_{
b
} = *K*
_{1}, which contradicts that *x** is the local minimum. Thus, (15) is an active constraint at *x**.

To check for (16), first assume that (16) is inactive at *x**, which means *W** − *K*
_{2}
*M** > 0, by applying the KKT necessary conditions (page 316, Proposition 3.3.1 in [20]), we have the unique Lagrange multiplier for (16) *μ*
_{2} = 0 and \( {\displaystyle {\sum}_{j=0}^{K-1}}{\left.\frac{\partial {P}_{\upvarepsilon, \mathrm{j}}}{\partial W}\right|}_{{\mathrm{x}}^{*}}=0 \). However, we should also observe that by (33), \( {\displaystyle {\sum}_{j=0}^{K-1}}{\left.\frac{\partial {P}_{\upvarepsilon, \mathrm{j}}}{\partial W}\right|}_{{\mathrm{x}}^{*}}\ne 0 \) as *P*
_{nj} ≠ *P*
_{fj}, which leads to a contradiction. Thus, (16) is also an active constraint at *x**.

After specifying the type of each inequality constraint, we can obtain Theorem 1 by applying the KKT conditions in Lemma 1 to problem (21).

Remark 1
*We can similarly obtain second-order sufficiency conditions of the problem by applying the following KKT sufficient conditions (page 320, Proposition 3.3.2 in* [20]*): If*
(27)
*to*
(37)
*hold for some x and μ*
_{
i
} > 0, *i* = 1, … *m*, *then x is a strict local minimum.*

Remark 2
*Once the solution is found, the integer constraints need to be imposed. For example, assume M*, *N*, *W*, *L are positive integers; first, round one parameter (*e.g., *N) to the nearest integer, then plug it back to the problem and re-compute the solution. After that, round the rest of the parameters in a similar fashion.*

Theorem 1 states the necessary conditions for the optimization problem by employing a general PN-sequence. Now, we will employ the MLS code as the PN-sequence in the HSS system and reformulate Theorem 1 explicitly.

*M*,

*N*). Considering the upper bound \( {I}_{\mathrm{j}}^{\mathrm{u}} \) in (11), Theorem 1 remains the same, with the exception that \( \frac{\partial {I}_{\mathrm{j}}^{\mathrm{k}}}{\partial N} \) is replaced with \( \frac{\partial {I}_{\mathrm{j}}^{\mathrm{u}}}{\partial N} \). After performing some derivations, we obtain:

Plugging the above equations back into Theorem 1, then we can obtain necessary conditions for the local minimum of the upper bound of the error probability for the MLS code. Sufficient conditions can also be obtained from Remark 1.

Remark 3
*Note that in an MLS code, N is an integer such that N* = 2^{n} − 1 *where n is a positive integer. Thus, after obtaining solutions of the local minimum of the problem, N in each solution should be rounded to the closest integers in the form of* 2^{n} − 1 *(usually two integers correspond to N in one solution), and the rest of the parameters in the solution should be re-computed and rounded. Then, by comparing the error probabilities resulted from these two sets of parameters, we employ the parameter set with the lower error probability as the local minimum of the problem after re-applying the integer constraints.*

In the next section, our specific design and implementation of a hybrid DS/FFH radio transceiver using a single FPGA are presented.

## 6 ORNL specific hybrid DS/FFH design and implementation

The hybrid DS/FFH prototype was designed to demonstrate the fundamental advantages of the HSS system, such as jamming resistance, difficulty of unwanted interception, robust performance, and reasonable cost. The prototype operates in the unlicensed 902 to 928 MHz ISM band, although target applications such as the SG may ultimately use a dedicated frequency band. The system parameters for the prototype are selected based on the available ISM bandwidth and FPGA capabilities and using the analysis presented in the previous section. The selected parameters are considered to be nearly optimal for a typical smart grid environment.

We decided to use the Software Defined Radio (SDR) method for hardware implementation of the hybrid DS/FFH system because of its flexibility in changing the system to evaluate new concepts. This methodology has also proven to be very powerful in that the vast majority of the signal processing components can be placed in a single FPGA. The entire HSS band is down-converted to an intermediate frequency, digitized, and sent to the FPGA. Within the FPGA, look-up-table-based local oscillators down-convert the individual FH channels to baseband. These baseband signals are then decoded using DS correlators and stored in a buffer for subsequent delivery to a host computer.

The bit-shift number refers to the number of bits that the local DS code has been shifted for performing the correlation. To prevent ambiguous results from a correlation being between two bits, only every other bit position is used, which results in 31 positions available for each code word. Four bytes of blank data are sent at the beginning of the packet as a preamble to set the reference DS start time.

A different interpretation of this methodology would be that the DS code is shifted because of a different time-of-flight, similar to GPS or continuous wave radar. Similar to the way GPS can achieve precise time-of-flight resolution, it can be expected that this methodology can be further developed to obtain higher bit capacity. The work in [28] explores this method for multiple users occupying a channel simultaneously.

The HSS channel capacity is calculated by dividing the chip rate, or 1.25 MHz, by the 63-bit code length to get 19,841 DS sequences per second. Since the data is replicated three times for redundancy, the actual throughput is 6,613 DS sequences per second. Since each DS sequence contains 8 bits of data, the data throughput is 52,910 bits per second. The HSS prototype is optimized for reading household utility meters for smart grid applications and thus only requires 32 bytes, although the system has operated successfully with 256-byte packets.

To acquire the packet preamble, a spread-spectrum correlator continually looks for the preamble pattern on all channels. Once the preamble is detected, an internal timing sequence compares the signal with shifted copies of the DS code via a simple correlator. The shifted copy of the DS code that provides the strongest correlation then demodulates the actual data. To make the signal detection independent of the carrier phase, both phases of the carrier (I and Q) are correlated with the preamble’s code. However, the phase relationship must remain consistent during the duration of the DS sequence.

## 7 Numerical and experimental results

### 7.1 Hybrid DS/FFT system performance

We first demonstrate the performance of a hybrid DS/FFH system over Rician time-selective fading channels, progressively including the effects from wide- and partial-band jamming, multi-user interference, and varying degrees of Rician fading. The performance measure is the upper bound of BER described in (14) by employing (11). The parameters of the reference system model considered in this numerical example are total number of users is *K* = 100; number of hops per bit is *L* = 5; number of frequency-hopping channels is *M* = 30; period of PN-sequence in DSSS is *N* = 127; jamming-to-noise ratio (JNR) is 13 dB; number of channels fully jammed is 5; the Rician channel coefficient *γ* = 0.1 (represents the channel fading part); channel covariance function scaling factor *λ* = 10.8; and the portion of the channel partially corrupted is 0.4. The parameter space of the HSS system is explored to demonstrate its effectiveness under different conditions and scenarios. In the following analysis, we successively vary one parameter in the reference system model while fixing the other parameters.

*γ*.

The presented results demonstrate the effectiveness of the proposed hybrid DS/FFH scheme under severe channel conditions and, therefore, indicate that there is a high potential for employing it in complex smart grid communications.

### 7.2
*Optimizing hybrid DS/FFT system performance*

We now provide numerical examples to illustrate the results derived in Section 5. For convenience, we only test the necessary conditions that apply to the MLS code. We compute the solutions of the first-order necessary conditions (27) to (31) and impose the integer constraints. Then, the upper bound of the BER, *P*
_{
E
}, is plotted for different MLS code lengths *N* using (14) and (11) to verify the results computed from the derived first-order necessary conditions.

The parameters of the reference hybrid DS/FFH system model considered is the same as described in the previous section (*K* = 100; JNR = 13 dB; *γ* = 0.1; *and λ* = 10.8), in addition to a signal-to-noise ratio (SNR) of 20 dB; finally, the portion of the channel partially corrupted is \( q={W}_{\mathrm{J}}^{\mathrm{p}}/\left(N{W}_{\mathrm{b}}\right)=30/N \). Note that the parameters *M*, *N*, *W*, *L* need to be computed for assessing the optimal performance. From the previous analysis, the number of hops per bit is chosen as *L* = (1/*K*
_{3}
*W*
_{b}) = 5.

*K*

_{1}= 2600

*W*

_{b}and

*K*

_{2}= 0.2. Then, by applying (27) to (31), we obtain

*N*= 42. Because of the integer power-of-two constraint of

*N*(

*N*= 2

^{n}− 1), it is rounded to the nearest two integers, 31 and 63. Then by applying (30) and (31) for each integer of

*N*and comparing the corresponding BER of both integers, we see that

*N*= 31,

*M*= 83, and

*W*= 17 results in a smaller BER. The upper bound of the BER in (14) for different PN-code lengths,

*N*, is demonstrated in Figure 15, in which we can now observe that at

*N*= 31, the BER reaches its minimum. This coincides with the result from the first-order necessary conditions.

Now, we consider *K*
_{1} = 3600 *W*
_{b}, with *K*
_{2} unaltered. Through a similar procedure, we obtain *N* = 48.3. After rounding *N* to 31 and 63, it can found that *N* = 63, *M* = 57, and *W* = 11 results in smaller BER values. Figure 15 demonstrates the upper bound of the BER for different PN-code lengths, *N*, for this scenario. It can now be observed that the BER reaches its minimum at *N* = 63, which also coincides with the result from our analysis.

### 7.3 Experimental evaluation*s*

^{−6}.

## 8 Conclusion

In this paper, the performance of a hybrid DS/FFH system over Rician fading channels was considered. We derived the average BER for a hybrid DS/FFH system that includes the effects from wide- and partial-band jamming, multi-user interference, and/or varying degrees of Rician fading. Numerical results exploring the parameter space of the HSS system have also been presented to demonstrate its effectiveness under different conditions and scenarios. We have also demonstrated a novel non-convex optimization technique that minimizes the bit-error probability of a hybrid DS/FFH communication system under multiple constraints. By employing the Karush-Kuhn-Tucker conditions, the process solves for the optimal system design parameters. In addition, a hardware FPGA-based hybrid DS/FFH prototype was implemented successfully and optimized for a typical smart grid utility application. Experimental results indicate that high resistance of hybrid DS/FFH systems to other jamming and interference signals allows the possibility of intentionally operating several HSS radios in the band simultaneously. For smart grid applications, this would enable a base station to service several clients at the same time, provided the system arranged for different clients to use different hop patterns and DS codes, and possibly even coordinated transmission time windows. The use of hybrid DS/FFH waveform in wireless networks as employed in the smart grid is recommended, as it offers superior resistance to jamming attacks and improves the reliability of transmission compared to existing SS techniques like DS, FH, and hybrid DS/SFH systems.

## Declarations

### Acknowledgment

This manuscript has been authored by UT-Battelle, LLC under Contract No. DE-AC05-00OR22725 with the U.S. Department of Energy. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes. The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan.

In addition, this work has been partially supported by NSF grant CMMI-1334094.

## Authors’ Affiliations

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