In this section, the performance analysis of the proposed scheme is analyzed in AWGN channel environment to predict the BER performance. The chaotic sequence is generated from the symmetric Tent map, given by the equation \({x_{k + 1}} = 1 - 2|{x_k}|\). The chaotic sequence *x* is having uniform distribution between \(-1\) and 1 with zero mean and computed variance \(V(x)=1/3\) and \(V(x^2)=4/45=4/5V^2(x)\) [11]. For the rest of the analysis, the following assumptions are considered (1) The correlation between the chaotic sequence and its permutated version is decaying quickly for a sufficient correlation window. (2) The correlation between the chaotic sequence and the noise sample is statistically independent.

The Bit Error Probability of the proposed scheme is determined by calculating (1) erroneous permutation estimation \(Pr_{\mathrm{{map}}}\), (2) erroneous probability of bits detection which largely depends on the number of bits *n* which calculated by the following

$$\begin{aligned} P_{r_{\mathrm{{map}}}} = \frac{{{2^{(n - 1)}}}}{{{2^n} - 1}}{P_{{r_{\mathrm{{ed}}}}}} \end{aligned}$$

(7)

Correct estimation of the transmitted sequence is done by selecting the maximum absolute value of one of the correlator outputs. This output results from the correlation between the delayed reference signal and inverse permutated information-bearing signal. Each correlator output can be modeled as a Gaussian random variable \(D_m\). For equiprobable transmitted sequence, the error probability of permutation index estimation conditioned by \(P_j\) is given by

$$\begin{aligned} {P_{\mathrm{{red}}}} = ({P_r}|{D_{j1}}| < \max {P_r}|{D_m}|)~\text {for}~1 \le {{m}} \le {2^n},~ m \ne j_1 \end{aligned}$$

(8)

where the \(D_{j1}\) and \(D_m\) are decision variable at *m*th and \(j_1^{th}\) correlator output. The error will occur only if any value of \(D_m\) can have a magnitude larger than \(D_{j1}\). To detect the first information-bearing signal, the output of *m*th correlator can have two values and can be rewritten as

$$\begin{aligned} {\mathrm {D_{m}}} = {\left\{ \begin{array}{ll} \mathrm{{SS}}+ \mathrm{{SN}} + \mathrm{{NN}} & m= j_1 \\ \mathrm{{SI}}+ \mathrm{{SN}} + \mathrm{{NN}} & m \ne j_1. \end{array}\right. } \end{aligned}$$

(9)

The correlation components can be calculated as

$$\begin{aligned} \mathrm{{SS}} = {X_\beta }{P_{j_1}}^{ - 1}{P_{j_1}}{X_\beta } = {X_\beta }{X_\beta }^T = \sum \limits _{k = 1}^\beta {{x_k}} {x_k} \end{aligned}$$

(10)

$$\begin{aligned} \mathrm{{SN}} = {X_\beta }{W_0} + {W_\beta }{P_{j_1}}^{ - 1}{P_{j_1}}X = {\sum \limits _{k = 1}^\beta {{x_{k - \beta }}} }{w_k} + {\sum \limits _{k = 1}^\beta {{w_{k - \beta }}} }{x_k} \end{aligned}$$

(11)

$$\begin{aligned} \mathrm{{NN}}=W_\beta P_{j_1}^{-1}W_0=\sum _{k=1}^{\beta }w_{k-\beta }w_k^{'}, \end{aligned}$$

(12)

where \(w_i^{'}\) is the permutated noise sample.

Based on the assumption that the correlation between noise sample \(w_i\) and chaotic samples \(x_j\) for all *i* and *j* is negligible, for sufficient value of \(\beta\), then

$$\begin{aligned} E({D_{j_1}}) = E(\mathrm{{SS}}){\mathrm{+ E(\mathrm{{SN}}) + E(\mathrm{{NN}}) = E}}\left( \sum \limits _{k = 1}^\beta {{x_k}} {x_k}\right) + 0 + 0 = \beta V(x) = \frac{L}{{L + 1}}n{E_b} \end{aligned}$$

(13)

$$\begin{aligned} V(\mathrm{{SS}}) = V\left( \sum \limits _{k = 1}^\beta {{x_k}} {x_k}\right) = V\left( \sum \limits _{k = 1}^\beta {{x^2}_k} \right) = \frac{4}{5}\beta V(x)V(x) = \frac{{4{{(L)}^2}}}{{5\beta {{(L + 1)}^2}}}{n^2}E_b^2 \end{aligned}$$

(14)

$$\begin{aligned} V{\mathrm{(SN)}} = 2\beta V(x)V(w) = 2\frac{{Ln}}{{L + 1}}{E_b}{N_0} \end{aligned}$$

(15)

$$\begin{aligned} V{\mathrm{(NN) = }}V(w).V(w') = \frac{\beta }{4}N_o^2. \end{aligned}$$

(16)

While \(E(D_{m}) \approx 0\), similarly

$$\begin{aligned} V({D_{m}})&= V(\mathrm{{SI}}){\mathrm{+ }}V{\mathrm{(SN) + }}V{\mathrm{(NN)}} \end{aligned}$$

(17)

$$\begin{aligned}&= \frac{{{{(L)}^2}{n^2}}}{{\beta {{(L + {1})}^2}}}E_b^2 + 2\frac{{Ln}}{{(L + 1)}}{E_b}{N_0} + \frac{\beta }{4}N_o^2 \end{aligned}$$

(18)

The output of each \(2^n-1\) correlators \(D_m\) for \(m \ne j_1\) are statistically independent random values characterized by a normal distribution with zero mean and can be given as

$$\begin{aligned} {f_{{D_{m}}}}(y) = \frac{1}{{\sqrt{2\pi V({D_{m}})} }}{e^{ - \frac{{^{{{(y)}^2}}}}{{2V({D_{m}})}}}} \end{aligned}$$

(19)

While the correlator output conditioned by correct permutation can be given as

$$\begin{aligned} {f_{{D_{j_1}}}}(y) = \frac{1}{{\sqrt{2\pi V({D_{j_1}})} }}{e^{ - \frac{{^{{{(y - E({D_{j_1}}))}^2}}}}{{2V({D_{j_1}})}}}} \end{aligned}$$

(20)

It is easier to calculate the probability of correct permutation which is occurred only if magnitude of \(D{j_1} > D_1\), \(D{j_1} > D_2\) and \(\ldots \, Dj_1>D_M\), therefore the probability of correct map detection can be given as [22]

$$\begin{aligned} p_{\mathrm{{map}}}=1-\int _{0}^{\infty } F_{Dm}(y)^{2^n-1}f_{j_1}(D{_{j_1}})(y)\mathrm{{d}}y, \end{aligned}$$

(21)

where \(F_{Dm}(y)\) is the commutative distribution function given by

$$\begin{aligned} {F_{{D_{m}}}}(y) = erf\left( \frac{y}{{\sqrt{2\pi V({D_{m}})} }}\right) \end{aligned}$$

(22)

The overall BER of the system can be given by substitution on

$$\begin{aligned} \mathrm{{BER}}_{\text{{SR-PI-DCSK}}}&= \frac{2^{(n - 1)}}{{2^n} - 1} \left[ 1 - \frac{1}{{\sqrt{2\pi \frac{{4{{(L)}^2}{n^2}}}{{5\beta {{(L + 1)}^2}}}E_b^2 + 2\frac{{Ln}}{{L + 1}}{E_b}{N_0} + \frac{\beta }{4}N_o^2} }} \right. \\&\int \limits _0^\infty erf\left( \frac{y}{\sqrt{2\pi \frac{{(L)^2}{n^2}}{{\beta {{(L + 1)}^2}}}E_b^2 + 2\frac{{Ln}}{{L + 1}}{E_b}{N_0} + \frac{\beta }{4}N_o^2} }\right) ^{{2^n} - 1} \\&\left. \mathrm{{e}}^{ - \frac{^{{{(y - E({D_{j_1}}))}^2}}}{{2V({D_{j_1}})}}}\mathrm{{d}}y \right] \end{aligned}$$

(23)

When *L* is very large, the overall BER of the system can be given by

$$\begin{aligned} \mathrm{{BER}}_{\text{{SR-PI-DCSK}}}&= \frac{2^{(n - 1)}}{{2^n} - 1} \left[ 1 - \frac{1}{{\sqrt{2\pi \frac{{4\,{n^2}}}{{5\beta {\,}}}E_b^2 + 2nE_b N_0 + \frac{\beta }{4}N_o^2} }} \right. \nonumber \\&\int \limits _0^\infty erf\left( \frac{y}{\sqrt{2\pi \frac{\,{n^2}}{{\beta {\,}}}E_b^2 + 2nE_b N_0 + \frac{\beta }{4}N_o^2} }\right) ^{{2^n} - 1} \nonumber \\&\left. \mathrm{{e}}^{ - \frac{^{{{(y - E({D_{j_1}}))}^2}}}{{2V({D_{j_1}})}}}\mathrm{{d}}y \right] \end{aligned}$$

(24)