1 Introduction

Orthogonal frequency division multiple access (OFDMA) systems attract a lot of attention these days in the area of digital communications for the right reasons [1]. In OFDMA, sub-carriers are grouped into sub-channels which are assigned to multiple users for simultaneous transmissions. Since the bandwidth of a single sub-channel is generally smaller than the coherence bandwidth, a frequency selective fading channel is converted to a flat fading channel.

In subsequent work [2, 3], Ranging is an uplink synchronization process which helps to maintain orthogonality among the sub-carriers in the uplink channel of OFDMA systems by making sure that the signals from all users arrive at the base station (BS) synchronously. In the later stages of the Ranging process, the mobile stations (MSs) will have to adjust transmission time instants and transmit powers so that at the BS, and their Ranging signals synchronize with the mini-slot boundary of the BS and have equal power. By means of the Ranging process, the system compensates near-far problems (different propagation delays, received powers) in larger cells.

At the beginning of the initial Ranging process, MS notifies the request of initial wireless access to the BS by transmitting Ranging signal. If the BS detects the Ranging signal of the MS successfully, BS allocates downlink and uplink resources to MS for further communication. Then, the MS transmits identification information (Ranging code) to the BS through the allocated channels. Finally, authorisation and registration is performed. If multiple MSs transmit their Ranging signals simultaneously, BS has to recognize each MS by ranging signal identification. Due to diverse mobility and location, each MS has different transmission time delays (TTDs). The BS has to perform uplink time synchronization to compensate TTD of each MS. Both code identification and uplink time synchronisation are based on peak detection of received signals by exploiting the cross-correlation property of CDMA codes. Under the benign channel environment, interference is supposed to be removed by the cross-correlation property of the CDMA codes. However, it is prone to be affected by the channel characteristics of the multipath fading channel [4].

In order to preserve this cross-correlation property, we introduce spatial diversity to take advantage of multipath and interference rich environment [5]. At the receiver side, there will be multiple antennas separated at distance more than half the signal wavelength, so that it could achieve spatial diversity. The signals from multiple antennas can be combined with the help of different combining techniques like maximal ratio combining (MRC), selection combining (SC), and equal gain combining (EGC).

Jamming is a special kind of interference (usually deliberate) that disrupts communications by decreasing the signal-to-noise ratio (SNR) we intend to disrupt [6]. Whenever only a small portion of the signal bandwidth is attacked, narrowband jamming (Entire output of the Jammer is concentrated in a very narrow bandwidth, identical to the signal bandwidth) is said to have occurred.

Ranging process in IEEE 802.16 system provides a number of functionalities, such as initial network entry, uplink synchronization, power adjustment, and system coordination. In further work [7], a novel ranging transmission power control algorithm which reduces the interference between MS-BS ranging and MS-RS ranging was discussed. Computer simulations show the performance of ranging process in terms of ranging detection, success probability, and validate the efficiency of the proposed transmission power control algorithm.

Further, handover delay of several fast handover schemes, including single neighbour BS scanning, fast ranging and pre-registration have been proposed, analyzed, and compared [8]. Simulation results show that fast handover schemes can reduce handover delay, thus improving the QoS of IEEE 802.16e broadband wireless networks.

The performance of a direct sequence code division multiple access (DS-CDMA) system was investigated in the presence of multiple access interference (MAI) and additive white Gaussian noise (AWGN) in the presence of partial-band jamming [9]. Two types of receivers were studied: Single user matched filter receiver and a decorrelating receiver. As technology becomes increasingly able to meet the requirements, interest in faster, non-coherent frequency hopping rates to reduce the jamming of communication has increased. The focus is on the performance of a fast frequency hopping spread spectrum system operating in the presence of partial-band jamming [10].

Due to its capability to resist jamming signals, chirp spread spectrum (CSS) technique has attracted much attention in the area of wireless communications. In ensuing work [11], the authors presented analytical results on the performance of a CSS system by deriving symbol error rate (SER) expressions for a CSS M-ary Phase Shift Keying (PSK) system in the presence of broadband and tone jamming signals, respectively. The performance of WiMAX based system was found to greatly differ with the use of different jamming signals [12]. The issues related to single-carrier jamming and multi-carrier jamming are also discussed.

The work done [13] applies the idea of iterative minimum mean squared error (MMSE) multiuser interference suppression. The soft-decoded feedback estimates of the interfering signals are employed to perform soft MMSE multiuser interference suppression in conjunction with successive interference cancellation. Closed-form expressions for interference power of narrowband jamming signal on time-hopping ultra wide band (UWB) system with binary pulse position modulation, assuming that the jamming signal as a continuous wide sense stationary zero mean random process and the time hopping codes as a pseudo-random sequence, were derived [14].

The details of initial ranging and algorithm are proposed to carry out a successful ranging process. Performance results and the comparison of computational complexity with traditional methods were also presented [15]. New ranging designs which enable multiuser diversity gain and facilitate new efficient low complexity algorithms for multiuser ranging signal detection, timing estimation, and power estimation were studied [16].

This paper is organized as follows: Sect. 2 describes the system model. Section 3 includes derivations and discusses performance measures like detection miss rate under jamming for various fading environments. Section 4 includes detailed analysis of the expressions and the graphs. Finally, Sect. 5 presents the Conclusions.

2 System Model

Using cross-correlation of the Ranging process, helps BS identify the MS. Compared to earlier work on the Ranging process with a single antenna [17, 18], the current work uses multiple antennas (antenna spacing greater than half the wavelength), and employs combining techniques like SC, MRC, and EGC to obtain the received signal with the highest SNR. This high quality received signal leads to high probability of identifying MS by the BS.

2.1 Transmitter Block Description of a Ranging Process

Figure 1a, shows the block diagram of the transmitter block in a Ranging process, which implements SC, EGC and MRC diversity. There are multiple MSs and a single BS. Each MS has a total of \({\text{ N }}_\mathrm{{c}}\) sub-carriers, out of which SF number of sub-carriers are allocated for the Ranging process. Each MS will be allotted a unique code and a transmit antenna to transmit the signal. The code will undergo Inverse Fast Fourier Transform (IFFT), which will convert the signal from frequency domain to time domain. After this, Guard Time Interval (GTI) is added to make sure that the signals are not affected by inter symbol interference (ISI). The resultant signal is sent to the BS through the channel. The channel could be any fading environment like Rayleigh, Rician, or Nakagami.

Fig. 1
figure 1figure 1

a Transmitter block of Ranging process [17]. b Receiver block of a Ranging process with SC diversity [17]. c Receiver block of a Ranging process with EGC diversity. d Receiver block of a Ranging process with MRC diversity

Full size image

2.2 Ranging Process with SC Diversity

With SC diversity, the main idea is to work with the branch that sees the best channel conditions for any given transmission. Among all transmissions available, the branch with the highest SNR is chosen.

At the BS for SC diversity, there could be multiple antennas as shown by Fig. 1b. The received signals from each of the antennas must be combined together with the help of the SC technique. Then guard time removal (GTR) is incorporated. After this process, the signal is converted to frequency domain from time domain through the Fast Fourier Transform (FFT) block [17]. In order to identify the MS, it is necessary to estimate the unique Ranging code TTD.

TTD is described as the difference between the time at which the MS starts transmitting information and the time at which the BS receives information. Code estimation is based upon peak detection, exploiting cross-correlation property of the ranging code. TTD estimation is done randomly. Performance degradation is caused through deterioration of cross-correlation property of the Ranging process. The amount of performance degradation depends on channel characteristics, SNR, and interfering users.

2.3 Ranging Process with EGC Diversity

In EGC technique, the signals of all branches are co-phased and summed together to form the equivalent channel output. This combining technique is better than SC, but worse than MRC. In Fig. 1c, the receiver block of the Ranging process with EGC diversity is shown.

At the BS, there could be multiple antennas. The received signals from each antenna must be combined together with the help of the EGC technique. To do this, the received signals are made to pass through a co-phasing and summing block. Co-phasing involves placing two or more identical antennas side-by-side or one over another (“stacking”) at a certain distance apart (usually greater than or equal to half the wavelength) and feeding antennas in-phase. Co-phasing or “stacking” has long been a way to get high gain from the antennas. After co-phasing, the signals are summed together to get the EGC output. The GTR, FFT, and code estimation process is similar to that of the SC diversity Ranging process.

2.4 Ranging Process with MRC Diversity

In MRC technique, received signals are optimally combined. Channels with larger gains are more emphasized than others. This is intuitive since the received signals through better channels are more reliable, and thus provide us with more accurate information. This is the best combining technique. Figure 1d shows the block diagram containing multiple MSs and a single BS of a ranging process which implements MRC diversity.

The received signals from each of the antennas must be combined together with the help of the MRC technique. To do this, the received signals are first weighted with the respective channel gains. Then, the signal is treated with a co-phaser and a summer. The co-phaser will feed the signal in-phase while the summer provides the MRC output.

3 Mathematical Analysis of Detection Miss Rate

Two things are to be estimated in order to identify the MS by the BS described [17], one is the Ranging process and other is the TTD. If the code and TTD are estimated successfully, the state of the Ranging process is known as detection success [7, 17]. If the code is not estimated successfully, the state of the Ranging process is known as detection miss [7, 17]. If the code is estimated successfully and TTD is not estimated successfully, the state of the Ranging process is known as detection error. Detection miss rate is also defined as the probability of estimating the code wrongly. Detection miss rate is considered for analysis in this paper.

Here, 1,000 samples of the maximum correlation values of the codes are taken in a Rayleigh fading channel with 5 users [2]. Consider Eqs. (22) and (23) of earlier work [19] as shown below:

$$\begin{aligned}&p_{D_i } (r)\approx \frac{1}{\sigma _i^{2\gamma _R } 2^{\gamma _R }\Gamma \left( {\gamma _R } \right) }r^{\gamma _R -1}\exp \left( {-\frac{r}{2\sigma _i^2 }} \right) ;\, r\ge 0,\,\end{aligned}$$
(1)
$$\begin{aligned}&\sigma _i^2 =\left\{ {\begin{array}{ll} \frac{A_i^2 \sigma _h^2 }{2}+\frac{\sigma _w^2}{2M};&{}i\in I_R \\ \frac{\sigma _w^2 }{2M};&{} \text{ otherwise } \\ \end{array}} \right. \end{aligned}$$
(2)

It is found out that the probability density function (PDF) curves of maximum correlation matches well with that of a classical Rayleigh distribution. Thus, maximum correlation can be approximated as a Rayleigh random variable, which can be obtained from the PDF shown in (1) (Equation 22 of [19]). Further, the detection miss rate without interference of any kind, which is one of the terms involved in (1), can be calculated from (2) (Equation (23) of [19]). The detection miss rate, \(\sigma _i^2\), is also called the variance of the \(i\)th ranging signal.

Here, \(D_i \) is the summation of squares of \(2\upgamma _R \) approximately i.i.d. real-valued Gaussian random variables, \(_{ }2\upgamma _\mathrm{{R}}\)is the number of degrees of freedom, \(M\) is the number of symbol intervals, \(\sigma _\mathrm{{w}}^2\) is the noise power, \(A_i\) is the amplitude of the \(i\)th ranging signal, and \(\upsigma _\mathrm{{h}}^2\) is the statistical average energy of channel taps [2].

3.1 Detection Miss Rate in a Rayleigh Fading Channel Suffering from Narrowband Jamming

Let \(\varepsilon _\mathrm{{c}},T_\mathrm{{c}},\, w_\mathrm{{m}}\), and \(J_\mathrm{{avg}}\) represent the energy per chip, the chip time, the weight of the \(m\)th code and the average jamming power spectral density, respectively. The PDF of a Rayleigh faded random variable,\(r\), is given by Equation (5.2-3) on page 314 of [20] as

$$\begin{aligned} f_{Rayl} \left( r \right) =\frac{1}{\overline{\gamma }}\exp \left( {-\frac{r^{2}}{\overline{\gamma }}} \right) \; \forall r>0, \end{aligned}$$
(3)

where \(\bar{{\upgamma }}\) is the average received SNR.

From Equation (13.2-30) on page 737 of [6], the probability of code word error on the \(m^\mathrm{{th}}\) code due to narrowband jamming can be expressed as

$$\begin{aligned} P_2 \left( m \right) =Q\left( {\sqrt{\frac{2\varepsilon _c W_m }{T_c J_{avg} }}} \right) . \end{aligned}$$
(4)

Then, the probability of detection miss rate in a Rayleigh fading channel due to narrowband jamming is derived by combining (3) and (4). So,

$$\begin{aligned} P_{nj}^{Rayl} =\frac{1}{4\sqrt{\bar{{\gamma }}}}\sqrt{\pi }-\frac{1}{2\bar{{\gamma }}\sqrt{\pi }}\sum _{l=0}^\infty {\frac{2^{l}\left( {\mu /{\sqrt{2}}} \right) ^{2l+1}l !}{(2l+1)!\left( {{\mu ^{2}}/2+1/{\bar{{\gamma }}}} \right) ^{l+1}}}, \end{aligned}$$
(5)

where \(\mu =\sqrt{\frac{2\varepsilon _c w_m }{J_{avg} T_c }}\).

3.2 Detection Miss Rate in a Rayleigh Fading Channel with SC Diversity Suffering from Narrowband Jamming

The PDF of a Rayleigh faded random variable \(r\) undergoing \(M\)-branch SC diversity is given by Equation (5.4.83) on page 364 of [20] as

$$\begin{aligned} f_{Rayl}^{SC} (r)=\frac{M}{\overline{\gamma }}\sum _{k=0}^{M-1} {(-1)^{k}\frac{(M-1)!}{(M-k-1)! \,k!}} \exp \left( {-\frac{r^{2}(1+k)}{\overline{\gamma }}} \right) \quad \forall \,r>0. \end{aligned}$$
(6)

The probability of detection miss rate due to narrowband jamming in \(M\)-branch SC Rayleigh fading channel is derived by combining (4) and (6). Hence,

$$\begin{aligned}&P_{\text{ nj }} ^\mathrm{{ray,SC}}\nonumber \\&\quad =\frac{M}{4\sqrt{\overline{\gamma }}}\sum _{k=0}^{M-1} {\frac{\left( {M-1} \right) !}{\left( {M-k-1} \right) !k!}\,\cdot \frac{1}{\left( {k+1} \right) ^{1/2}}} \sqrt{\pi } \nonumber \\&\qquad -\frac{M}{2\sqrt{\pi \overline{\gamma }}}\sum _{l=0}^\infty {\frac{\left( {\mu /{\sqrt{2}}} \right) ^{2l+1}2^{l}}{\left( {2l+1} \right) !}\sum _{k=0}^{M-1} {\left( {-1} \right) ^{k}} \frac{\left( {M-1} \right) !}{\left( {M-k-1} \right) !k!}\cdot \frac{1}{\left( {{\mu ^{2}}/2+{\left( {k+1} \right) }/{\overline{\gamma }}} \right) ^{l+1}}l!}\nonumber \\ \end{aligned}$$
(7)

3.3 Detection Miss Rate in a Rayleigh Fading Channel with EGC Diversity Suffering from Narrowband Jamming

The PDF of a Rayleigh faded random variable \(r\) undergoing \(M\)-branch EGC diversity is given by Equation (5.4.121) on page 376 of [20] as

$$\begin{aligned} f^{EGC}_{ray} \left( r \right) =\frac{\Gamma Mb^{M}r^{2M-2}}{12\left( {br^{2}+\Gamma } \right) ^{M+1}}\;\forall \,r>0, \end{aligned}$$
(8)

where \(\Gamma \) is the average received SNR, and \(b=\frac{M}{2}\left( \frac{\left( M-\frac{1}{2}\right) !}{\sqrt{\pi }}\right) ^\frac{-1}{M}\).

The probability of detection miss rate due to narrowband jamming in \(M\)-branch EGC Rayleigh fading channel is derived by combining (4) and (8). So,

$$\begin{aligned} P_{nj} ^{ray,SC}&= \frac{M\sqrt{b}}{4\sqrt{\Gamma }}\left( {\frac{2}{3}+\sum _{k=1}^\infty {\frac{\left( {-1} \right) ^{k}}{k!}\,\frac{2}{2k+3}\prod _{l=0}^{k-1} {\left( {M-1.5-l} \right) } } } \right) \nonumber \\&\quad -\frac{M}{2\sqrt{\pi }}e^{{\beta ^{2}\Gamma }/{4b}}\sum _{l=0}^\infty {\frac{\left( {\mu /{\sqrt{2}}} \right) ^{2l+1}\left( {2/b} \right) ^{l}}{\left( {2l+1} \right) !}} \sum _{k=0}^{M+l-1} {\left( {-1} \right) } ^{k}\nonumber \\&\quad \times \Gamma ^{\frac{k+1}{2}}\frac{\left( {M+l-1} \right) !}{\left( {M+l-k-1} \right) !k!}\left( {\frac{2b}{\mu ^{2}}} \right) ^{\frac{l-k}{2}} W_{\left( {-1-k/2+l/2,-1/2-k/2+l/2} \right) } \left( {\frac{\beta ^{2}\Gamma }{2b}} \right) ,\qquad \end{aligned}$$
(9)

where \(W_{k,m} (z)=\frac{e^{-z/2}z^{k}}{\Gamma (\frac{1}{2}-k+m)}\int _o^\infty {t^{-k-1-m}} \left( {1+t/z} \right) ^{k-1/2+m}e^{-t}dt\) is WhittakerW function (page 343 of [21]).

3.4 Detection Miss Rate in a Rayleigh Fading Channel with MRC Diversity Suffering from Narrowband Jamming

The PDF of a Rayleigh faded random variable \(r\) undergoing \(M\) branch MRC diversity is given by Equation (5.4.91) on page 367 of [20] as:

$$\begin{aligned} f_{ray} ^{MRC}\left( r \right) =\frac{r^{2M-2}}{\left( {M-1} \right) !\overline{\gamma }^{M}}\exp \left( {-\frac{r^{2}}{\overline{\gamma }}} \right) \forall \,r>0. \end{aligned}$$
(10)

The probability of detection miss rate due to narrowband jamming in \(M\) branch MRC Rayleigh fading channel is derived by combining (4) and (10). Hence,

$$\begin{aligned} P_{nj}^{Rayl, MRC}&= \frac{1}{4\sqrt{\bar{{\gamma }}}(M-1) !}\Gamma \left( {M-\frac{1}{2}} \right) \nonumber \\&\quad -\frac{1}{2\sqrt{\pi }\bar{{\gamma }}^{M}\left( {M-1} \right) !}\sum _{l=o}^\infty {\frac{\left( {\mu /{\sqrt{2}}} \right) ^{2l+1}2^{l}}{\left( {2l+1} \right) !}} .\frac{1}{\left( {{\mu ^{2}}/2+1/{\bar{{\gamma }}}} \right) ^{l+M}}\Gamma \left( {l+M} \right) \nonumber \\ \end{aligned}$$
(11)

3.5 Detection Miss Rate in a Nakagami Fading Channel Suffering from Narrowband Jamming

For a Nakagami fading channel with parameter \(d\), the PDF of the fading channel with \(r\) representing the Nakagami random variable is given by Equation (2.147) on page 47 of [6] as

$$\begin{aligned} f_{Nak} (x)=C(d)x^{2d-1}\exp \left( {-\frac{dx^{2}}{\overline{\gamma }}} \right) ;\quad d\ge 0.5, \end{aligned}$$
(12)

where \(C(d)=\frac{2d^{d}}{\Gamma (d)\bar{{\gamma }}^{d}}.\) Here, \(\overline{\upgamma }=E\left( {\text{ x }^{2}} \right) \), and the parameter \(d\) is defined as the ratio of moments, called the fading figure, \(d=\frac{\overline{\gamma }^{2}}{E\left[ {\left( {x^{2}-\overline{\gamma }} \right) ^{2}} \right] }\) [Equation (2.1-148) on page 47]. When \(d\) = 1, Nakagami distribution becomes the Rayleigh distribution, and when \(d\) = 0.5, it becomes a one-sided Gaussian distribution, and when \(d\) tends to infinity, the distribution becomes an impulse (no fading). Even the Rician distribution can be closely approximated using the Nakagami fading parameter \(d\).

The probability of the detection miss rate due to narrowband jamming in a Nakagami fading channel is derived by combining (4) and (12). Hence,

$$\begin{aligned} P_{nj}^{Nak}&= C(d)\cdot \frac{2^{d-2}}{d^{d}}\Gamma (d)\nonumber \\&\quad -\frac{C(d)}{\sqrt{\pi }}\sum _{l=o}^\infty {\frac{\left( {\mu /{\sqrt{2}}} \right) ^{2l+1}2^{d+2l-1/2}}{\left( {2l+1} \right) !}} \frac{1}{\left( {\mu ^{2}+d} \right) ^{d+l+1/2}}\Gamma \left( {d+l+\frac{1}{2}} \right) . \end{aligned}$$
(13)

3.6 Detection Miss Rate in a Nakagami Fading Channel with SC Diversity Suffering from Narrowband Jamming

The PDF of Nakagami fading channel with SC diversity is given as

$$\begin{aligned} f_{Nak}^{SC} (x)=\frac{2L}{\left( {\Gamma (m)} \right) ^{L}}\left( {\Gamma \left( {m,\frac{mx^{2}}{\overline{x^{2}} }} \right) } \right) \left( {\frac{m}{\overline{x^{2}} }} \right) ^{m}x^{2m-1}\quad \forall \,x\ge 0, \end{aligned}$$
(14)

where \(\Gamma \left( {u, \alpha } \right) =\int _0^\alpha {e^{-t}t^{u-1}dt} \) is the incomplete Gamma function of first kind (Page 890 of [21]).

The probability of the detection miss rate due to narrowband jamming in a Nakagami fading channel is derived by combining (4) and (14). Hence,

$$\begin{aligned}&P_{nj}^{Nak,\,SC}\nonumber \\&\quad = \frac{2Lm^{m}}{\left( {\overline{x^{2}} } \right) ^{m}\left( {\Gamma (m)} \right) ^{L}}\int \limits _0^\infty {\left( {1-\text{ erf }\left( {\sqrt{\frac{2\varepsilon { }_cx^{2}w_m }{T_c J_{avg} }}} \right) } \right) } \left( {\sum _{n=0}^\infty {\frac{\left( {-1} \right) ^{n}\left( {\frac{mx^{2}}{\overline{x^{2}} }} \right) ^{m+n}}{n!\,(m+n)}} } \right) x^{2m-1}dx.\nonumber \\ \end{aligned}$$
(15)

In (15), \({\text{ erf }}(\cdot )\) is the error function defined as \(\text{ erf }(u)=\frac{1}{\sqrt{2\pi }}\int _u^\infty {\exp \left( {-\frac{y^{2}}{2}} \right) dy}\).

3.7 Detection Miss Rate in a Nakagami Fading Channel with EGC Diversity Suffering from Narrowband Jamming

The PDF of Nakagami fading channel with EGC diversity is given as

$$\begin{aligned} f_{Nak}^{EGC} (x)=C_0 (d_0 )x^{2d_0 -1}\exp \left( {-\frac{d_0 x^{2}}{\overline{\gamma _0 } }} \right) ;\quad d_0 \ge 0.5, \end{aligned}$$
(16)

where \(C_0 (d_0 )=\frac{2d_0^{d_0 } }{\Gamma (d_0 )\overline{\gamma }_0^{d_0 } }\). Here, \(d_0 =\frac{\left( {\overline{x^{2}} } \right) ^{2}}{\left( {x^{2}-\overline{x^{2}} } \right) ^{2}},\quad \overline{\gamma _0 } =M\overline{\gamma }+M(M-1)\overline{\gamma }\frac{\Gamma ^{2}\left( {d+0.5} \right) }{d\Gamma ^{2}(d)}\).

The probability of detection miss rate due to narrowband jamming in a Nakagami fading channel with EGC diversity is derived by combining (4) and (16). Hence,

$$\begin{aligned} P_{nj}^{Nak, EGC}&= C_0 (d_0)\cdot \frac{2^{d_0 -2}}{d_0^{d_0 } }\Gamma (d_0 )\nonumber \\&\quad - \frac{C_0 (d_0 )}{\sqrt{\pi }}\sum _{l=o}^\infty {\frac{\left( {\mu /{\sqrt{2}}} \right) ^{2l+1}2^{d_0 +2l-1/2}}{\left( {2l+1} \right) !}} \frac{1}{\left( {\mu ^{2}+d_0 } \right) ^{d_0 +l+1/2}}\Gamma \left( {d_0 +l+\frac{1}{2}} \right) .\nonumber \\ \end{aligned}$$
(17)

3.8 Detection Miss Rate in a Nakagami Fading Channel with MRC Diversity Suffering from Narrowband Jamming

The PDF of Nakagami fading channel with MRC diversity is given as

$$\begin{aligned} f_{Nak}^{MRC} (x)={C}^{\prime }(d)x^{2dM-1}\exp \left( {-\frac{dx^{2}}{\overline{\gamma }}} \right) \quad \forall d\ge 0.5, \end{aligned}$$
(18)

where \({C}^{\prime }(d)=\frac{C(d)\Gamma (d)}{\Gamma (Md)}\).

The probability of the detection miss rate due to narrowband jamming in a Nakagami fading channel with MRC diversity is derived by combining (4) and (18). Hence,

$$\begin{aligned}&P_{nj}^{Nak,\,MRC}\nonumber \\&\quad ={C}^{\prime }(d)\cdot \frac{2^{Md-2}}{d^{Md}}\Gamma (Md)\nonumber \\&\qquad -\frac{{C}^{\prime }(d)}{\sqrt{\pi }}\sum _{l=o}^\infty {\frac{\left( {\mu /{\sqrt{2}}} \right) ^{2l+1}2^{Md+2l-1/2}}{\left( {2l+1} \right) !}} \frac{1}{\left( {\mu ^{2}+d} \right) ^{Md+l+1/2}}\Gamma \left( {Md+l+\frac{1}{2}} \right) .\qquad \end{aligned}$$
(19)

3.9 Detection Miss Rate in a Rician Fading Channel Suffering from Narrowband Jamming

Consider a Rician fading channel, where \(y\) and \(\sigma \) represent the LOS component and the RMS voltage of the received signal, respectively. The PDF of a Rician faded random variable, \(y\), is given by Equation (2.147) on page 47 of [6] as

$$\begin{aligned} f_{Ric} (y)=\frac{y}{\sigma ^{2}}\exp \left( {-\frac{y^{2}+s^{2}}{2\sigma ^{2}}} \right) I_0 \left( {\frac{sy}{\sigma ^{2}}} \right) \quad \forall \,y\ge 0, \end{aligned}$$
(20)

where \(s\) denotes the peak amplitude of the dominant or line of sight (LOS) signal, and \(I_{0}(\cdot )\) is the zeroth order modified Bessel function of the first kind. As \(s\) tends to zero, the dominant path decreases in amplitude, and the Rician distribution degenerates to a Rayleigh distribution.

The probability of detection miss rate in a Rician fading channel due to narrowband jamming is derived by combining (4) and (20). So,

$$\begin{aligned} P_{nj}^{Ric}&= \frac{1}{2}\sum _{k=0}^\infty {\left( {-1} \right) ^{k}\frac{y^{2k}}{2^{k}\sigma ^{2k}}\frac{e^{{-y^{2}}/{2\sigma ^{2}}}}{k\,!}} \nonumber \\&\quad -\frac{e^{{-y^{2}}/{2\sigma ^{2}}}}{2\sigma ^{2}\sqrt{\pi }}\sum _{k=o}^\infty {\sum _{l=o}^\infty {\left( {-1} \right) ^{k}\frac{2^{k}\left( {\mu /{\sqrt{2}}} \right) ^{2l+1}}{(2l+1) !}\frac{\left( {y/{2\sigma ^{2}}} \right) ^{2k}}{\left( {k !} \right) ^{2}}} } \frac{\Gamma (l+k+1.5)}{\left( {\frac{\mu ^{2}}{2}+\frac{1}{2\sigma ^{2}}} \right) ^{l+k+1.5}}.\nonumber \\ \end{aligned}$$
(21)

3.10 Detection Miss Rate in a Rician Fading Channel with SC Diversity Suffering from Narrowband Jamming

For SC diversity, the PDF of a Rician faded random variable \(y \)undergoing \(L\)-branch SC diversity is given by replacing \(L\) by \(\sum _{i=1}^L {\frac{1}{i}} \). The probability of detection miss rate in a Rician fading channel due to narrowband jamming incorporating SC diversity is derived by replacing \(L\) by \({L}^{\prime }\) where \({L}^{\prime }=\sum _{i=1}^L {\frac{1}{i}}\). So,

$$\begin{aligned} P_{nj}^{Ric, SC}&= \frac{1}{4}\sum _{k=o}^\infty {(-1)^{k}\frac{y^{2k}\left( {2\sigma ^{2}} \right) ^{-1/2-k}}{k !\Gamma \left( {{n{L}^{\prime \prime }}/2+k} \right) } } e^{-{y^{2}}/{2\sigma ^{2}}}\Gamma \left( {\frac{n{L}^{\prime \prime }}{2}-\frac{1}{2}+k} \right) \nonumber \\&\quad -\frac{1}{\sqrt{8\pi }}e^{{-y^{2}}/{2\sigma ^{2}}}\sum _{k=0}^\infty \sum _{l=o}^\infty \left( {-1} \right) ^{k}\nonumber \\&\quad \times \frac{\left( \mu \right) ^{2l+1}y^{2k}}{\left( {2\sigma ^{2}} \right) ^{{n{L}^{\prime \prime }}/2+2k}\left( {2l+1} \right) !} \frac{1}{k !\Gamma ({n{L}^{\prime \prime }}/2+k)}\frac{1}{\left( {{\mu ^{2}}/2+\frac{1}{2\sigma ^{2}}} \right) ^{l+\frac{n{L}^{\prime \prime }}{2}+k}}\nonumber \\&\quad \times \Gamma \left( {\frac{n{L}^{\prime \prime }}{2}+l+k} \right) .\nonumber \\ \end{aligned}$$
(22)

3.11 Detection Miss Rate in a Rician Fading Channel with EGC Diversity Suffering from Narrowband Jamming

For EGC diversity, the PDF of a Rician faded random variable \(y\) undergoing \(L\)-branch EGC diversity is given by replacing \(L\) by \(\left( {1+(L-1)\frac{\pi }{4}} \right) \). The probability of detection miss rate in a Rician fading channel due to narrowband jamming incorporating EGC diversity is given by

$$\begin{aligned} P_{nj}^{Ric, EGC}&= \frac{1}{4}\sum _{k=o}^\infty {(-1)^{k}\frac{y^{2k}\left( {2\sigma ^{2}} \right) ^{-1/2-k}}{k !\Gamma \left( {{n{L}^{\prime }}/2+k} \right) } } e^{-{y^{2}}/{2\sigma ^{2}}}\Gamma \left( {\frac{n{L}^{\prime }}{2}-\frac{1}{2}+k} \right) \nonumber \\&\quad -\frac{1}{\sqrt{8\pi }}e^{{-y^{2}}/{2\sigma ^{2}}}\sum _{k=0}^\infty \sum _{l=o}^\infty \left( {-1} \right) ^{k}\nonumber \\&\quad \times \frac{\left( \mu \right) ^{2l+1}y^{2k}}{\left( {2\sigma ^{2}} \right) ^{{n{L}^{\prime }}/2+2k}\left( {2l+1} \right) !} \frac{1}{k !\Gamma ({n{L}^{\prime }}/2+k)}\frac{1}{\left( {{\mu ^{2}}/2+\frac{1}{2\sigma ^{2}}} \right) ^{l+\frac{n{L}^{\prime }}{2}+k}}\nonumber \\&\quad \times \Gamma \left( {\frac{n{L}^{\prime }}{2}+l+k} \right) .\nonumber \\ \end{aligned}$$
(23)

3.12 Detection Miss Rate in a Rician Fading Channel with MRC Diversity Suffering from Narrowband Jamming

The PDF of a Rician faded random variable \(y\) undergoing \(L\)-branch MRC diversity is given by Equation (3) of [22] as

$$\begin{aligned} f_{Ric}^{Nak} (y)=\frac{1}{2\sigma ^{2}}\left( {\frac{y^{2}}{s^{2}}} \right) ^{\frac{nL-2}{4}}\exp \left( {-\frac{s^{2}+y^{2}}{2\sigma ^{2}}} \right) I_{\frac{nL}{2}-1} \left( {\frac{ys}{\sigma ^{2}}} \right) \quad \forall \,y\ge 0. \end{aligned}$$
(24)

The probability of detection miss rate in a Rician fading channel due to narrowband jamming incorporating MRC diversity is derived by combining (4) and (24). So,

$$\begin{aligned} P_{nj}^{Ric, MRC}&= \frac{1}{4}\sum _{k=o}^\infty {(-1)^{k}\frac{y^{2k}\left( {2\sigma ^{2}} \right) ^{-1/2-k}}{k !\Gamma \left( {{nL}/2+k} \right) } } e^{-{y^{2}}/{2\sigma ^{2}}}\Gamma \left( {\frac{nL}{2}-\frac{1}{2}+k} \right) \nonumber \\&\quad -\frac{1}{\sqrt{8\pi }}e^{{-y^{2}}/{2\sigma ^{2}}}\sum _{k=0}^\infty \sum _{l=o}^\infty \left( {-1} \right) ^{k} \nonumber \\&\quad \times \frac{\left( \mu \right) ^{2l+1}y^{2k}}{\left( {2\sigma ^{2}} \right) ^{{nL}/2+2k}\left( {2l+1} \right) !} \frac{1}{k !\Gamma ({nL}/2+k)}\frac{1}{\left( {{\mu ^{2}}/2+\frac{1}{2\sigma ^{2}}} \right) ^{l+\frac{nL}{2}+k}}\nonumber \\&\quad \times \Gamma \left( {\frac{nL}{2}+l+k} \right) .\nonumber \\ \end{aligned}$$
(25)
Fig. 2
figure 2

Comparison of theoretical and simulated results on CER versus SIR in a Rayleigh fading channel suffering from narrowband jamming for no diversity, SC, EGC, and MRC diversity techniques

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Fig. 3
figure 3

Theoretical CER versus Average received SNR in a Rayleigh fading channel suffering from narrowband jamming without diversity, and using SC, EGC, and MRC diversity schemes with diversity orders 2

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3.13 Diversity Techniques for Computing Code word Error Rate Suffering from Narrowband Jamming

Figures 2, 3, 4, 5, 6 and 7 show the performance of code word estimation error under narrowband jamming for the case with no diversity, MRC diversity, EGC diversity, and SC diversity. Code word error rate (CER) decreases when diversity techniques are employed when compared to the no diversity technique case. Among the diversity techniques, improvement (reduction in CER) is observed in the following order: SC, EGC, and MRC. Also, it is observed that as the order of diversity increases, the code word estimation error decreases.

Fig. 4
figure 4

Comparison of theoretical results on CER versus Chip time in a Rayleigh fading channel (\(d =\) 1) suffering from narrowband jamming for SC, EGC, and MRC diversity techniques with diversity order 3

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Fig. 5
figure 5

Comparison of theoretical and simulated results on CER versus SIR in a Rician fading channel suffering from narrowband jamming for two Rician fading parameters (\(y =\) 0, 1) for no diversity, SC, EGC, and MRC diversity techniques

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Fig. 6
figure 6

CER versus RMS voltage of the received signal in a Rician fading channel suffering from narrowband jamming for two Rician fading parameters (\(y =\) 0, 1) without diversity, and using MRC diversity technique

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Fig. 7
figure 7

Comparison of theoretical and simulated results on CER versus Chip time in a Rician fading channel suffering from narrowband jamming for two Rician fading parameters (\(y =\) 0, 1) for no diversity, SC, EGC, and MRC diversity techniques

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4 Numerical Results

In the simulation results discussed in this section, the number of bits simulated is mostly 10,000, but in certain cases, like the \(d\) = 5 curve of Fig. 8, we have used \(10^{7}\) bits. The simulation parameters/system specifications used in each Figure is provided in the legend/gtext of the corresponding Figures itself.

Fig. 8
figure 8

Comparison of theoretical and simulated results on CER versus SIR in a Nakagami fading channel suffering from narrowband jamming for various Nakagami fading parameters (\(d\))

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Figrue 2 shows the relationship between the detection miss rate or CER and Energy per chip to Average Jammer power spectral density Ratio (Signal to Interference plus Noise Ratio, SINR) in a Rayleigh fading channel under narrowband jamming scenario. As SINR increases, the probability that the signal reaches the destination without error increases. The probability that the codeword is properly estimated increases, thus reducing the CER. It is also concluded from analytical results that with the increase in number of receive antennas, combined with SC, EGC, and MRC techniques, the probability of the signal being received properly will increase, and hence the codeword is estimated properly. Thus, the CER decreases with an increase in the number of receiver antennas. Also, the CER decreases with increase in diversity order.

The analytical and simulated graphs showing the performance of diversity techniques in a Rayleigh fading channel have a higher slope as compared to those for the no diversity technique. This can be inferred from the fact that for an SIR of 6 dB, the SC \(\left( {N_r =2} \right) \) case provides a CER of \(2\times 10^{-3}\) as compared to the no diversity case which provides a CER as high as \(2.5\times 10^{-1}\). Further, the CER obtained for the MRC diversity technique (best among the other techniques discussed in this paper) is around\(10^{-4}\). Hence, the SC diversity technique provides 2 orders \(( {O( {10^{-1}} )\,\text{ to }\,O( {10^{-3}} )} )\) of reduction in CER as compared to the no diversity case, and the MRC diversity technique provides 3 orders \(( {O( {10^{-1}} )\,\text{ to }\,O( {10^{-4}} )} )\) reduction in CER as compared to the no diversity case.

The above relationship between CER and SINR holds true for the Nakagami and Rician fading channel scenarios under narrowband jamming as shown in Figs. 8 and 5, respectively. In the case of Nakagami fading, it is concluded that with increase in the number of degrees of freedom, the probability of the signal being received properly will increase, and hence the codeword is estimated in a better way. Thus, CER decreases with an increase in the number of degrees of freedom. When \(d\) = 0.5, Nakagami fading becomes equal to Rayleigh fading.

In the case of Rician fading (Fig. 5), with an increase in the LOS component, the probability of the signal being received properly will increase, and hence the codeword is estimated in a better way. Thus, CER decreases with an increase in the LOS component as shown by the results. It is also concluded from analytical results that with an increase in the number of receiver antennas, whose signals are combined with the help of MRC diversity technique, the probability that the signal being received properly will increase, and hence the codeword is estimated properly. Thus, CER decreases with an increase in the number of receive antennas. CER is also plotted for different diversity cases, like MRC, EGC, and SC for \(y = 1\). The simulation parameters used are \(T_c =2, w_m =1, \sigma =2.1\). The simulated results agree very closely with the analytical results. It is found that MRC performs better than EGC, which in turn performs much better than SC. Simulation results with the same specifications as those used for the analytical results are plotted for the three diversity cases, and they are in close agreement with the analytical results. When \(y\) = 0, Rician fading becomes Rayleigh fading.

The analytical and simulated graphs showing the performance of diversity techniques in Rayleigh \(( {y=0} )\) and Rician \(( {y=1} )\) fading channels have a higher slope as compared to those for a no diversity technique. For an SIR of 6 dB, the EGC \(( {N_r =2} )\) case provides a CER of almost \(10^{-3}\) for Rician fading channels, and the MRC \(( {N_r =2} )\) case provides CER of around \(7\times 10^{-4}\) for Rayleigh fading channels, respectively. For the no diversity technique, the CER for a Rayleigh fading channel is \(3\times 10^{-1}\) and that for a Rician fading channel is \(2\times 10^{-1}\). Thus, the EGC diversity technique provides 2 orders \(( {O( {10^{-1}} )\,\text{ to }\,O( {10^{-3}} )} )\) of reduction in CER as compared to the no diversity case, and the MRC diversity technique provides 3 orders \(( {O( {10^{-1}} )\,\text{ to }\,O( {10^{-4}} )} )\) reduction in CER as compared to the no diversity case.

Fig. 9
figure 9

Comparison of theoretical and simulated results on CER versus Average received SNR in a Nakagami fading channel suffering from narrowband jamming for various Nakagami fading parameters (\(d\))

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The graphs of CER versus SNR are shown in Figs. 3, 9 and 6 for the case of narrowband jamming in Rayleigh, Nakagami and Rician fading channels, respectively. As the average received SNR increases, the probability that the signal will be detected properly increases. Diversity techniques aid in detecting the signal properly, which means lower CER is obtained for diversity techniques as compared to the no diversity technique. As the diversity order increases, the CER decreases. This means the chances that the codeword will be estimated properly increases. Thus, the CER decreases with an increase in the average received SNR. In Fig. 6, the simulation parameters used are \(\text{ T }_\mathrm{{c}} =2, \text{ w }_\mathrm{{m}} =1, \frac{\text{ E }_\mathrm{{c}}}{\text{ J }_\mathrm{{avg}} }=1\). The simulated results agree very closely with the analytical results.

Fig. 10
figure 10

Comparsion of theoretical and simulated results on CER versus Chip time in a Nakagami channel suffering from narrowband jamming for various Nakagami fading parameters (\(d\))

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The relationship between CER and chip time under narrowband jamming is given by the results in Fig. 4 (Rayleigh fading with \(d = 1\)), Figs. 10 and 7. As the chip time increases, the number of bits that could be used for spreading the codeword decreases. This means that the probability that the codeword is properly estimated decreases with increase in chip time. So, an increase in chip time increases the CER. This is shown by the analytical results. With the employment of diversity techniques, the CERs are plotted for MRC, EGC, and SC schemes. The graphs show that the CER is higher for SC scheme as compared to EGC scheme, which in turn is higher as compared to MRC scheme. In Fig. 7, the simulation parameters used are \(\sigma =2.1, w_c =1, \frac{E_c }{J_{avg} }=1\). Simulation results are also found to be in close agreement with the analytical results with the same specifications chosen as that in the analytical results.

From Fig. 7, it can be observed that the maximum CER obtained for a Rayleigh fading channel \(( {y=0} )\)with no diversity is around \(8\times 10^{-1}\)and for a Rician fading channel \(( {y=1} )\) with no diversity is around \(5\times 10^{-1}.\) Compared to this, the CER values for a Rayleigh fading channel with EGC technique is \(9\times 10^{-2}\)and those for a Rician fading channel with MRC technique is \(5\times 10^{-2}.\) Thus, for both Rayleigh and Rician fading channels, there is one order of magnitude \(( {O( {10^{-1}} )\,\text{ to }\,O( {10^{-2}} )} )\) reduction in CER.

From Fig. 8, the CER achieved at SIR = 6 dB is around \(3\times 10^{-1}\) for a Nakagami parameter, \(d=0.5;\)around \(2\times 10^{-3}\) for \(d=1\); and around \(3\times 10^{-7}\) for \(d=5.\) Thus, there are two orders of improvement \(( {O( {10^{-1}} )\,\text{ to }\,O( {10^{-3}} )} )\)in CER between \(d=0.5\) and \(d=1,\)and six orders of improvement\(( {O( {10^{-1}} )\,\text{ to }\,O( {10^{-7}} )} )\) in CER between \(d=0.5\) and \(d=5\).

From Fig. 9, the CER achieved at SNR = 5 dB is around \(4\times 10^{-1}\) for \(d=0.5,\, 8\times 10^{-2}\) for \(d=1\), and \(1.5\times 10^{-2}\) for \(d=2\). Thus, there is one order of improvement \(( {O( {10^{-1}} )\,\text{ to }\,O( {10^{-2}} )} )\) in CER for both \(d=1\) and \(d=5\) when compared to \(d=0.5.\)

From Fig. 10, it can be observed that the maximum CER of \(8\times 10^{-1}\) for Nakagami fading occurs at \(d=0.5;\) maximum CER of \(2\times 10^{-1}\) occurs at \(d=1\); maximum CER of \(4\times 10^{-2}\) occurs for \(d=2\); and maximum CER of \(9\times 10^{-4}\) occurs at \(d=5\).

5 Conclusions

Jamming destroys the orthogonality of the ranging code set which affects the estimation of the codes. This in turn affects performance measures like detection miss rate and detection success rate. In this paper, performance measure like code word error estimation rate is derived and plotted for diversity schemes like SC, EGC, and MRC, for a fixed number of receive antennas under the presence of Rayleigh, Rician, and Nakagami fading channels in the presence of narrowband jamming. Special cases when Nakagami and Rician channels degenerate to Rayleigh channel conditions are also shown. A small increase in low values of chip time proves to be a sensitive measure in the evaluation of CER, as a small increase in chip time corresponds to a considerable increase in CER.

Diversity techniques like MRC, SC, and EGC, considerably improve the performance of Ranging process used in various fading channel models like Rayleigh, Nakagami and Rician suffering from narrowband jamming scenario. The analysis made in this paper helps us to understand by how many orders of magnitude do the diversity techniques improve over the no diversity technique in terms of computation of the code word error rate.