Abstract: The effects of Gaussian filtering on the processing gain of Acousto-Optic Correlators (AOC) in direct-sequence spread-spectrum receivers are considered. Analytical expressions are derived for ideal AOC’s. The results are given in terms of the ratio β of the code chip rate to the filter bandwidth. For real AOC’s, results are obtained from numerical simulation and are compared to previous results obtained for other types of filters.
INTRODUCTION
Acousto-optic devices are still considered attractive for processing signal and images as they offer real-time operation and high dynamic range (Berg and Lee, 1991; Balakshy and Kostyuk, 2009; Korpel, 2009). Acousto-Optic Correlators (AOC) used with Spread-Spectrum (SS) signals figure among the valuable applications. Different architectures of AOC are described in literature (Bazzi et al., 1994; Kim et al., 2004; Gurevich et al., 2004). The effects of integrating the AOC in a spread-spectrum receiver are reported by Bazzi et al. (2006) where the receiver equivalent model in the baseband is reduced to a low-pass filter acting on the received signal. The results show a processing gain reduction which is filter dependent.
The effects of Gaussian filtering in the receiver are presented in this study. Gaussian filtering is particularly effective when used in conjunction with Minimum Shift Keying (MSK) modulation or other modulations which are suited for power-efficient nonlinear amplifiers (Rappaport, 2002). The receiver model is first briefly reviewed. Analytical results are then derived for ideal AOC. For real AOC’s, results are obtained from numerical simulation and are compared to previous results obtained for two other types of filters: ideal low-pass and RC filters.
THE SS RECEIVER MODEL
Figure 1 shows a view of the equivalent SS receiver diagram using an AOC. The received SS signal is down-converted into baseband before performing correlation with a reference code. We distinguish the two filters of frequency responses H1(f) and H2(f) where:
Filter H1(f) is proper to the AOC. It represents the acousto-optic interaction frequency response and affects both the received and the reference signals. The equivalent normalized baseband representation of H1(f) is (Bazzi et al., 1996):
| (1) |
| Fig. 1: | Equivalent diagram for the SS receiver with an embedded AOC |
| Fig. 2: | AO normalized correlator response: (a) Frequency response and (b) impulse response |
where,
f is the acoustic frequency and Δf is the cutoff frequency at -4dB corresponding to half the first-null bandwidth. The normalized response of the AOC is presented in Fig. 2a and b.
Filter H2(f) is the receiver filter which characterizes the receiver channel. This filter affects the received signal. The baseband Gaussian characteristics, considered in this study, is given by:
| (2) |
where, the parameter a is related to the 3 dB bandwidth B of the Gaussian shaping filter by Rappaport (2002) :
| (3) |
| Fig. 3: | Gaussian filter response: (a) Frequency response and (b) impulse response |
where, β is the inverse of the filter bandwidth.
Figure 3a and b show the frequency response H2(f) and the time response h2(t) of the Gaussian filter.
ANALYSIS OF THE RECEIVER PERFORMANCE
The analysis is done under the conventional pseudo-random sequence correlation where the received signal is correlated in the receiver with a locally generated reference code c(t) as shown in Fig. 1. The received signal s(t) consists of a desired signal Ac(t) corrupted by an additive zero-mean white Gaussian noise of Power Spectral Density (PSD) equal to η:
s(t) = Ac(t)+n(t)(4) |
(4) |
where, A is the amplitude of the received signal, n(t) denotes a sample function of the noise process.
At the receiver input the signal to noise ratio is defined as (Proakis, 2000):
| Table 1: | Code characteristics |
| (5) |
where,
The processing gain is defined as the enhancement obtained in the Signal to Noise Ratio (SNR) as the signal goes through the receiver. In an ideal receiver (infinite bandwidth), the processing gain is equal to the spreading factor N i.e., the number of chips in the code period (Dixon, 1976). When N is large, the autocorrelation of the code can be approximated by Dixon (1976):
| (6) |
where, the chip rate is normalized to one and tri(t) = 1-abs(t) for abs(t)<1 and 0 otherwise. Table 1 shows the autocorrelation characteristics of the code chip ε(t).
At the receiver output, referring to Fig. 1, the signal obtained is:
| (7) |
where, * is the convolution operator, the reference code is time-reversed in order to obtain correlation. The subscript (ff) refers to a two-stage filtering of signals: first in the receiver filter and second in the AO correlator. The subscript (f) refers to AO filtering alone. The superscript N refers to a single period of the code given by:
| (8) |
where, rect(x) = 1 for abs(x)<0.5 and 0 otherwise:
Eq. 7 can be split as:
| (9) |
where, s0(t) and n0(t) represent, respectively signal and noise at the correlator output.
The Signal to Noise Ratio (SNR)0 at the correlator output is defined
as the ratio of the signal correlation peak power
| (10) |
Using Eq. 5 and 10, the SNRE can be written as:
| (11) |
Signal and noise analysis: The detailed analysis of s0(t) and n0(t) is found by Bazzi et al. (2006). The expressions for s0(t) and Cn0n0(t) are given by:
| (12) |
and
| (13) |
where, h1(t) and h2(t) are the impulse responses of the AO correlator and the receiver filter, respectively. Cxx(t) denotes the autocorrelation of x(t).
The output noise variance is given by the autocorrelation peak value Cn0n0(0) in Eq. 13.
Analysis of the correlator performance: The performance is studied in terms of two parameters:
| • | The ratio β of the code chip rate (normalized to one) to the cutoff frequency Δf of the AOC: |
| (14) |
| • | The ratio β of the code chip rate to the receiver filter bandwidth |
α and β represent the filtering acuteness of the AOC and the receiver filter, respectively. Table 2 shows the impulse response characteristics of the AO correlator and the Gaussian filter used in the receiver.
ANALYTICAL RESULTS
Analytical results of the SNRE in the system are found in the case of ideal AOC (H1(f)=1).
Eq. 12 and 13 reduce, in this case, to:
| (15) |
| Table 2: | AO correlator and Gaussian receiver filter characteristics |
and
| (16) |
The autocorrelation function Ch2h2(t) for the Gaussian filter is found in Table 2. The correlator output signal is given from Eq. 15 as:
| (17) |
The maximal signal value s0(0) is given by:
| (18) |
where, Erfc(x) is the Complementary Error function defined as:
| (19) |
Similarly, the noise variance at the correlator output is computed as:
| (20) |
The SNRE enhancement ratio is obtained, in terms of β, by replacing Eq. 18 and 20 in Eq. 11.
The SNRE loss compared to the ideal system is presented in Fig. 4 along with the signal and noise power reductions at the correlator output. In Fig. 5, the Gaussian filter effect is compared to the effects obtained previously for RC and ideal low-pass receiver filters (Bazzi et al.,2006).
| Fig. 4: | Effects of the receiver Gaussian filter (ideal AOC) |
| Fig. 5: | SNRE losses for different receiver filters (ideal AOC) |
The figure shows that for β<2.5 the Gaussian filter and the RC filter have almost the same performance. For β = 5, an approximate SNRE loss of 1.8 dB is obtained with the Gaussian filter as compared to respective losses of 1.2 and 4.2 dB for RC and ideal low-pass filters.
NUMERICAL RESULTS
Results in the general case of a real AOC are obtained from numerical simulation of the system (receiver + correlator) as defined in Fig. 1.
The value N has been taken equal to 1023 with 100 samples per chip.
| Fig. 6: | SNRE loss in real AOC with Gaussian receiver filter |
Results are presented in Fig. 6 in terms of the Gaussian filter parameter β for different values of the AOC acuteness factor α. For α = 0 (ideal AOC), the simulation results agree well with the analytical results.
We also see that as α increases the curves become broader and the AOC filtering effects become dominant.
CONCLUSION
Analysis of the effects of signal filtering on Acousto-Optic correlator performance is extended to the case of Gaussian filtering commonly used with MSK modulation. Analytical results were derived for ideal AOC in terms of the ratio β of the chip rate to the filter bandwidth. Numerical results were found in the general case. The detailed performance obtained will help optimize an equalizer for this kind of filter in order to reduce the processing gain loss in the correlator.
ACKNOWLEDGMENT
This work was supported by the National Council for Scientific Research - Lebanon.