1 Introduction

The hydroxyl free radical (OH) is one of the most important oxidants in the atmosphere. It is the primary agent responsible for removal of atmospheric trace gases and volatile organic compounds (VOCs). It also initiates the reactions leading to the production of O3 in the atmosphere and impacts the formation of aerosol. The source/sink of OH radicals and the related chemical process play a crucial role in key environmental issues, such as local and global climate change, atmospheric ozone chemistry and acid atmospheric deposition. Because of its very high reactivity, very short life time (≤1 s) and very low concentration in the atmosphere (∼106 OH radicals/cm3), interference-free high sensitivity in-situ real time quantitative assessment of OH radicals represents a great challenge [1].

In 1980, Litfin et al. [2] reported the use of magnetic rotation spectroscopy (Faraday effect) to efficiently suppress excess laser noise (of a color center laser) and thus improve the detection sensitivity by a factor of 30 compared with simple Zeeman modulation spectroscopy. As an application example of Faraday rotation spectroscopy (FRS) technique, they presented the first observation of the fundamental vibrational band of the OH radical at 2.69 μm (∼3708 cm−1). In 1981, Pfeiffer et al. [3] made further improvements in a similar FRS approach and reported a detectivity of 1011 OH radicals/cm3. Since then, FRS has been widely used in detection of molecular ions [4], O2 [5, 6], NO2 [7–9] and NO [10–17]. Significant progress has been achieved over the past decade in technologies of diode lasers and highly sensitive room temperature detectors, which has led to the development of FRS based spectroscopic instruments that are more compact, highly selective, and sensitive (e.g. NO detection with path-bandwidth normalized sensitivity of 340 ppt-m/Hz−1/2 and long term stabilities up to 4000 s [14]).

Recently, we demonstrated an FRS instrument using a distributed feedback (DFB) diode laser operating at 2.8 μm for sensitive detection of OH radicals [18]. The instrument prototype showed a shot-noise dominated performance. In a point-by-point wavelength tuning scheme, a 1σ detection limit of 8.2×108 OH radicals/cm3 was achieved with an optical path of 25 cm and a lock-in time constant of 100 ms. Using a DFB quantum cascade laser (QCL), Kluczynski et al. [16] implemented wavelength sweeping technique for fast detection of NO at 5.33 μm. The fast wavelength sweeping method can eliminate drifts in baseline and laser frequency.

In this paper, fast wavelength scanning approach is implemented for the OH radical detection using a DFB diode laser operating at 2.8 μm. The Allan variance analysis showed improved immunity to a long-term baseline drift. Performances of the FRS measurements using two different wavelength scanning techniques were compared. Experimental investigation of the effects of strong H2O vapor absorption on FRS detection of OH as well as the effects of laser power on the optimum analyzer offset angle and hence the FRS detection sensitivity will be presented.

2 FRS detection

Faraday rotation spectroscopy relies on the Faraday effect observed in paramagnetic species. Under a longitudinal magnetic field, the magnetic circular birefringence occurs in the vicinity of Zeeman-split absorption lines, and the polarization axis of a linearly polarized beam is rotated. For detection of the FRS rotation angle, two polarizers with high extinction ratio are usually employed. The first linear polarizer is placed between the laser beam and the gas cell to establish a polarization axis of the incident laser radiation, and the second polarizer, placed between the gas cell and the photodetector, acts as a polarization analyzer. Polarization axis of the 2nd polarizer is set at 90∘+φ angle with respect to the polarization of the incident light (where φ is a small analyzer offset angle from the crossed position). This nearly crossed polarizers scheme provides a significant laser excess noise reduction. FRS signal is detected as modulation of the light intensity transmitted through the analyzer. The light power, P(φ), emerging from the analyzer is given by [2, 10]

$$ P ( \varphi ) = \frac{P_{0}}{2}\exp ( - 2LI_{A} ) \bigl[ \cosh ( I_{\Delta} L ) - \cos ( LR_{\Delta} + 2\varphi ) \bigr] $$
(1)

where P 0 is the light power incident on the gas sample, L is the absorption path length within the magnetic field. I A =k 0[(κ ++κ −)/2], I Δ=k 0(κ +−κ −), R Δ=k 0 (n +−n −), where κ + and κ − are the absorption coefficients, n + and n − are the indices of refraction of the gas medium for the right-handed (RHCP, +) and left-handed (LHCP, −) circularly polarized waves propagating through the sample, respectively. k 0=ω/c=2πν/c is the wave vector with ν being the frequency of the laser radiation, and c is the speed of light.

In the case of weak absorption and small φ (two polarizers nearly crossed), the absorption terms, I A and I Δ can be neglected, and Eq. (1) can be simplified to [2, 10]

$$ P(\varphi) = \frac{P_{0}}{2} \bigl( 1 - \cos 2\varphi + R_{\Delta} (\nu)L\sin 2\varphi \bigr) $$
(2)

R Δ is responsible for creation of the FRS signal, and is used to acquire the spectral information of the paramagnetic sample. It can be calculated as a sum of the spectral signals from all the allowed transition components (Fig. 1). R Δ can be expressed in terms of integrated line strength, Voigt line shape function and modulation amplitude [18, 19]:

$$ R_{\Delta} ( \nu ) = A\sum_{M_{J}',M_{J}''} ( - 1)^{M_{J}' - M_{J}''}\operatorname {Re}\bigl[Z(z)\bigr] $$
(3)

where M′ and M″ are the magnetic quantum numbers for the upper and lower states. For the configuration in which the magnetic field is parallel to the optical propagation axis, M′−M″=±1, and

$$ Z( z ) = \frac{1}{\sqrt{\pi}} \int_{ - \infty}^{ + \infty} \frac{\exp ( - t^{2} )}{t - z}\,dt $$
(4)

is the plasma dispersion function with z=x+iy, and

(5)
(6)
(7)

where N is the molecule concentration (molecule/cm3), S is the line intensity (cm−1/(molecule cm−2)); γ c and γ D are the collisional width and Doppler width (HWHM) in cm−1, respectively. In the presence of magnetic field the line center frequency ν M′M″ is given by

(8)

where ν 0 is the line center frequency (in cm−1), g J is the rotational gyromagnetic ratio, μ B is the Bohr magneton, B is the magnetic-field strength (T), and h is the Planck constant.

Fig. 1
figure 1

(a) Zeeman splitting energy pattern for the Q(3/2) transition of OH (2 Π 3/2 state, υ=1←0). ΔM J =+1: RHCP; ΔM J =−1: LHCP; (b) contribution of the refraction indices of each \(M_{J}''\) component (n +: RHCP; n −: LHCP). The absolute value of the Zeeman splitting is the same for all \(M_{J}''\) components of the Q(3/2) transition. For clarity an additional frequency offset was introduced in the figure to show all contributing \(M_{J}''\) transitions; (c) final demodulated 1f FRS signal (black line) that is proportional to n +−n −

Full size image

In FRS experiments, the Zeeman splitting of the molecular absorption line (and thus the magnetic circular birefringence) is usually modulated at frequency f m by applying an AC magnetic field, which provides an “internal modulation” of the sample. Phase-sensitive detection is then carried out at harmonics of f m to perform demodulation of the FRS signal within a narrow lock-in pass band. The external noise like interference fringes and baseline drift due to variations of light intensity can be efficiently suppressed [2]. By implementation of such a background-free derivative spectroscopy technique, the detectivity of the FRS technique is further improved. The FRS signal (H 1) from 1f phase sensitive detection can be expressed as follows:

$$ H_{1} = \frac{2}{\pi} \int_{0}^{\pi} \biggl[ \frac{P_{0}}{2}R_{\Delta} L\sin( 2\varphi) \biggr]\cos( \theta) \,d\theta $$
(9)

The frequency-dependent portion of Eq. (3) can be expressed as a Fourier series [19] and the Fourier component of the signal at f m can be calculated by numerical integration.

3 Selection of optimum transition lines

The absorption intensity, magneto-optic interaction pathlength, polarization analyzer offset angle and the number of the allowed transition components are of importance in increasing FRS signal amplitude [18]. It is essential to work with the transitions providing the largest R Δ. The Q(1.5e) and Q(1.5f) double lines of OH of the 2 Π 3/2 state (υ=1←0) at 3568.52382 cm−1 and 3568.41693 cm−1 have the highest absorption line intensity (9.556×10−20 and 9.553×10−20 cm−1/(molecule cm−2) at 296 K) in the infrared region with the largest g J value. For the Q branch, the g J values of the upper and lower states, as well as the magnetic modulation sensitivity for each transition are the same, which makes the Q(3/2) transition at ∼3568.5 cm−1 the most favorable for sensitive OH detection.

Figure 1(a) shows the Zeeman energy level diagram appropriate for the Q(3/2) transition of OH. There are three ΔM J =+1 (RHCP) and three ΔM J =−1 (LHCP) transitions. The corresponding contribution of the dispersion spectra of each \(M_{J}''\) are shown in Fig. 1(b). For small magnetic field modulation depth the FRS signal can be well approximated by a difference between RHCP and LHCP dispersion profiles (Fig. 1(c)).

4 Experimental set-up

The experimental set-up used in the present work is schematically shown in Fig. 2. A 2.8 μm continuous-wave DFB diode laser (nanoplus GmbH) operating at room temperature was employed to probe the Faraday rotation effects via measurements of the Q(1.5) double lines of OH near 3568.5 cm−1. The laser temperature and current were controlled by a LDC 501 laser diode controller (Stanford Research System). For point-by-point tuning method, a GPIB card was used to control injection current of the laser diode [18]. For fast wavelength sweeping method, frequency tuning of the diode laser was performed by scanning the injection current with a 15 Hz ramp provided by a function generator (Agilent 33220A).

Fig. 2
figure 2

Schematic diagram of a fast wavelength scan Faraday rotation spectroscopic system. PM: parabolic mirror; F: lens; PC: personal computer

Full size image

Output of the laser was collimated by a 90∘ off-axis parabolic mirror (PM1) with an effective focal length (EFL) of 25 mm. The laser beam was then transformed into a quasi-parallel beam with a diameter of ∼6 mm by using a combination of two anti-reflection (AR) coated CaF2 lenses: F1 (f1=25 mm), and F2 (f2=50 mm). The collimated laser beam was then directed to the FRS setup: a 38-cm long gas cell (made of a quartz tube terminated with two 1 degree wedged CaF2 windows), located inside a 25 cm long solenoid, was placed between two polarizers (with an extinction ratio of ξ<5×10−6). The magnetic field B was provided by a solenoid made of ∼437 m (2323 turns) of 16 gauge, enamel-coated copper wire wrapped around a 25-cm long PVC pipe. The pipe inner diameter was 50 mm and the outer diameter 60 mm. To operate the solenoid in AC mode, a series resonant RLC circuit (with a resonant frequency f m =1.302 kHz) was constructed. A sine-wave voltage at f m produced by an internal sine generator in a phase sensitive lock-in amplifier (Signal Recovery 7270 DSP Lock-in) was amplified with an audio amplifier (RMX 1850HD) to drive the coil for generation of a sinusoidal magnetic field.

The laser beam emerging from the analyzer was focused by a second parabolic mirror (PM2, EFL=50 mm) onto a thermoelectrically (TEC) cooled (HgCdZn)Te photovoltaic detector (Vigo PVI-4TE-3.4) with a detectivity D ∗=1.3×1012 cm Hz1/2/W and a responsivity of R=1.8×106 V/W). The DC signal (laser intensity) from the detector was directly digitalized and recorded with a laptop via a National Instruments data acquisition card (DAQ card, Model 6062E), while the AC component (FRS signal) was fed to the lock-in amplifier and demodulated at f m . The demodulated FRS signal was then acquired by the DAQ card. In our experiment, the OH radicals were produced in a discharge cell by a 2.45 GHz micro-wave (MW) discharge (Sairem Wavemat) in a water vapor flow at a pressure of a few mbar [18]. The generated OH free radical concentration was estimated using cross-calibration of OH spectrum with respect to a close-by H2O line with known concentration given that the line intensities are known for both OH and H2O lines [18].

5 Results and discussion

5.1 Optimization of magnetic field

A maximum FRS signal occurs when the Zeeman splitting becomes comparable with the spectral line width of the transition at B=0 [5]. A series of experiments were performed to determine the optimum magnetic field strength B opt. The magnetic field inside the coil was measured with a Gauss meter. A magnetic field generation of 94.4 Grms/A (Gauss root-mean-square per Ampere) was deduced from the linear relation between the magnetic field intensity and the AC solenoid current shown in Fig. 3(a). Figure 3(b) shows spatial distribution of the magnetic field strength inside the coil (coil length L=25 cm). The magnetic field is relatively uniform (within ∼6 %) inside the coil, while it decreases rapidly within a 5-cm distance from both ends of the coil (more than 8 Grms/cm). The magnetic field non-uniformity must be taken into account for accurate spectroscopic modeling of OH FRS spectra [18]. A continues generation of OH at sample pressure of 0.6 mbar (limited by the MW generation process) was used to optimize the sample magnetic field. The largest FRS signal was found (Fig. 3(c)) at an AC solenoid current of 1.85 A, corresponding to B opt=177 Grms. All experiments were then performed under this optimum magnetic field.

Fig. 3
figure 3

(a) Magnetic field strength B vs. coil current I; (b) axial distribution of the magnetic field strength B inside a 25-cm long coil; (c) optimum magnetic field strength B opt was found at an AC solenoid current of 1.85 A and corresponds to B opt=177 Grms measured inside the coil

Full size image

5.2 The effects of strong H2O absorption

For measurements of OH at atmospheric pressure one important issue is spectral interference from strong absorption of diamagnetic species, in particular H2O. Indeed, the FRS relies on the magneto-optic effect observed only for paramagnetic species; therefore the major atmospheric constituents such as H2O and CO2 which are diamagnetic do not produce any significant Faraday Effect. However, strong absorption of H2O vapor outside the FRS cell might have an effect on amplitude of the FRS signal of OH. Figure 4(a) shows HITRAN database [20] simulation spectra of direct absorption of H2O vapor, CO2 and OH near 2.8 μm. The simulations were performed with 2 % H2O, 400 ppmv CO2 and 1 ppmv OH at 296 K. The used absorption pathlengths of 120 cm and 40 cm corresponded to the optical pathlengths outside and inside the FRS cell respectively. Effects of ambient H2O vapor and CO2 absorptions outside the FRS cell (between the laser source and the detector) were taken into account by the simulation at 760 Torr using their atmospheric concentrations, while OH and H2O vapor absorptions inside the FRS cell were calculated for a gas pressure of 400 Torr. For reference only, the OH absorption is shown for absorption pathlength of 100 m and concentration of 1 ppm. As shown in Fig. 4(b), the OH Q (1.5) double lines might be affected by the strongest H2O absorption line at 3568.289 cm−1 (ν 3 band, 624←725 transition with a line intensity S=2.65×10−20 cm−1/(molecule cm−2)) at atmospheric pressure (red curve). It should be noted that the nature of this interference is solely related to the power drop due to the absorption which also reduces FRS signal. The spectral simulation at reduced pressures suggests that after power normalization an H2O-interference-free FRS detection of OH could be realized at pressures below 400 Torr (green line). In the current system due to limitations imposed by the OH generation method (low pressure is required by the microwave discharge system), the working pressure for the FRS experiment is set at 0.6 mbar (0.45 Torr). Under these conditions, the H2O and OH absorption lines are well separated and the effects of H2O absorption on the FRS signal from OH radicals are virtually eliminated. At higher pressures (up to 400 Torr) the system is expected to show similar or even higher sensitivity performance (due to increase in number density of the target species within the Doppler limited regime).

Fig. 4
figure 4

(a) The HITRAN simulated (at 296 K) direct absorption spectra of 2 % H2O (red) and 400 ppm CO2 (black) at P=760 Torr in a 120 cm absorption pathlength outside the FRS cell. Inside the FRS cell, the simulations were performed with 2 % H2O (in a 40 cm long cell) at P=400 Torr. For comparison OH was simulated with 1 ppm concentration 100 m optical path. (b) A magnified spectrum of the Q(3/2) double lines used for FRS detection shows H2O vapor interference at different pressures

Full size image

Figure 5(a) shows a direct absorption spectrum (DC detector signal) of an ambient H2O vapor absorption outside the FRS cell (broad feature) and partially from the H2O vapor absorption in the cell (narrow line at low pressure). An experimental FRS spectrum of OH radical is shown in Fig. 5(b). The FRS spectrum was recorded using point-by-point wavelength tuning. The lock-in time constant was set at τ=100 ms with f m =1.302 kHz. By providing 0.02 mA steps to the laser current, 300 spectral points were collected within one scan through the Q(1.5e) and Q(1.5f) lines. Each spectral point was an average of 4000 sample data acquired with 1 s acquisition time. The wait time before each spectral point was 250 ms yielding a full spectrum acquisition time of 375 s. The optimum polarization analyzer offset angle was set to φ=6∘. As visible in Fig. 5(b), the ambient water absorption background causes unequal signal intensities for the Q(1.5e) and the Q(1.5f) lines. This is primarily due to variations in the laser intensity (P 0) resulting from strong H2O absorption. In order to correct for this effect, the DC output (P 0) of the detector was recorded (Fig. 5(a)) together with the FRS signal (Fig. 5(b)) and was used to normalize the FRS spectrum in the post-processing. Figure 5(c) shows the processed OH spectrum after normalization and the Q(1.5) lines show identical FRS signal strengths.

Fig. 5
figure 5

(a) Laser power as a function of optical frequency in the absence of magnetic field; FRS spectra at low pressure (∼1 mbar) in the cell: (b) raw data acquired for OH; (c) OH spectrum after normalization to the laser power (a)

Full size image

Based on the results for point-by-point frequency tuning mode shown in Fig. 5(c), a 1σ noise level of 826 nV (or bandwidth normalized 639.5 nV Hz−1/2) was measured in the baseline of the FRS spectrum, which resulted in a detection limit of 8.2×108 radicals/cm3 of OH. This corresponds to a minimum detectable Faraday rotation angle of Θ min=1.2×10−7 rad/Hz1/2.

5.3 Fast wavelength scanning

In a fast scanning mode, frequency tuning across OH absorption line was achieved by sweeping injection current of the laser with a scan repetition rate of 15 Hz. Fast wavelength scan makes the FRS system less susceptible to slow fluctuations caused by the drifts in the system such as variations of the magnetic field intensity, shift in the resonant frequency of the magnetic coil circuit etc. Figure 6 shows a fast scanned FRS spectrum measured with the developed instrument. The scan rate is currently limited by the relatively low magnetic field modulation frequency f m of ∼1.45 kHz. A 15 Hz ramp (Fig. 6(a)), which is approximately f m /100, was used to sweep laser wavelength via current tuning from 128 to 131 mA. This corresponded to an optical frequency tuning range of ∼0.13 cm−1. The lock-in time constant was set to 1 ms and no spectrum distortion was observed for scan rates lower than 15 Hz. In order to improve the SNR, 5 consecutive laser scans were averaged. Figure 6(b) shows the DC output from the detector, which demonstrates the variation in laser intensity as a function of diode laser current. Figure 6(c) and (d) show lock-in demodulated in-phase and quadrature component of the FRS signal. Since a total harmonic signal demodulated by the lock-in amplifier is a vector sum of the in-phase and quadrature components (X–Y model) [21], the quadrature Y signal can be used to monitor the lock-in demodulation performance. Due to fluctuation in the magnetic field related to thermal effects in the coil, the phase of the magnetic field modulation may vary with respect to the lock-in reference phase. This effect has also been observed as a drift in frequency locking feedback loop used in FRS [14]. To suppress such drifts in the present work (especially changes in the RLC resonant frequency of the coil circuit) the lock-in reference phase angle was adjusted based on the recorded Y-signal.

Fig. 6
figure 6

(a) 15 Hz ramp for sweeping laser injection current; (b) DC output of the detector; (c) FRS signal (lock-in in-phase 1f component); (d) lock-in Y-component allows for monitoring of the FRS signal phase drifts. A single FRS spectrum is a result of five laser scans averaged

Full size image

The measurement precision and stability of the FRS system operating in fast scanning mode were evaluated using Allan variance analysis. Time series measurement (over a 27 min period) of the FRS signal for OH radicals produced under stable MW discharge conditions (to ensure a constant OH concentration) is shown in the upper panel of Fig. 7. The system exhibits a standard deviation of FRS signal readings of 6.72 μV with a 0.33 s sampling rate. According to the Allan variance plot (Fig. 7 lower panel), the system showed random noise dominated operation for averaging times up to ∼50 s. At this optimum averaging time the corresponding ultimate minimum detection limit of σ Allan=559.5 nV was determined. After ∼25 s averaging the performance of fast scanning mode approaches that of point-by-point scan method (which required 375 second to perform a single scan). Another advantage of the fast scanning method is that it enables a relatively simple suppression of a slow system drift by performing baseline correction (DC, linear or parabolic) on a scan-to-scan basis. Such drifts that can occur in FRS systems (e.g. due to electromagnetic pickup) are difficult to suppress if a slow point-by-point laser scanning method is used.

Fig. 7
figure 7

Allan variance plot (bottom) calculated for a time series of the OH FRS signal measured over a 27 min period (top). The laser was scanned at a rate of 15 Hz, and a spectrum was calculated by averaging five laser scans

Full size image

5.4 Laser power effects

In FRS experiments, the major noise contributions affecting the detection sensitivity are: (1) detection system noise originating from detector and amplifiers, which is independent of the optical intensity; (2) laser amplitude excess noise, proportional to the laser power; (3) shot noise proportional to the square root of the detected laser intensity. For small ϕ the laser power transmitted through the analyzer (hence the laser noise) is proportional to ∼ϕ 2. Given the FRS signal is proportional to ϕ, we have: (1) the FRS signal-to-detection system noise is proportional to ϕ; (2) the FRS signal-to-laser noise varies as 1/ϕ; and (3) the signal-to-shot noise is independent of ϕ. Therefore the analyzer offset angle ϕ should be experimentally optimized to achieve an optimal signal-to-noise ratio under different experimental conditions, depending on individual contributions of each noise component. A detailed discussion of the FRS signal and noise sources was given in Refs. [14, 18] and references therein. In the present work, we investigated the effects of the laser intensity (and the resulting analyzer offset angle ϕ) on FRS detection sensitivity. The experiments were conducted with point-by-point method.

As demonstrated in Ref. [14], the SNR signal is proportional to \(\sqrt{P_{0}}\) when the term ξP 0 is negligible in comparison with P(φ). This is achieved by using high extinction ratio ξ and small φ. In the present work, after a careful optical alignment, the laser power reaching the detector was increased by a factor of 8.2 as compared to our previous work [18] and the results in Sect. 5.2. This yielded a decrease in the optimum offset angle φ from 6∘ down to 2.25∘ (Fig. 8). As shown in Fig. 9, the total noise of the instrument was 0.875 μV/Hz1/2, which corresponds to Θ min=4.7×10−8 rad/Hz1/2. The detection sensitivity was then improved by a factor of ∼2.6, which was in good agreement with the enhancement factor of \(\sqrt{8.2}=2.86\) expected from the optical power increase. It is worth noting that in the laser noise limited case the SNR is proportional to 1/ϕ, and the decrease in optimum offset angle φ from 6∘ to 2.25∘ (by a factor 2.67) corresponds approximately to the magnitude of the SNR enhancement.

Fig. 8
figure 8

Effects of the laser power (with the same relative intensity noise) on the optimum analyzer offset angle φ. (a), (c) Detected laser intensity determination before (a) and after (c) improvement of optical alignment. (b), (d) Calculated SNR as a function of the analyzer offset angle. (d) Results obtained with an improved optical alignment

Full size image
Fig. 9
figure 9

FRS signal (without power normalization) acquired at an offset angle of 2.25∘ which resulted in a Θ min of 4.7×10−8 rad/Hz1/2, corresponding to an improvement in detection sensitivity by a factor of ∼2.6 compared to the case in Fig. 5

Full size image

6 Conclusions

Based on a prototype FRS spectrometer for OH radical detection, the performance of the instrument using two different laser wavelength scanning modes, point-by-point wavelength tuning and fast wavelength sweeping, were experimentally compared. Both modes can be used for sensitive OH radical detection, but owing to a better temporal resolution and the ability to suppress longer-term system drifts (after applying fast baseline correction algorithms) the fast scan mode is potentially more robust. Further improvement could be also realized using high scanning rate by implementation of higher resonant frequencies of the magnetic coil circuit.

The effects of laser power variation as well as influence of strong atmospheric H2O vapor absorption were investigated experimentally. We demonstrated that in case of H2O vapor absorption, the amplitude of OH FRS signal is affected due to laser power drop, especially when working at atmospheric pressure. We have demonstrated that with the present FRS instrument this effect is negligible at reduced pressures (below 400 Torr), and after power normalization H2O interference-free FRS detection of OH radical can be reliably performed. It was also shown that an increased laser power in the current instrument resulted in improvement of the system SNR performance that is consistent with the theoretical prediction.