2.1.

Using Single Photodetector

The SNR for the time-varying signal in a single photodiode is defined as⁷

2.2.

Using Single Photodetector and Polarizer

This system depends on the idea that the polarizer blocks all light of perpendicular polarization and passes some but not all of the light of one polarization. The receiver is composed of a photodiode (PD) and a linear polarizer (PL). If a linear polarizer is placed in a linearly polarized beam, the transmittance $T(\theta )$ will vary between a maximum value $T_{\max }$ and a minimum value $T_{\min }$ according to Malus’s law.⁷

From this equation, one can see that the improvement in SNR depends mainly on the transmittance of the polarizer.

2.3.

Using a Differential Detector with a Single Polarizer

In this section, a different technique to combat the effects of the artificial light interference is analyzed. A similar solution was described in Ref. 6, using an equation of optical filter. Here, we used the same idea with the polarizer equations described in Ref. 7. The lower front-end in Fig. 1 receives the transmitted signal with much higher interference amplitude since no polarizer is used. The result is that the interference-to-signal ratio is much higher at the output of the lower front-end than at the upper front-end. If the output of the lower front-end is attenuated so that the amplitude of the interference received by the two stages is equal and the two outputs of the lower and upper front-ends are subtracted, the interference is totally cancelled while the transmitted signal is only partially attenuated. The output current is obtained as

Fig. 1

Optical receiver using a single polarizer.

2.4.

Using a Differential Detector with Two Polarizers

Another efficient method to eliminate the interference from ambient noise light is the differential detection, in which two photodetectors (PD1 and PD2) and two linear polarizers (PL1 and PL2) are used.⁷ For orthogonal polarizers, PL1 is copolarized (${\theta }_1=0\mathrm{deg}$) and PL2 is cross-polarized (${\theta }_2=90\mathrm{deg}$) to the transmitted signal; the transmittance of PL1 for the signal is $T_{\max }$ and that of PL2 is $T_{\min }$, as in the transmittance equation. The differential output current is the difference between the two photocurrents⁷ and ${\mathrm{SNR}}_{\mathrm{DD}\text{-}\mathrm{OP}}$ is⁷

The SNR in a differential detector with nonorthogonal polarizers (DD-non-OP) is derived in the form

2.5.

Using L-PPM Modulation and I&D Demodulation Filter

The demodulator in our study is assumed to be a maximum likelihood I&D demodulator. The influence of flicker and other intensity variations can be modeled after considering the demodulation in Ref. 8. One can define the flicker voltage ${\mathrm{\Delta }}_i(V)$ and the noise voltage ${\sigma }_n(V)$ as follows:⁸

In these expressions, $C$ is the value of the capacitor in I&D filter, $L$ is the number of slots for L-PPM, ${[(\mathrm{d}{I}_{\mathrm{amb}})/(\mathrm{d}t)]}_{\max }$ is the maximum slope of the ambient noise current,⁸ and $N_{\mathrm{amb}}$ is the noise caused by ambient radiation with double-sided spectral density, where $N_{\mathrm{amb}}=e·I_{\mathrm{amb}}$. The frame with duration $T_{\text{frame}}$ is divided into $L$ slots with duration $T_{\text{pulse}}$; only one of these slots contains an optical pulse. For $L=4$ we can use the following equations⁸

This means a reduction in the value of noise voltage and flicker voltage. The SNR equations will be the same for all systems. If the I&D demodulator filter and half pulse L-PPM are used in the previous techniques, described in Sec. 2, the design of the differential detector shows the cancellation of ambient current noise for the tungsten lamp to be zero, but the noise due to the photodetectors will increase, which makes the value of flicker and noise voltage (noise voltage due to ambient radiation) equal to zero.

3.1.

Comparison Between $\mathsf{\Delta }\mathsf{SNR}$ for the Different Systems

From Fig. 2, one can observe that $\mathrm{\Delta }{\mathrm{SNR}}_{\mathrm{SD}\text{-}\mathrm{SP}}$ increased suddenly from 0 dB at $I_n=0\mathrm{mA}$ to 6 dB at $I_n=0.1\mathrm{mA}$; then, the value will remain constant at 6 dB until $I_n=1\mathrm{mA}$. The reason for this is the effect of the polarizer on the value of noise current. It is multiplied by a fraction equal to $T_n$, which caused this improvement, and it will saturate because the polarizer will remove the noise current by the same factor that fixes the value of $\mathrm{\Delta }{\mathrm{SNR}}_{\mathrm{SD}\text{-}\mathrm{SP}}$. Also it is noticed that increasing the noise current leads to an increase in $\mathrm{\Delta }{\mathrm{SNR}}_{\mathrm{DD}\text{-}\mathrm{SP}}$, where its level changes from $-6.99\mathrm{dB}$ at a noise current equal to 0 mA to 28.33 dB at a noise current equal to 1 mA. The reason for this observation is as follows: at zero noise current, there will be no improvement, but the value of $\mathrm{\Delta }{\mathrm{SNR}}_{\mathrm{DD}\text{-}\mathrm{SP}}$ in this case will be worse than the value of $\mathrm{\Delta }{\mathrm{SNR}}_{\mathrm{SD}\text{-}\mathrm{SP}}$ because the amplitude of the current signal will be decreased by a factor equal to ($T_{\max }$- $T_n$), which decreases ${\mathrm{SNR}}_{\mathrm{DD}\text{-}\mathrm{SP}}$. For $i_{\text{amb}}>0$ the systems that used DD-OP and DD-SP will remove this current to improve SNR. It is clear that the best $\mathrm{\Delta }\mathrm{SNR}$ is obtained in the differential detector with a two orthogonal polarizers system because it has the highest value addition; $\mathrm{\Delta }{\mathrm{SNR}}_{\mathrm{DD}\text{-}\mathrm{OP}}$ is equal to zero at zero current noise, which means that the SNR of a single photodetector will be equal to ${\mathrm{SNR}}_{\mathrm{DD}\text{-}\mathrm{OP}}$.

Fig. 2

Comparison between $\mathrm{\Delta }\mathrm{SNR}$ for different systems.

3.2.

Effect of Using a Differential Detector with Two Nonorthogonal Polarizers

In Fig. 3, the SNR has three different values at $\theta$ equal to 0, 60, and 90 deg and the value of SNR is better in the case of ${\theta }_2=90\mathrm{deg}$. This is because when ${\theta }_2=0\mathrm{deg}$, the transmission axes of PL1 and PL2 will be in the same direction of the transmitted signal, which yields a zero value for the amplitude of the transmitted signal at the output of the differential detector.

Fig. 3

Effect of the angle between the two polarizers and the transmittance difference on ${\mathrm{SNR}}_{\mathrm{non}\text{-}\mathrm{DD}\text{-}\mathrm{OP}}$.

3.3.

Effect of Detector Parameters on $\mathsf{\Delta }\mathsf{SNR}$

For the receiver bandwidth, Fig. 4 shows its effect on the ${\mathrm{SNR}}_{\mathrm{DD}\text{-}\mathrm{OP}}$. It can be seen that an increase in the detector bandwidth (BW) decreases the values of ${\mathrm{SNR}}_{\mathrm{DD}\text{-}\mathrm{OP}}$ and ${\mathrm{SNR}}_{\mathrm{SD}}$. But the change is too small (it is nearly constant) in ${\mathrm{SNR}}_{\mathrm{SD}}$ compared to ${\mathrm{SNR}}_{\mathrm{DD}\text{-}\mathrm{OP}}$ due to the existence of the two photodetectors, which leads to a decrease in the level of $\mathrm{\Delta }\mathrm{SNR}$ as described in Table 1. This can be explained by the wide bandwidth of the receiver, which leads to a high concentration of noise that produces a decrease in ${\mathrm{SNR}}_{\mathrm{DD}\text{-}\mathrm{SP}}$. We are here referring to shot noise not ambient current noise, because an increase in ambient current noise will increase ${\mathrm{SNR}}_{\mathrm{DD}\text{-}\mathrm{SP}}$. But increasing the shot noise current will decrease ${\mathrm{SNR}}_{\mathrm{DD}\text{-}\mathrm{SP}}$ due to existence of the two photodetectors.

Table 1

Comparison between the values of ΔSNR with the changing photodetector parameters on ΔSNR.

Fig. 4

Effect of receiver bandwidth on (a) ${\mathrm{SNR}}_{\mathrm{SD}}$ and ${\mathrm{SNR}}_{\mathrm{DD}\text{-}\mathrm{OP}}$ and (b) $\mathrm{\Delta }{\mathrm{SNR}}_{\mathrm{DD}\text{-}\mathrm{OP}}$.

It is also found that an increase in the excess noise factor in Fig. 5(a) and the current gain of the detector in Fig. 5(b) will lead to a decrease in $\mathrm{\Delta }{\mathrm{SNR}}_{\mathrm{SD}\text{-}\mathrm{OP}}$ due to the two photodetectors. This is because a higher excess noise factor and bigger current gain lead to a high concentration of noise. Table 1 indicates that $\mathrm{\Delta }{\mathrm{SNR}}_{\mathrm{SD}\text{-}\mathrm{SP}}$ is constant. This is because the change in photodetector parameters will not affect the system that uses only a single detector.

Fig. 5

Effect of receiver parameters on (a) excess noise factor (b) current gain on $\mathrm{\Delta }{\mathrm{SNR}}_{\mathrm{DD}\text{-}\mathrm{OP}}$.

3.4.

Using L-PPM and I&D Filter

3.4.1.

Effect of L-PPM and I&D filter parameters on ${\mathsf{\Delta }}_{\mathsf{i}}(\mathsf{V})$ and ${\mathsf{\sigma }}_{\mathsf{n}}(\mathsf{V})$

As one can observe from Fig. 6, increasing the modulation index (by changing lamp type or manufacturer) decreases the level of flicker voltage and noise voltage. The reason for that is mainly due to the level of IR radiation that produces noise that exceeds the effect of the very low modulation index (10%) with the low flicker operating frequency (100 Hz), which both control the level of flicker voltage. This is only valid in incandescent lighting. Also, we can see that the polarizer decreases the level of flicker voltage by a factor equal to $T_n$ and the noise voltage by a factor equal to $\sqrt{T_n}$ (effect of polarizer). For both the differential systems, noise and flicker are zero because these systems canceled the effect of ambient noise (effect of DD). An increase in demodulator capacitance decreases the level of flicker voltage and noise voltage as shown in Fig. 7. This is because a large capacitance means a large capability of storing ambient radiation, including flicker. This can give a reason for the large change in the level of flicker and noise voltages seen at 1 and 100 μF. Also, the polarizer decreased those effects and this decrease is not constant because ambient noise current does not depend on capacitance.

Fig. 6

Effect of modulation index of L-PPM and integrate-and-dump (I&D) filter on ${\mathrm{\Delta }}_i(V)$ and ${\sigma }_n(V)$.

Fig. 7

(a) Effect of capacitance of I&D filter on ${\mathrm{\Delta }}_i(V)$ and ${\sigma }_n(V)$. (b) Effect of frame time on ${\mathrm{\Delta }}_i(V)$ and ${\sigma }_n(V)$.

Similar to the previous discussion, Fig. 7(b) indicates that decreasing the frame time from 20 to 1 μs will increase the level of flicker and noise voltage. This is mainly because decreasing the frame time in a low flicker frequency and low modulation index environment, like incandescent lighting, leads to a high concentration of noise and flicker levels in a short period of data time. Using a maximum-likelihood I&D demodulator capacitance will lead to a sudden increase in the shot noise generated (especially at 1 μs) and, hence, increases both flicker and noise voltages.

Now, the effect of number of slots per frame is investigated. Figure 8 shows that increasing the number of slots per frame increases the level of flicker interfering voltage, but the level of noise voltage remains constant. Increasing the number of slots per frame will make the capacitor charge several times to get data slots with noise and flicker. Hence, this increases their levels, noting that the frame time is constant.

Fig. 8

Effect of number of slots per frame on noise and flicker voltages.

3.4.2.

Effect of L-PPM and I&D filter parameters on $\mathsf{\Delta }\mathsf{SNR}$ for different systems

Figure 9(a) shows that an increase in the modulation index decreases $\mathrm{\Delta }{\mathrm{SNR}}_{\mathrm{DD}\text{-}\mathrm{OP}}$ with a very small change. This can be explained as follows: an increase in the modulation index leads to a decrease in the ambient noise current. Using a DD-OP, which removes the ambient current noise, makes its effect very small on $\mathrm{\Delta }{\mathrm{SNR}}_{\mathrm{DD}\text{-}\mathrm{OP}}$. But it affects only the shot noise current of the two photodetectors, which is very small. For Fig. 9(b), an increase or decrease in the value of the capacitance has no effect on both differential $\mathrm{\Delta }{\mathrm{SNR}}_{\mathrm{DD}\text{-}\mathrm{OP}}$ and $\mathrm{\Delta }{\mathrm{SNR}}_{\mathrm{DD}\text{-}\mathrm{OP}}$. This is because the capacitance has no effect on the ambient current noise equation, and also, it will not affect both $\mathrm{\Delta }{\mathrm{SNR}}_{\mathrm{DD}\text{-}\mathrm{OP}}$ and $\mathrm{\Delta }{\mathrm{SNR}}_{\mathrm{DD}\text{-}\mathrm{OP}}$. Increasing $T_{\text{frame}}$ in Fig. 9(c) will increase ${\mathrm{SNR}}_{\mathrm{DD}\text{-}\mathrm{OP}}$ and decrease $\mathrm{\Delta }{\mathrm{SNR}}_{\mathrm{DD}\text{-}\mathrm{OP}}$. This is because increasing the frame time leads to a decrease in the ambient noise current and the shot noise current, which affects ${\mathrm{SNR}}_{\mathrm{DD}\text{-}\mathrm{OP}}$. It is clear from Fig. 9(d) that an increase in $L$ decreases ${\mathrm{SNR}}_{\mathrm{DD}\text{-}\mathrm{OP}}$ and increases $\mathrm{\Delta }{\mathrm{SNR}}_{\mathrm{DD}\text{-}\mathrm{OP}}$. This is because an increase in $L$ increases the ambient noise current and the shot noise current, which affects ${\mathrm{SNR}}_{\mathrm{DD}\text{-}\mathrm{OP}}$ and $\mathrm{\Delta }{\mathrm{SNR}}_{\mathrm{DD}\text{-}\mathrm{OP}}$.

Fig. 9

Effect of (a) modulation index, (b) capacitance, (c) time frame, and (d) number of slots on $\mathrm{\Delta }{\mathrm{SNR}}_{\mathrm{DD}\text{-}\mathrm{OP}}$.

One can notice from Table 2 that the variation of $\mathrm{\Delta }{\mathrm{SNR}}_{\mathrm{SD}\text{-}\mathrm{SP}}$ and $\mathrm{\Delta }{\mathrm{SNR}}_{\mathrm{DD}\text{-}\mathrm{OP}}$ is very low because the noise current is very low, as the analysis considered that the noise source is not very closed. The level of $\mathrm{\Delta }{\mathrm{SNR}}_{\mathrm{DD}\text{-}\mathrm{SP}}$ is negative because the noise current is very low and this system decreases the value of the transmitted signal. The whole result can be summarized in Figs. 10 and 11; it is seen that the SD system just adds a noise signal to the input signal, the SD-SP system decreases the noise signal by a factor equal to $T_n$, the DD-SP system removes noise but it decreases input signal by a factor equal to ($T_s-T_n$), and the DD-OP system removes noise without decreasing the value of input signal.

Table 2

Effect of L-PPM and I&D demodulation filter parameters on ΔSNR for different systems.

Parameters	SD-SP	DD-SP
$m_i$ (10% : 20%)	$4.56\times {10}^{-8}:2.85\times {10}^{-9}$	$-6.9895:-6.9897$
$C$ (1 μf:100 μf)	Const at $4.56\times {10}^{-8}$	Const at $-6.9895$
$T_{\text{frame}}$ (1:20) μs	$1.5:\approx 0$	$-5.8400:-6.9895$
$L$ (4:20)	$(0.0046:0.1140)\times {10}^{-5}$	$-6.9895:-6.9888$

Fig. 10

Input signal and noise in the systems described in Sec. 2.

Fig. 11

Output signal from different systems.