# Shot Noise and Bit-Error Rate (BER) for Coherent Demodulation and Delay Demodulation

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The signal-to-noise ratio (SNR) and the resulting BER for a specific modulation format depend on the demodulation scheme employed. This is so because the noise added to the signal is different for different demodulation schemes. In this tutorial we consider the shot-noise limit and discuss BER for the three demodulation schemes discussed in the demodulation schemes tutorial. In the next tutorial we will focus on a more realistic situation in which system performance is limited by other noise sources introduced by lasers and optical amplifiers employed along the fiber link.

#### 1. Synchronous Heterodyne Receivers

Consider first the case of the binary ASK format. The signal used by the decision circuit is given

with φ = 0. The phase difference φ = φ_{s} - φ_{LO} generally varies randomly because of phase fluctuations associated with the transmitter laser and the local oscillator. We consider such fluctuations later in the next tutorial but neglect them here as our objective is to discuss the shot-noise limit. The decision signal for the ASK format then becomes

where I_{p} ≡ 2R_{d}(P_{s}P_{LO})^{1/2} takes values I_{1} and I_{0} depending on whether a 1 or 0 bit is being detected. We assume no power is transmitted during the 0 bits and set I_{0} = 0.

Except for the factor of 1/2 in this equation, the situation is analogous to the case of direct detection discussed in the optical receiver sensitivity tutorial. The factor of 1/2 does not affect the BER since both the signal and the noise are reduced by the same factor, leaving the SNR unchanged. In fact, one can use the same result

where the Q factor can be written as

In relating Q to SNR, we used I_{0} = 0 and set σ_{0} ≈ σ_{1}. The latter approximation is justified for coherent receivers whose noise is dominated by shot noise induced by the local-oscillator and remains the same irrespective of the received signal power. As shown in the coherent detection tutorial, the SNR can be related to the number of photons N_{p} received during each 1 bit by the simple relation SNR = 2ηN_{p}, where η is the quantum efficiency of the photodetectors employed.

The use of the two equations above with SNR = 2ηN_{p} provides the following expression for the BER:

One can use the same method to calculate the BER in the case of ASK homodyne receivers. The last BER equation and the Q equation above still remain applicable. However, the SNR is improved by 3 dB in the homodyne case.

The last BER equation can be used to calculate the receiver sensitivity at a specific BER. Similar to the direct-detection case discussed in the optical receiver sensitivity tutorial, we define the receiver sensitivity as the average received power required for realizing a BER of 10^{-9} or less. From the two equations above, BER = 10^{-9} when Q ≈ 6, or when SNR = 144 (21.6 dB). We can use the following equation to relate SNR to

if we note that

simply because signal power is zero during the 0 bits. The result is

In the ASK homodyne case, is smaller by a factor of 2 because of the 3-dB homodyne-detection advantage. As an example, for a 1.55-μm ASK heterodyne receiver with η = 0.8 and Δf = 1 GHz, the receiver sensitivity is about 12 nW and reduces to 6 nW if homodyne detection is used.

The receiver sensitivity is often quoted in terms of the number of photons N_{p} using the BER expression above because such a choice makes it independent of the receiver bandwidth and the operating wavelength. Furthermore, η is also set to 1 so that the sensitivity corresponds to an ideal photodetector. It is easy to verify that for realizing a BER of 10^{-9}, N_{p} should be 72 and 36 in the heterodyne and homodyne cases, respectively. It is important to remember that N_{p} corresponds to the number of photons within a single 1 bit. The average number of photons per bit, , is reduced by a factor of 2 in the case of binary ASK format.

Consider now the case of the BPSK format. The signal at the decision circuit is given by

The main difference from the ASK case is that I_{p} is constant, but the phase φ takes values 0 or π depending on whether a 1 or 0 is being transmitted. In both cases, I_{d} is a Gaussian random variable but its average value is either I_{p}/2 or - I_{p}/2, depending on the received bit. The situation is analogous to the ASK case with the difference that I_{0} = - I_{1} in place of being zero. In fact, we can use

for the BER, but Q is now given by

where I_{0} = - I_{1} and σ_{0} = σ_{1} was used. By using SNR = 2ηN_{p}, the BER is given by

As before, the SNR is improved by 3 dB, or by a factor of 2, in the case of PSK homodyne detection.

The receiver sensitivity at a BER of 10^{-9} can be obtained by using Q = 6. For the purpose of comparison, it is useful to express the receiver sensitivity in terms of the number of photons N_{p}. It is easy to verify that N_{p} = 18 and 9 for heterodyne and homodyne BPSK detection, respectively. The average number of photons/bit, , equals N_{p} for the PSK format because the same power is transmitted during 1 and 0 bits. A PSK homodyne receiver is the most sensitive receiver, requiring only 9 photons/bit.

For completeness, consider the case of binary FSK format for which heterodyne receivers employ a dual-filter scheme, each filter passing only 1 or 0 bits. The scheme is equivalent to two complementary ASK heterodyne receivers operating in parallel. This feature allows us to use

for the FSK as well. However, the SNR is improved by a factor of 2 compared with the ASK case because the same amount of power is received even during 0 bits. By using SNR = 4ηN_{p}, the BER is given by

In terms of the number of photons, the sensitivity is given by

The figure below shows the BER as a function of ηN_{p} for the ASK, PSK, and FSK formats, demodulated by using a synchronous heterodyne receiver.

It is interesting to compare the sensitivity of coherent receivers with that of a direct-detection receiver. The table below shows such a comparison.

As discussed in the optical receiver sensitivity tutorial, an ideal direct-detection receiver requires 10 photons/bit to operate at a BER of ≤ 10^{-9}. This value is considerably superior to that of heterodyne schemes. However, it is never achieved in practice because of thermal noise, dark current, and many other factors, which degrade the sensitivity to the extent that

is usually required. In the case of coherent receivers, below 100 can be realized because shot noise can be made dominant by increasing the local-oscillator power.

#### 2. Asynchronous Heterodyne Receivers

The BER calculation for asynchronous receivers is more complicated because the noise does not remain Gaussian when an envelop detector is used. The reason can be understood from this equation showing the signal processed by the decision circuit.

In the case of an ideal ASK heterodyne receiver, φ can be set to zero so that (subscript d is dropped for simplicity)

Even though both i_{c} and i_{s} are Gaussian random variable with zero mean and the same standard deviation σ, where σ is the root mean square (RMS) noise current, the probability density function (PDF) of I is not Gaussian. It can be calculated by using a standard technique and is found to be given by

where I_{0}(x) represents a modified Bessel function of the first kind and I varies in the range 0 to ∞ because the output of an envelop detector can have only positive values. This PDF is known as the *Rice distribution*. When I_{p} = 0, the Rice distribution reduces to a *Rayleigh distribution*, well known in statistical optics.

The BER calculation follows the analysis of the optical receiver sensitivity tutorial with the only difference that the Rice distribution needs to be used in place of the Gaussian distribution. The BER is given by

with

where I_{D} is the decision level and I_{1} and I_{0} are values of I_{p} for 1 and 0 bits. The noise is the same for all bits (σ_{0} = σ_{1} = σ) because it is dominated by the local oscillator power. The integrals in the equations can be expressed in terms of Marcum's Q function, defined as

The result for the BER is

The decision level I_{D} is chosen such that the BER is minimum for given values of I_{1}, I_{0}, and σ. It is difficult to obtain an exact analytic expression of I_{D}. However, under typical operating conditions, I_{0} ≈ 0, I_{1}/σ ≫ 1, and I_{D} is well approximated by I_{1}/2. The BER then becomes

Using SNR = 2ηN_{p}, we obtain the final result,

A comparison with the following equation,

obtained in the case of synchronous ASK receivers shows that the BER is larger in the asynchronous case for the same value of ηN_{p}. However, the difference is so small that the receiver sensitivity at a BER of 10^{-9} is degraded by only 0.5 dB. If we assume η = 1,

shows that BER = 10^{-9} for

while

in the synchronous case.

Consider next the PSK format. As mentioned earlier, asynchronous demodulation cannot be used for it. However, DBPSK signals can be demodulated by implementing the delay-demodulation scheme in the microwave regime. The filtered current in this following equation:

is divided into two parts, and one part is delayed by exactly one symbol period T_{s}. The product of two currents depends on the phase difference between any two neighboring bits and is used by the decision current to determine the bit pattern.

To find the PDF of the decision variable, we write this equation in the form

where

Here, n = i_{c} + ii_{s} is a complex Gaussian random process. The current used by the decision circuit can now be written as

If ω_{IF}T_{s} is chosen to be a multiple of 2π and we can approximate ψ with φ, then I_{d} = ±r(t)r(t - T_{s}) as the phase difference takes its two values of 0 and π. The BER is thus determined by the PDF of the random variable r(t)r(t - T_{s}).

It is helpful to write this product in the form

where

Consider the error probability when φ = 0 for which I_{d} > 0 in the absence of noise. An error will occur if r_{+} < r_{-} because of noise. Thus, the conditional probability is given by

This probability can be calculated because the PDFs of r_{±}^{2} can be obtained by noting that n(t) and n(t - T_{s}) are uncorrelated Gaussian random variables. The other conditional probability, P(0|π), can be found in the same manner. The final result is quite simple and is given by

A BER of 10^{-9} is obtained for ηN_{p} = 20. As a reminder, the quantity ηN_{p} is just the SNR per bit in the shot noise limit.

#### 3. Receivers with Delay Demodulation

In the delay-demodulation scheme shown in the demodulation schemes tutorial, one or more MZ interferometers with one-symbol delay are used at the receiver end. In the DBPSK case, a single MZ interferometer is employed. The outputs of the two detectors in this case have average currents given by

The decision variable is formed by subtracting the two currents such that

The average currents for 0 and 1 bits are R_{d}P_{0} and - R_{d}P_{0} for Δφ = 0 and π, respectively.

To see how the noise affects the two currents, note that from this equation

that I_{d} can be written in the form

where

is the optical field entering the receiver. Here, n(t) represent the noise induced by vacuum fluctuations that lead to the shot noise at the receiver. A comparison of this equation with

obtained in the case of heterodyne detector with delay implemented in the microwave domain, shows the similarity between the two cases. Following the discussion presented here, one can conclude that the BER in the DBPSK case is again given by

As before, the SNR per bit, ηN_{p}, sets the BER, and a BER of 10^{-9} is obtained for ηN_{p} = 20.

The analysis is much more involved in the case of the DQPSK format. Proakis has developed a systematic approach for calculating error probabilities for a variety of modulation formats that includes the DQPSK format. Although his analysis is for a heterodyne receiver with delay implemented in the microwave domain, the results apply as well to the case of optical delay demodulation. In particular, when the DQPSK format is implemented with the Gray coding, the BER is given by

where I_{0} is the modified Bessel function of order zero and Q_{1}(a, b) is Marcum's Q function introduced earlier.

The following figure shows the BER curves for the DBPSK and DQPSK formats and compares them with the BER curve obtained in the case in which a heterodyne receiver is employed to detect the BPSK or the QPSK format (without differential encoding).

When DBPSK is used in place of BPSK, the receiver sensitivity at a BER of 10^{-9} changes from 18 to 20 photons/bit, indicating a power penalty of less than 0.5 dB. In view of a such a small penalty, DBPSK is often used in place of BPSK because its use avoids the need of a local oscillator and simplifies the receiver design considerably. However, a penalty of close to 2.4 dB occurs in the case of DQPSK format for which receiver sensitivity changes from 18 to 31 photons/bit.

Because of the complexity of the BER expression above, it is useful to find its approximate analytic form. Using the upper and lower bounds on Marcum's Q function, this BER expression can be written in the following simple form

This expression is accurate to within 1% for BER values below 3 x 10^{-2}. If we now employ asymptotic expansions

valid for large values of x, and use a and b from

we obtain

This expression is accurate to within a few percent for values of ηN_{p} > 3.