Application Note:

HFAN-9.0.2
Rev.1; 04/08

Optical Signal-to-Noise Ratio and the Q-Factor in

Fiber-Optic Communication Systems

Functional Diagrams

Pin Configurations appear at end of data sheet.

Functional Diagrams continued at end of data sheet.
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AVAILAB
 Optical Signal-to-Noise Ratio and the Q-Factor in
Fiber-Optic Communication Systems
1 Introduction v L2 v2
S EL = = i L2 R or S EH = H = i H2 R (3)
The ratio of signal power to noise power at the R R
receiver of a fiber-optic communication system has a
direct impact on the system performance. Many where SE represents electrical signal power and the
electrical engineers are familiar with signal-to-noise subscripts L and H represent the low or high power,
ratio (SNR) concepts when referring to electrical voltage, or current levels associated with a binary
signal and noise powers, but have less familiarity zero or one respectively.
with the equivalent optical signal and noise powers.
The purpose of this application note is to show the Now we will repeat the above derivations for the
relationship between the electrical and optical case of optical signals using electromagnetic vector
signal-to-noise ratio (SNR), and then introduce the notation. Using this notation, the power in an optical
Q-factor. signal can be defined as the magnitude of the vector
cross product of the electric and magnetic fields,
While the principles outlined in this application note which can be written and simplified as follows:
may be applied to many types of systems, the scope
of the discussion is limited to binary digital 2
E (t ) E (t )
communications over optical fiber. Within this PO (t ) = E (t ) × H (t ) = E (t ) = (4)
scope, there are only two possible symbols that can η η
be transmitted, where these symbols represent a
binary one or a binary zero. Thus, the symbol rate where the notation | X | represents the magnitude of
and the bit rate are equivalent. µ
X, and η = is the optical impedance of the
ε
2 Signal Power fiber (µ = permeability and ε = permittivity).
Recognizing that there are only two discrete power
The power in an arbitrary electrical waveform can be levels leads to the optical equivalent of equation (3),
defined as the voltage multiplied by the current, i.e.,
which is written mathematically as: 2 2
EL EH
PE (t ) = v(t )i(t ) (1) S OL = and S OH = (5)
η η
Using ohm’s law, we can substitute v(t) = i(t)R, or
alternately i(t) = v(t)/R, into equation (1) to get: where SO represents optical signal power and the
subscripts L and H represent the low and high power
or low and high electric field strengths associated
PE (t ) = v 2 (t ) / R = i 2 (t ) R (2) with a binary zero or one respectively.

where R = voltage/current is the resistance in ohms.

In binary digital communications, the signal is 3 Noise Power
limited to two discrete levels. Based on this, we can
represent the electrical signal power at any given Noise can be defined as any unwanted or interfering
time by either: “signal” other than the one that is intended or
expected. The various types of noise and their
sources are beyond the scope of this application
Application Note HFAN-9.0.2 (Rev.1; 04/08) Maxim Integrated
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note. For purposes of illustration we will model the The answer lies in the fact that when we add the
noise power as random, normally distributed, zero signal (a constant) and the noise (an average value),
mean, and additive (the most common type of we compute the result as an average. (We don’t need
noise). to know the value of the signal plus noise at every
instant of time–we only care about the average
The random nature of the noise means that the
instantaneous value of the noise amplitude is value.) In the average, the cross term 2 i H σ i is equal
unpredictable. Thus, instead of classifying the noise to zero. The reason for this is that the probability
in terms of its actual value at any given time, we use density function (pdf) of the noise was defined as
statistical averages and probabilities. We will zero mean and normally distributed. Since this pdf is
classify the noise amplitude in terms of its root- symmetric about the mean, multiplication by a
mean-square (rms) average, which is commonly constant will not change the mean, which will
given the symbol σ. The noise power is similarly remain zero, i.e., in the average, the result will
expressed in terms of its mean-square average always be zero.
(equivalent to the statistical variance), which is
given the symbol σ2. In general, the noises 5 Signal-to-Noise Ratio (SNR)
associated with the high and low signal levels in
binary optical digital communications each have a Knowledge of the ratio of the signal power to the
different value. noise power (signal-to-noise ratio or SNR) is
The mean-square average electrical and optical noise important because it is directly related to the bit
powers can be computed mathematically using the error ratio (BER) in digital communication systems,
following equations: and the BER is a major indicator of the quality of the
overall system.
T 2
N E = T1 iN (t ) R dt = σ i2 R Drawing from the results of the preceding sections,
0
we can mathematically express the electrical SNR as
T 1 1
= T1 v N2 (t ) dt = σ v2 (6) SE v2 R v2
0 R R SNR E = = 2 =
N E σ v R σ v2
T 2 1 1
N o = T1 E N (t ) dt = σ o2 (7)
0 η η i2R i2
= = (8)
σ i2 R σ i2
where N is the noise power, T is the integration
period, σ2 is the mean-square average power, and the Similarly, the optical SNR is
subscript N signifies that the associated current,
voltage or electric field is classified as noise. 2 1
E 2
S η E
SNRo = o = = (9)
4 Signal Plus Noise No 1 σ o2
σ o2
Addition of the signal and noise amplitudes versus η
addition of the signal and noise powers can
sometimes cause mathematical confusion. For In practice, optical powers are rarely measured
example, if the combined signal and noise amplitude directly. Instead, the optical power is converted to a
is written as i(t ) = (i H + σ i ) , then the power would proportional electric current using a device such as a
PIN photodiode, and then the current is measured.
seemingly be i 2 (t ) R = (i H + σ i ) 2 R , which, when The ratio between the output current and the incident
multiplied out, is equal to (i H2 + 2 i H σ i + σ i2 ) R . optical power is called the responsivity
(mathematically represented using the symbol R ),
But, addition of the results of equations (3) and (6)
which has the units of Amperes per Watt (A/W). It is
gives SE + NE = (i H2 + σ i2 ) R . So, why is there a important to note that the conversion between
difference in the two results? optical power (units of Watts) and electrical current

Application Note HFAN-9.0.2 (Rev.1; 04/08) Maxim Integrated

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(units of Amperes) essentially results in a square synchronization between the bit stream and the bit
root operation. In other words, as we recall from clock, the major obstacle to making the correct
equation (2), electrical power is related to the square decision is noise added to the received data.
of the voltage or current, and, as we recall from
equation (5) optical power is related to the square of
the magnitude of the electric field. The result is that
Decision Circuit
the conversion between optical signal or noise DATA
VCC
D Q
power, (So or No – both related to | E |2 ) and
electrical current results in what is essentially a + +
square root relationship, i.e., – –
CLK
PLL
i signal = S o R and inoise = N o R (10)

Also, the optical SNR, when converted to an

Figure 1. Block-diagram of a fiber-optic receiver
electrical SNR, is equal to the square root of the
equivalent electrical SNR. This is illustrated
mathematically by combining equations (8) and (10) If we assume that additive white Gaussian noise
as follows: (AWGN) is the dominant cause of erroneous
decisions, then we can calculate the statistical
SE i signal S o R probability of making such a decision. The
SNR E = = = = SNRo (11) probability density function for v(t) with AWGN can
NE σi N oR be written mathematically using the Gaussian
probability density function (pdf) as follows:
6 The Q-Factor 2
1 v (t ) − v s
−
As discussed previously, there are only two possible 1 2 σx
signal levels in binary digital communication PROB [v(t ), σ x ] = e (12)
systems and each of these signal levels may have a 2πσ x2
different average noise associated with it. This
means that there are essentially two discrete signal-
to-noise ratios, which are associated with the two where vS is the voltage sent by the transmitter (the
possible signal levels. In order to calculate the mean value of the density function), v(t) is the
overall probability of bit error, we must account for sampled voltage value in the receiver at time t, and σ
both of the signal-to-noise ratios. In this section we is the standard deviation of the noise. Equation (12)
will show that the two SNRs can be combined into a is illustrated in Figure 2.
single quantity – providing a convenient measure of
overall system quality – called the Q-factor.
PROB [v(t)]
σ
In the following discussion, we will assume the
signals are electrical voltages, but, as demonstrated
in the previous sections, the concepts can easily be
extended to electrical current signals or optical
signals.
v(t)
To begin this discussion, we consider the decision vS
circuit in a fiber-optic receiver, which simply
compares the sampled voltage, v(t), to a reference Figure 2. AWGN probability density function
value, γ, called the decision threshold. If v(t) is
greater than γ, it indicates that a binary one was sent, If we assume that vS can take on one of two voltage
whereas if v(t) is less than γ, it indicates that a levels, which we will call vL and vH, then the
binary zero was sent. Assuming perfect probability of making an erroneous decision in the
receiver is:
Application Note HFAN-9.0.2 (Rev.1; 04/08) Maxim Integrated
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 P[ε] = P[v(t) > γ | vS = vL] P[vS = vL] +
P[ε] = P[v(t) > γ | vS = vL] × 0.5
P[v(t) < γ | vS = vH] P[vS = vH] (13) + P[v(t) < γ | vS = vH] × 0.5

where P[ε] is the probability of error and P[x | y] γ

1
represents the conditional probability of x given y. If = PROB [v(t ), σ L] dt
we further assume an equal probability of sending 2 −∞

vL versus vH (50% mark density), then P[vS = vL] = 1 ∞

P[vS = vH] = 0.5. Using this assumption, equation + PROB [v(t ), σ H ] dt (14)
2 γ
(13) can be reduced to:
where PROB[v(t),σx] is defined in equation (12).
This result is illustrated in Figure 3.
v(t) From Figure 3 and equations (13) and (14) we can
conclude that the probability of error is equal to the
area under the tails of the density functions that
P[v(t) | vS = vH]
extend beyond the threshold, γ. This area, and thus
the bit error ratio (BER), is determined by two
v(t) =Signal
+ Noise factors: (1) the standard deviations of the noise (σL
and σH) and (2) the voltage difference between vL
σH vH and vH.

It is important to note that for the special case when

γ σL = σH, the threshold is halfway between the low
and high levels (i.e., γ = (vH−vL)/2). But, for the
σL vL more general case when σL ≠ σH, the optimum
threshold for minimum BER will be higher or lower
than (vH−vL)/2.
P[v(t) | vS = vL]
In order to solve equation (14) we need a practical
PROB[v(t)] way to compute the result of the integrated Gaussian
pdf ( PROB[v(t),σx] ) that is defined in equation
(12). Since there is no known closed form solution
to this integral, it must be evaluated numerically. To
P[v(t) > γ | vS = vL]
maintain compatibility with existing numerical
solutions, equation (12) can be re-written in its
equivalent standardized (zero mean and standard
deviation of one) form. In order to convert to the
γ standardized form, we use the well known z = (x –
µ)/σ substitution, where x = v(t) and µ = vS in
equation (12). For example, we start with
1 x−µ
2

P[v(t) < γ | vS = vH] ∞ ∞ 1 −

PROB [ x, σ ] dx = e 2 σ dx
γ γ 2π σ
(from equations (12) and (14) )
Figure 3. Probability of error for binary signaling
x−µ
and then, substituting z= (so that
σ
x = zσ − µ and dx = σ dz ) results in:

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 −z2 By substituting the definition of Q from equation
∞ (18) into equation (16) we find that, when γ = γopt
1 2
e σ dz ,
2π σ x = zσ + µ = γ
Er [Q ] + Er [Q ] = Er[Q ]
1 1
P [ε ] = (19)
2 2
which is defined as the error function
Next, we solve equation (18) for γopt to get
−z2
∞
Er ( z ) =
1
e 2 v H σL + vLσH
dz (15) γopt = (20)
2π z =γ σL + σH

(Note that there are a number of variations of this and then substitute this expression for γopt back into
function published in the literature.) The error equation (18) to get
function gives the area under the tail of the Gaussian
pdf (mean = vS and standard deviation = σx) between vH − vL
Q= (21)
v(t) and infinity. This form of the error function is σL +σH
useful because numerical solutions are available in
both tabulated form1 and as built-in functions within It should be noted that multiplying the individual
many software utilities (e.g., Er(x) = 1- terms in equation (21) by resistance, impedance, or
NORMSDIST(x) in Microsoft Excel). In terms of responsivity will convert the expression for Q to
Er(z), equation (14) can be rewritten as2: equivalent terms of current or optical power, i.e.,

1 v −γ 1 γ −v vH − vL i −i P − POL
P [ε ] = Er Hσ + Er σ L (16) Q= = H L = OH (22)
2 H 2 L
σL +σH σL +σH σL +σH
It is interesting to note that the arguments of the Finally, we can substitute equation (21) into the
error functions in equation (16) represent the square result from equation (19) to get
root of the signal power divided by the square root
of the noise power, which, we recall from equation
vH − vL
(11), is equivalent to the optical signal-to-noise ratio. P [ε ] = Er[Q ] = Er (23)
Thus, equation (16) can be rewritten as follows: σL +σH

Er [SNROH ] + Er [SNROL ]
1 1
P [ε ] = (17) 7 Conclusions
2 2
The Q-factor defined in equations (18) and (21)
where SNROH and SNROL are the optical SNRs for
represents the optical signal-to-noise ratio for a
the high and low levels.
binary optical communication system. It combines
the separate SNRs associated with the high and low
The optimum threshold level, γopt, is defined as the
levels into overall system SNR. The form of the Q-
threshold level that results in the lowest probability
factor given in equation (21) simplifies both the
of bit error. Further, setting the optimum threshold
measurement of SNR and the calculation of the
level also results in the same probability of bit error
theoretical BER due to additive random noise.
when a high signal is transmitted as when a low
signal is transmitted. This means that for the special
For example, measurement of the Q-factor can be
condition of γopt, SNROH = SNROL, which leads to the performed with the vertical histogram function on
following definition of the Q-factor3: many communications oscilloscopes. This can be
done by displaying a portion of the data pattern and
v H − γ opt γ opt − v L alternately applying the vertical histogram to the
Q≡ = (18)
σH σL high (one) level and the low (zero) level. The
oscilloscope histogram function will estimate the
Application Note HFAN-9.0.2 (Rev.1; 04/08) Maxim Integrated
Page 6 of 7
mean (vH or vL) and the standard deviation (σH or numerical integration or the use of tabulated values.
σL), which can then be used directly to compute the A much simpler method of analyzing system
Q-factor. performance is to optimize the Q-factor, knowing
that this will result in optimized BER.
The Q-factor is also useful as an intuitive figure of
merit that is directly tied to the BER. For example,
the BER can be improved by either (1) increasing 1
B. Sklar, Digital Communications: Fundamentals and
the difference between the high and low levels in the Applications, Englewood Cliffs, New Jersey: Prentice
numerator of the Q-factor, or (2) decreasing the Hall, pp. 741-743.
noise terms in the denominator of the Q-factor. 2
G. Agrawal, Fiber-Optic Communication Systems, New
York, N.Y.: John Wiley & Sons, pp. 172, 1997.
Finally, the Q-factor allows simplified analysis of
system performance. The most direct measure of 3
N.S. Bergano, F.W. Kerfoot, and C.R. Davidson,
system performance is the BER, but calculation of "Margin Measurements in Optical Amplifier Systems, " in
the BER requires evaluation of the cumulative IEEE Photonics Technology Letters, vol.5, no. 3, pp. 304-
normal distribution integral. Since this integral has 306, Mar. 1993.
no closed form solution, evaluation requires

Application Note HFAN-9.0.2 (Rev.1; 04/08) Maxim Integrated

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