Introduction

The bit error rate (BER) is the most significant performance parameter of any digital communications system. It is a measure of the probability that any given bit will have been received in error. For example a standard maximum bit error rate specified for many systems is 10^-9. This means that the receiver is allowed to generate a maximum of 1 error in every 10⁹ bits of information transmitted or, putting it another way, the probability that any received bit is in error is 10^-9.

The BER depends primarily on the signal to noise ratio (SNR) of the received signal which in turn is determined by the transmitted signal power, the attenuation of the link, the link dispersion and the receiver noise. Measurement of the BER is not a trivial process and requires sophisticated and expensive equipment to achieve accuracy, particularly at high bit rates. However, the effects of noise and other signal degradation processes can be investigated qualitatively and perhaps even in a pseudo quantitative manner by generating the “eye pattern” or “eye diagram” for the system. Certainly signal and BER degradation trends can be readily seen due to the effects of signal attenuation and dispersion.

Recently OptoSci has introduced an extension module to its very popular ED-COM Fibre Optic Communications educator kit examining eye diagrams and bit error rates. This new module is called BER(COM) and the key educational objectives for the students during the experimental investigation in this module are

BER(COM) is an extension module to the ED-COM educator kit and must be used in conjunction with this kit (see Figure 2). BER(COM) consists of a PRBS generator with data and clock output ports. The non return to zero PRBS pattern repeats every 511 (2^N=9-1) bits and in random order all possible bit sequences for N bit words up to N = 9. The PRBS bit rate may be varied from 1Mbit/s to 40Mbit/s using the rotary control knob on the front fascia and the bit rate is displayed on the front panel LCD. When used with the ED-COM fibre reels, the additional 2km graded index fibre reel supplied allows fibre links of 1km to 5km to be investigated with the ED-COM laser, LED and receiver.

Figure 2:

ED-COM and BER(COM) units in operation

Q-factor Analysis software (Figure 3) is also provided with BER(COM) to enable a digital oscilloscope to sample the received signals in the centre of the bit period, transfer the samples to a PC and then analyse them. The analysis algorithms in the software enable the construction of a signal level histogram (i.e. a plot of the number of samples occurring in a narrow voltage range versus voltage) which is essentially the probability distribution of the signal levels around the mean 0 and 1 levels. Theoretical Gaussian distributions are curve fitted within the software to the measured distributions, the signal level (noise) variances are extracted and the Q-factors and BER are determined. As with all of OptoSci’s educational kits, a detailed student manual covering the relevant background theory and experimental procedure and a comprehensive instructor manual with full sample results for all the experiments are also provided.

Figure 3:

Q-factor analysis software showing voltage histogram and analysis results

Theory

In a digital optical telecommunications receiver, the incident signals are sampled in the centre of the bit period and the sampled level is compared to a threshold to determine the presence of a one or zero. With threshold detection of this nature errors arise when noise in the system pulls a one signal level below threshold at the sampling point and pushes a zero above threshold (see Figure 4).

Figure 4.:

Threshold detection in digital receivers, illustrating the possibility of errors in the presence of random noise

Clearly, due to the random nature of the noise the total signal can be above threshold for a 0 and below for a 1 giving rise to a non zero probability of errors occurring with an associated bit error rate (BER) defined by:-

where N_e is the number of errors occurring in time t and N_t is the number of bits arriving in time t. The bit error rate (BER) is simply the probability that an error will occur in a given bit period.

Clearly from Figure 4, if we have greater SNRs, we can set higher threshold levels (v_th in Figure 4) relative to the noise and hence reduce the probability of error. It is standard practice in digital telecommunications to ensure by design that a specified BER is achieved. For some applications the maximum specified BER is 10^-9 implying that only one error in 10⁹ received bits is tolerated e.g. 1 error per second at a bit rate of 1Gbit/s. For telecommunications applications the specified maximum BER falls in the range 10^-9 to 10^-12.

To calculate the probability of error, we need to establish the noise statistics and compute the probability that the noise level at any given sampling point pushes the signal to the wrong side of the threshold for a 1 or 0 transmitted. Figure 5 illustrates the basics of the calculation. Essentially, at the sampling point in any bit period, the signal is varying randomly about some mean value due to the random noise fluctuations i.e. the sampled signal level is a random statistical variable about its mean. To determine the probability of error we need to know the signal probability distribution functions [p_i(v)dv, i = 0 or 1] for the 0 and 1 levels i.e. the probability that the sampled signal level will fall between level v and v + dv as a function of v for a received 0 and 1 signal, where v₀ and v₁ are the mean levels for a received 0 and 1 respectively. Illustrations of such functions are presented in Figure 5. It should be noted that the probability distribution functions (PDFs) for 0 and 1 received signals may be different since different noise processes may dominate the two regimes. This is reflected in Figure 5. However, for many applications, particularly when thermal noise limits apply, the PDFs are identical for both 0 and 1 levels.

Figure 5:

PDFs [p_i(v) dv_i, i=0,1] for levels of 0 and 1 (v₀ and v₁) in the presence of random (Gaussian) noise. The probabilities of error on a 0 signal (P₀-shaded) and 1 signal (P₁-hatched) are also indicated

The total probability that a 0 signal is recorded in error as 1 is the area under the probability distribution function, p₀(v).dv, for v > v_th (shaded area of Figure). Also, the total probability that a 1 signal is recorded in error as 0 is the area under the probability distribution function, p₁(v).dv, for v < v_th (hatched area of Figure). Hence, the probabilities (P_i) that a received signal of level i is recorded wrongly for i = 0 and i = 1 respectively are:

In the interests of simplicity, we shall adopt a useful approximation in assuming that the noise is a Gaussian random variable. This means that the PDFs [p_i(v)dv, i = 0 or 1] for the sampled signal voltage at the receiver output are Gaussian functions with a mean voltage of v_i and a variance of , where v_N is the root mean square (rms) noise voltage. Assuming equal numbers of 1s and 0s in the bit stream and evaluating the integrals of equations (2) and (3), the BER, or error probability, P_e can now be written as⁴

where, for thermal noise limited detection (equal noise on 0 and 1 levels)

For this parameter set, Q is in fact the signal to noise ratio and the threshold voltage is half of the received average 1 level. Hence we can easily determine the BER from equation (4).

Figure 6 shows the variation in the BER as a function of Q according to equation 4. The maximum acceptable bit error rate in a telecommunications system is 10^-9 and system specifications often demand BERs down to 10^-12. Such specifications imply the need for Q values of 6 and 7 respectively or SNRs of 12 and 14.

Figure 6:

Error probability P_e versus error probability factor Q

In addition to the quantitative approach above, much qualitative information may be gained from generating what is known as the eye diagram. To observe statistical variations in the received signal we need to be able to display many bits (hundreds or even thousands) of both 1s and 0s in various sequences all superimposed on one another. This can be done by transmitting a square pulse non return to zero (NRZ) pseudo-random bit stream (PRBS) and displaying the receiver output on the oscilloscope which is triggered by the PRBS clock. In such an arrangement, the oscilloscope displays the signal every few clock cycles or indeed every clock cycle depending on the timebase setting. If we now utilise the persist function many signal sequences are displayed superimposed on each other and we can see the effects of the statistical phenomena, such as noise and jitter, on the signal features.

BER (COM) in conjunction with ED COM enables the generation and investigation of eye diagrams for various conditions of signal attenuation, dispersion and noise. In addition, it enables the measurements and recording of the 0 and 1 sampled signal levels for a large number of bit periods. From these the noise probability distribution functions and the rms noise voltage may be determined allowing the Q factors and BER to be calculated.

Experimental Programme

The BER(COM) educator kit enables the generation of eye diagrams for all the system permutations of OptoSci’s Fibre Optic Communication educator kit (ED-COM). Under the experimental programme the students perform the following investigation:

Some of the results from a typical set of measurements obtained using ED-COM and BER(COM) will be examined in more detail in the following section.

Typical Eye Diagram Measurements

Figure 7 illustrates the typical measurements which are made on an eye diagram in order to provide semi-quantitative information about the transmission quality of a communications link.

Figure 7:

Sample measurements on eye diagrams.

Investigation of Eye Diagrams for LED Transmitter

The eye diagrams for LED transmission at 20 and 40 Mbit/s through each of 2 km, 3km and 4km of fibre are displayed in Figure 8.

Figure 8:

Eye diagrams for LED transmitter at 20 and 40 Mbit/s over 2km, 3km, and 4km.

As the transmission length increases at 20Mbit/s, the signal strength and SNR decrease due to signal attenuation. The effects of dispersion also become more evident with the rise time increasing with longer link lengths and the 80% pulse width decreasing as a percentage of the bit period. Pulse spreading with increasing rise times and decreasing width as a fraction of the bit period serves to increase the BER as the available sampling window in the centre of the pulse reduces. However, at 20Mbit/s the effects of dispersion on the BER are limited since the bit period is long and there is significant tolerance for the timing of the sampling point. The threshold level and the Q-factor decrease with length and of course this will be accompanied by an increase in the BER from this effect.

At 40Mbit/s as the transmission length increases, the signal strength and SNR similarly decrease due to signal attenuation. However, the effects of dispersion now become very significant. The rise time increases to equal the bit period and the 80% pulse width reduces to a small percentage of the bit period. Coupled with increasing pulse jitter and jitter on the sampling clock there is an increasing probability that the sampling point will be off centre and not coincident with the peak of the signal, resulting in increased BER. In addition, for distances of 3km and beyond, the effects of inter symbol interference (ISI) can be clearly seen. The pulses have spread significantly into adjacent bit periods resulting in a reduction of the peak signal strength and an increase in the noise level. These effects serve to take the noise closer to the threshold level, with concomitant degradation in the BER.

Investigation of Eye Diagrams for Laser Transmitter

Figure 9 depicts the eye diagrams for Laser transmission at 20 and 40 Mbit/s through each of 2 km, 3km and 4km of fibre.

Figure 9:

Eye diagrams for Laser transmitter at 20 and 40 Mbit/s over 2km, 3km, and 4km.

The trends as regards SNR, Q-factor and BER are very similar to those for the LED except that the laser power at the input is considerably greater. The main differences arise from the effects of dispersion. The rise time and the 80% pulse width do not change as significantly with length for the laser relative to those for the LED. This is indicative of much lower dispersion. The LED linewidth is very broad (>30nm) whilst the laser linewidth is narrow (~1nm). Hence, pulses from the LED experience a significant level of both material and inter-modal dispersion and spread much more quickly than the laser pulses for which material dispersion is insignificant due to the narrow laser linewidth. As a result of the reduced dispersion, it can be seen that the laser eye diagrams do not show any signs of significant ISI compared with the LED operating at 40Mbit/s at distances over 3km.

Investigation of Q-factor and BER as a function of increasing link length with the Q-factor Analysis Software

Using the Q-factor Analysis software described previously, the students establish the best fit Gaussian functions to the sampled data around the mean 0 and 1 signal levels in order to determine the Q-factor and the BER. Typical Q-factors and BER values for both the LED and laser transmitters operating at 40Mb/s over 1-5km are listed in Table 1.

Table 1:

Q-factor & BER variation with link length at 40Mb/s

High Q-factors in excess of about 10 and the corresponding BER values are pretty much meaningless essentially the BER is 0 in such situations. From Table 1 it can be seen that the Q-factor and BER both degrade as a function of increasing link length for both the LED and Laser transmitters, but those for the LED degrade much faster, particularly beyond 2km. Degradation occurs in both due to the attenuation in the fibre resulting in decreasing received signal power and SNR with increasing length. However, the degradation for the LED is more severe due to increased levels of pulse spreading and inter symbol interference. This is supported by the earlier risetime measurements, etc. and the evidence of ISI in the eye diagrams. There is no evidence of significant pulse spreading and ISI in risetime measurements or in the eye diagrams for the laser transmitter, so BER degradation for the laser arises mainly from attenuation. Pulses from the LED suffer greater spreading and ISI than those for the laser as a result of the increased linewidth of the LED output giving rise to a significant level of material dispersion in addition to intermodal dispersion. Material dispersion for the laser pulses is insignificant relative to intermodal dispersion due to the laser’s narrow linewidth and pulse spreading does not significantly degrade the BER at lengths up to 5km.

Investigation of Q-factor and BER as a function of increasing bit rate using the Q-factor Analysis Software

The measured Q-factors and BER values for both the LED and laser transmitters operating over 4km at bit rates of 10, 20, 30 and 40Mb/s are presented in Table 2.

Table 2:

Q-factor & BER variation with bit rate over 4km

For a fixed link length the total fibre attenuation is fixed and degradation of the Q-factor and BER with increasing bit rate can only be caused by pulse spreading and ISI. Clearly from Table 2 there is significant degradation for the LED transmitter but little or no degradation for the laser. Both the laser and LED output pulses are affected similarly by intermodal dispersion, but only the LED output experiences significant levels of material dispersion due to its large linewidth. The additional contribution from material dispersion leads to the Q and BER degradation seen for the LED transmitter, whereas the laser transmission, with only intermodal dispersion being significant, exhibits little degradation.