The bit error rate can be measured and plotted in terms of Carrier/Noise ( C/N ), or it can be in terms of E b /N o..

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Colin Ezra Andrews
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1 BER Testing - NRZ-MSB-Doubled 6/22/12 This report contains data obtained from measurements made in 2000 and 2004 using VMSK, MCM and NRZ-MSB ( degree ). New data is then added from June 2012 for NRZ-MSB doubled. Chapter 14. Modified from UNB Textbook Measuring Bit Error Rate ( BER ) The BER, or quality of the digital link, is calculated from the number of bits received in error related to the number of bits transmitted. BER= (Bits in Error)/(Total bits received) The bit error rate can be measured and plotted in terms of Carrier/Noise ( C/N ), or it can be in terms of E b /N o.. In order to obtain a level playing field for the comparison of digital modulation methods, engineers have adopted the E b /N o standard. That is - bit energy divided by the noise power that passes the bandpass of the filter. The bit energy E b is = (Signal Power)/(Data Rate (b/s) ** ) The noise power per Hz of bandwidth ( N o, or η ) is = (Noise Power)/(Noise Filter Bandwidth(Hz)) Noise /Hz must be multiplied by BW to obtain total noise power. SignalPower Eb BitRate Es NoisePower FilterBW Eb SignalPower C NoisePower N EQ. 1. ( If the bit rate and filter BW are equal ) If the noise BW equals the bit rate then E b /N o = C/N. The letter Tau is the time for 1IF cycle. Since the true bit rate on a cycle per cycle basis is equal to the IF frequency, which is equal to the sampling rate, which is equal to the zero group delay Nyquist filter bandwidth, the above equation is valid for MSB. ( BR = BW ). This is cycle by cycle calculation. Traditionally this is "Energy per Transmitted Symbol", where the sample is 1 symbol period. When using MSB modulation, this changes to "Energy per IF cycle, or energy per sample", since the zero group delay filter and synchronous detector sample at the IF frequency and not at the data symbol rate. The noise BW is also much smaller, but with a Nyquist BW equal to the IF. This is compensated for in the equations if the bit rate is assumed to be equal to the IF frequency instead of the actual bit rate, since theoretically, the sampling on each cycle could determine a new bit difference each IF sampled cycle. See post detection correction factor in Equation 7. 1

2 A 2 SignalPower SNR sin m Eq. 2 NoisePower Signal Power/Noise Power = SNR = C/N, for m = +-90 degree, or missing cycle modulation. The detector reference tuning can keep m at 90 degrees, which causes QPSK to have the same SNR as BPSK. ( Ref. 11 ). The same applies for 3PSK and NRZ-MSB The measurements of signal power and noise power are made with a True RMS voltmeter. The load impedance can be ignored since it is the same for both. The measured ratio is C/N, or (carrier volts)/(noise volts). The standard method used to measure E b /N o is to use a white noise generator having an output bandwidth at least 4 times the bandwidth of the receiver filter to insure uniform noise distribution. ( Ref. 3 ). Receiver Filter Noise Filter Measuring Eb/No Fig Title {T itle} Siz e D oc u ment N umber R ev {D o c } {R e v C ode } D ate: Sun day, D ec e mber 0 5, She et 1 of 1 If the measurements are made after the receiver filter, the measured C/N ratio can be used without bothering to calculate the actual values of E b, or N o. Fig Test Set Noise Generator Output. Note 5 db per div. scale. This is the noise spectrum utilized for MSB testing. The noise bandwidth spread in Fig is approximately 18 khz at the 3 db points, which is considered the same as a rectangular 2

3 filter BW, which is many times as wide as the 500 Hz 3dB noise bandwidth of the ultra narrow band filters The filter is a series emitter filter with ceramic resonator... ( The filter overload threshold from noise = -- approximately 16 db ). Filter overload - the reason for limiting noise BW, is discussed in Textbook Appendix 5. Do not measure with a fixed data pattern Carrier to Noise Ratio and erfc values db 4 db 6 db 8 db 10 db MCM-MSB BPSK-erfc Differential BPSK Measured for MSB ` Fig The Measured C/N for MCM or MSB and the theoretical value for BPSK. The MCM/MSB curve does not follow the BPSK curve due to the post detection factor The MSB curve applies to VMSK as well. The C/N measured here is the post detection C/N,( Eq. 7.) ( Ref 5 ). which accounts for the 3dB difference between this curve and Fig All UNB methods are 1 bit/sec./hz methods where C/N = E b /N o = SNR. ( Measurements - VMSK 1999, MCM 2000, NRZ-MSB 2004, NRZ-MSB doubled 2012 ). Measurements using either VMSK or MSB show this post detection value to be valid. A 10-6 BER when Q = 2.23 ( = 6.9 db ) instead of 3.28 is measured using RMS values for both E an N as shown in Fig See Fig Note slight difference in data here where 7.5 db = Q = E/N = It was found the LG - SA 7270 spectrum analyzer could be used instead of a true RMS voltmeter when set for 1 khz RBW and 10Hz VBW. 3

4 E/N E peak/n rms RMS/RMS 2 1 Post Det. RMS/RMS Figure The approximate Q values plotted for Signal E s and Noise. Note the convergence of Q probability toward zero for a 50% BER, whereas SNR would converge to 1.0. Q =.477 for 50% BER. The upper line ( Epeak/Nrms ) corresponds to the Q function table. The bit error rate for Ultra Narrow band modulation follows the Q probability curve. If the voltage measurements are Es peak and the noise is RMS, use the upper curve, which is the Q function plot. If both are RMS, as would be obtained from a true RMS meter, use the center curve. For post detection measurements, the E/N is.7 that for pre -detection, use the lower curve. See Eq. 7 at end. ( Eq ).( Ref 5 ) It makes little or no difference what the correction factor for Q is to obtain a BER above Shannon s Limit. Shannon s Limit is still E/N = 1, or 0 db SNR. ( Equations 15.1 and 15.9 ). P e = ½ erfc [SNR] ½ = ½ erfc [E s /N o ] ½ Eq. 3 These are power ratios, not voltage ratios for A/N as used below.. A 1 A Pe Q erfc 2 N = Probability of Error (BER ) Eq. 4. A = V signal voltage peak, σ = RMS noise voltage, N = peak noise voltage. 4

5 Note the correction for Peak or RMS above.( post detection 2E b /N o not included ). 1 z 1 z Q( z) 1 erf erfc erfc( z) 2Q 2 z ****** erf ( z) 12Q 2z **** see discrepancy note at end. z = peak signal voltage/rms noise voltage. Noise is assumed to be RMS., which becomes N = peak noise when multiplied by 1.4. See Figure 15.3 for measurements when both are RMS. Q is the Gaussian probability density function. erf is the error function. erfc is the complimentary error function.. The erfc curve is plotted in db in Fig and numerically as the center curve in There is a probability of error in using these equations if power and volts are mistakenly interchanged. P E 1 S volts erfc 2 N volts Eq. 5. Eb ( IF) E s E s BitRate Ns Ns Eq. 6. When using negative group delay filters, the Nyquist bandwidth for the noise filter BW is equal to the intermediate frequency. Bit rate = Sampling rate is also at the IF. All values are on a cycle by cycle basis. ( See Eq. 1 ). RMS voltage values are measured with a true RMS meter. Change this as necessary to get power levels. These changes are necessary for the exact P E. to match C/N. UNB modulation is end to end pulse width amplitude modulation. It is not phase modulation as normally produced and understood. The signal is generated by pulsing on phase one for a digital one and pulsing on phase 2 for a digital zero. If only ones are pulsed using phase one ( no signal = zero ), the signal is the same as and usable as ordinary AM pulsing. ( As in RADAR ). If both are pulsed using different phases, the signal becomes similar to BPSK. BPSK has no carrier, therefore the 90 degree phase modulation method of NRZ-MSB is used to keep the carrier. It is then doubled to obtain 180 degrees of phase shift. The C/N necessary for a 10-6 BER using AM with carrier is published as 13.5 db. The C/N necessary for 10-6 BER using BPSK ( Suppressed carrier ) is published as 10.5 db. When the sidebands are not used, the C/N for a given BER can be lower than for BPSK. VMSK for example, measures at 7.5 db for 10-6 BER., as do some other single frequency UNB methods. 5

Output of AD8306 as limiter and TRUE RMS measuring chip.. The AD8306 is used when the levels are too low for the HP3403 True RMS meter to read reliably. ( below 5 mv.

6 Output of AD8306 as limiter and TRUE RMS measuring chip.. The AD8306 is used when the levels are too low for the HP3403 True RMS meter to read reliably. ( below 5 mv.) True RMS measurement is obtainable from the RSSI of the AD8306. The scale is 20 millivolts per db. Readings are available below those obtainable from the HP

7 erfc table to be used in calculating P E from P e = ½ erfc [SNR] ½ = ½ erfc [A/N] SNR is the power ratio. A/N is a voltage ratio. Both values measured as true RMS. 7

8 Apeak 1 A peak 1 A rms Pe Q erfc erfc rms 2 Npeak 2 Nrms Quote from Bellamy ( Ref. 5 ) on Post detection correction. Equation C.34 pp492. Since pre-detection SNRs are measured prior to band limiting the noise, a noise bandwidth must be hypothesized to establish a finite noise power. Commonly, a bit rate bandwidth 1/T s, or a Nyquist baseband bandwidth 1/2T s, is specified. The latter specification produces----. SNR = 2(E b / N 0 ), where SNR is measured at the detector. It is called post detection SNR because it is at the output of the signal processing circuitry. ( Bellamy [15.7] Eq. C34 ). Eb SNR (2) Since P e = ½ erfc [SNR] ½ the post detection level is improved 3 db. This can be seen in Figures 14.3 and Eq C.34 Eq. 7. 8

9 A way to visualize this is to note that a data pattern waveform has a frequency of 1/2 the sampling rate. O, the baseband bandwidth is 1/2 the RF bandwidth. Discrepancy.: Conversion from Q to erfc does not correspond to data entered. For BER = 10-3, Q = erfc = erfc x 1.4 = 3.05, or 3.05/1.4 =2.17 which matches formulas. ***** is from references. Epeak 1 Epeak 1 Epeak 1 Erms Q 1 erf erfc erfc Nrms 2 2N N rms Nrms rms erfc( z) 2Q 2 z ****** error erfc( z) Q / 2 Q 2 erfc( z) **** 2 x 3.05(1.4(z)) from references does not match erfc. Other formulas are correct. References: 1) K. Feher, Wireless Digital Communications, Prentice Hall 2) D. Pleasant, Practical Simulation of Bit Error Rates, Applied Microwaves & Wireless Magazine, Winter 94. 3) E. Franke & J. Wunderlich, Practical BER Measurements. Paper- R.F. Expo, West, Jan ) A.B. Carlson, Communications Systems, McGraw Hill M-ary OFDM 5) J.C. Bellamy, Digital telephony, John Wiley Details of Pre and Post detection. 6) H.R. Walker, Modulation Analysis Vol 13, Encyclopedia of Electrical and Electronic Engineering, John Wiley -also Applied Microwaves and Wireless magazine, July/Aug 1997 (7) Mischa Schwartz, Information Transmission, Modulation and Noise. McGraw Hill Sampling and Shannon Details. (8) Proakis and Saleh, Communications System Engineering Prentice Hall, (9) K. Feher, "Telecommunications Measurements, Analysis, and Instrumentation" Noble Publishing, Atlanta, Ga. (10) R. E. Best, "Phase Locked Loops" McGraw Hill. (11) B. Sklar, "Digital Communications", Prentice Hall, Equipment Used: 1)HP3780A Pattern Generator/Error Detector ( Useful to 50 Mb/s) 2)NoiseCom AWGN generator with output to 100 MHz. -5 dbm peak, 3)Series Emitter bandpass filter to reduce noise BW to prevent receiver filter overload.( Pegasus ) 4)AD8306 Test Board. 5) NRZ-MSB Doubled signal modulator driven by HP3780A. UNB with all sidebands reduced. 6) UNB receiver at 48 MHz with phase shift detection and sidebands reduced more than 26 db. Filter 3dB BW = 500 Hz.. ( filter overload threshold from noise = -- approximately 16 db ). 9

NRZ 1 NRZ 1 NRZ 0 NRZ 0 NRZ 0

NRZ 1 NRZ 1 NRZ 0 NRZ 0 NRZ 0 Minimum Sideband Keying with NRZ Input. GPSK/GMSK Equivalent without Sidebands Showing the effects of negative group delay filtering on abrupt phase change modulation H.R. Walker (Reviewed //0) Abstract: