When we talk about bit errors, we need to distinguish between two types of signals.

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1 All Aout Modulation Part II Intuitive Guide to Principles of Communications All Aout Modulation - Part II The main Figure of Merit for measuring the quality of digital signals is called the Bit Error Rate (BER) vs. E /N. Bit error rate is the ratio of numer of its received in error vs. total numer of its sent. When we talk aout it errors, we need to distinguish etween two types of signals.. A coded signal - This is a signal that has coding applied to it at the source so that only some of the its in the stream are important. We can tolerate a higher gross error rate in this channel, as long we can decode the information its with much etter integrity. The BER of the information its is much less than that of the coded signal. BER info its << BER coded its If E c /N is defined as the it energy of this signal, then E /N of the information its is approximately E /N = E c /N - Code rate (in dbs) A coded it stream may have a it error rate of - to -3, ut when coding is applied, the rate experienced y the information its is quite small, on the order of -4 to -.. An uncoded signal - There is no coding so all its are equally important and the stated BER is the BER of the information its, and E /N is equal to E c /N. Since most signals are coded, we need to keep in mind which BER we are talking aout. The BER of the received coded signal or the BER of the information its? Normally even that distinction is not there except in simulation and analysis ecause signals are soft-decision demodulated and this soft-decision is handed to the coding decoder for the final it decoding. For example, we have two QPSK links, of it rate R, one with coding of rate one-half and the other without. Both operate at an E /N of 7 db. Both have a gross BER of -3. Bit the one that is coded will ultimately give a BER of - on the information its. The only difference is that with link we get full R its of information whereas with the coded link the price of purity is that only half of the its are information its. Bit error statistics, although a discrete phenomena, are generally descried y continuous functions. In asence of unusual sources of interference, rf signals are suject primarily to Gaussian random errors. The Gaussian noise environment is coming from the hardware and from the air/medium in which the signal is traveling and these errors are generally random, coming in one's, and two's. A large numer of consecutive errors in a Gaussian environment are less likely. A lot of errors at once, called urst errors, are usually characterized as coming from fading, shadowing and Doppler effects. Errors don t actually occur, ut are made y the receiver. Prior to the receiver, the signal is in an analog waveform and in that context it errors have no meaning. How errors are made in the receiver is a function of modulation. Modulation determines how tightly various symols are packed in a signal space and how sensitive the receiver must e in making a decision. Let's egin y looking at the transmission of a polar NRZ signal through a communications channel. The signal has amplitude of ± volt, where v represents a it and a - v represents a it. Copyright Charan Langton

2 All Aout Modulation Part II Figure - Transmitted inary signal The signal picks up noise along the way and at the receiver we see instead of ± volts, amplitudes that include noise, sometimes higher than transmitted value, sometimes lower such that Received signal amplitude = + n, for it = s (t) Received signal amplitude = - + n, for it = s (t) where n represents the noise amplitude and is changing from sample to sample. (We are sampling each it more than once.) Figure - The inary signal has een acted upon y random noise. The received amplitude now varies at each sample. Making decision ased on the fluctuating values of Figure is like trying to read a tossed coin in a room where with each toss, the light level goes up or down randomly. Sometimes, the light is right enough and you can clearly read the coin and other times, it's too dark to read correctly. Each toss will e either a head or a tail, and equal numer of heads and tails are expected. The fact that an equal numer of heads and tails are expected is called the ase proaility of the process. But how we read the coin (or decode) depends on how right the light was at the moment of the toss. The fact that there are just two choices to pick etween (H or T) is akin to symol choices of a modulation. A modulation with fewer choices will lead to smaller errors, this is ovious. The value of noise, n varies with time and so is different from sample to sample. It is assumed to e normally distriuted with a zero mean and variance of quantityσ, where σ is the standard deviation of the noise process. Copyright Charan Langton

3 All Aout Modulation Part II 3 Figure 3 Effect of noise on the symol In Figure 3, we see two symols, a and symol with the transmitted shape and result of the addition of random noise at the receiver. Figure 4 Symols arrives in various states of distortion at the receiver. By sampling the amplitude at the decision instant, the receiver makes a decision aout what was sent. This decision is made strictly ased on the received amplitude. In Figure 4, we superimpose several received symols for it. We can see that the received amplitudes at the sampling instants vary. For most samples, the receiver would make a correct decision, ecause the added noise does not cause signal voltage to change polarity, ut there is one instance where noise is large enough that a positive transmitted voltage ecame a negative voltage in the eyes of the receiver. The receiver will make an error for this it. The receiver makes decisions y the following rules; If oserved voltage at the receiver is >, then transmitted it was it. If oserved voltage at the receiver is <, then transmitted it was it. We call this process Threshold Comparison. The selection of volts as a threshold is called the decision region oundary or Threshold, T. A threshold is like a net or a line etween the players in a game. The threshold is optimal when it is at the mean of the two values we are trying to distinguish etween. However this is true only if the apriori proaility of the two outcomes is equal. In a normal communication signal, the numer of 's sent (represented y a -v signal level) is approximately equal to numer of 's sent, so the optimum threshold is at volts (average of -v and +v). If at the time of decision, the value of noise n exceeds V volts, the transmitted level, then the oserved signal will flip from the correct decision region and a wrong decision will e made. We can write this proaility of error as Copyright Charan Langton

4 All Aout Modulation Part II 4 P( e ) = P( n < V ) Read this as: the proaility of making a decision error when a is sent is the proaility that noise at this moment is less than -V. It is ovious that the noise present must e negative in order to pull the positive voltage value into the negative region. The converse holds for the it. P ( e ) = P( n > V ) By Bayes Rule of proaility, which states that the overall rate of making a ad decision is P ( e) = P( e ) P() + P( e ) P() 3 This is called the conditional proaility. The proaility of making a decision error comes from making errors in interpreting oth and it. If equal numer of s and s are sent (s and s are equi-proale) then we can rewrite this equation as P ( e) = P( e ) + P( e ) 4 To evaluate this expression, we need to calculate error proailities P ( e ), P( e ) which maye assumed to e equal when the threshold is exactly in the middle. Gaussian distriution and density We know intuitively that proaility of an event is defined to e etween and. The mathematical definition of a Proaility Density Function, f(x), is that it satisfies the following properties.. The proaility that x, which can e any real quantity, positive or negative, is etween two points a and is Pa ( x ) = f( xdx ) 5 a. This proaility is a positive numer for all real x. 3. The area under the Proaility Density Function is. + f ( x) dx = 6 The variale x can e discrete or continuous. Height of a person is continuous, dice throws are discrete, as well as decoding of a or is considered discrete ut the received voltage may e thought of as a continuous variale. The Proaility Density Function (PDF), if you rememer from your statistics class, is a histogram. The y axis is the ration of desired events in the universe of events that take on the x value. For a Gaussian process, the proaility density function of Figure 5 is given y the expression ( x V) /σ Px ( ) = e 7 σ π where V is the average value and σ is the variance of the noise process. Two different distriutions are shown elow,. for height of women and. height of men in inches. Since the variance is different for them, the height distriution for men spreads out further. Copyright Charan Langton

5 All Aout Modulation Part II Height of men Height of women Figure 5 - Gaussian Proaility Density Function for variale x, the height of men and women in inches. If we integrate the PDF, we get its area. This integrated graph is called, the Cumulative Proaility Function, CDF is a very useful graph, in fact far more than the PDF. This is the graph we use to answer all kinds of useful questions such as aout BER of signals Figure 6 - Cumulative Proaility Function - Integral of the proaility density function Question: We ask, what is the proaility that a woman is etween 4 ' and 5'5" tall? By using the CDF in Figure 5 we can answer this question with ease. This is the difference etween the two y-axis values, one for 5'5" and the other 4'". P(4'" x 5'5") = F(5'5") F(4'").65 or (65%) The Cumulative proaility distriution of Figure 6 is given y x F( x) = f( x) dx 8 Integrating the PDF is difficult. The erf function which is approximated y Eq. can e used to develop the CDF. Here µ is the mean value of the function, and σ the standard deviation. Copyright Charan Langton

6 All Aout Modulation Part II 6 x µ µ x F( x) = erf erfc + = σ σ 9 The Error function is given y (an approximation) x n erf ( x) = e dn π The complementary error function, erfc function is equal to erfc( x) = erf ( x) The cumulative proaility function is always positive since proaility is always positive. This is intuitive. So are the erf and the erfc functions shown elow. We can however shift them around and normalize them to any mean value. The F(x) function shown elow is for mean value =..5 erfx erfcx F(x) 3 Figure 7 Plotting erf(x), erfc(x) and the Cumulative proaility distriution F(x) Bit Error Measurement To apply these ideas to determination of Bit Error rate, we start with the simplest of all modulations, BPSK. It has just two symols, one for and another for it. A simple link consists of a inary signal generator, a PSK modulator which multiplies the inary signal with a carrier which then picks up noise. The signals is then down converted to low-pass ut the noise remains. The signal is filtered y a filter called Integrate and Dump to give us the waveforms shown elow. Carrier Channel Noise Carrier Ideal Receiver Modulator Signal Signal Signal 3 Down Convert Signal 4 Integrate & Dump Signal 5 Threshold Comparator Signal with errors Copyright Charan Langton

7 All Aout Modulation Part II 7 Figure 8 A BPSK link (a) Signal - Baseand inary signal and its spectrum, () Signal - Modulated signal and its spectrum. Note that spectrum has shifted to carrier frequency. ( c) Signal 3 - Modulated signal with noise. The signal shape is the same as Signal and the spectrum is noisy. (d) Signal 4 - Converted signal at the front end of receiver, same as Signal ut has noise. (e) Signal 5 - Output of Integrate and Dump filter. Bit decision is made at then edge of the symol period ased on accumulated value. Figure 9 Signal transformation through the link Copyright Charan Langton

8 All Aout Modulation Part II 8 its its Figure - Threshold Comparator output =, Underlined its received in error. The Integrate and Dump filter works y adding up the individual voltage values at each sample during the it. In that it is an integrator. At the end of the it period it resets itself to start anew for the next it. The sum of all the values give a triangle-like shape and it decision is made ased on the last value. This asic link assumes that this system has infinite andwidth and has no other distortions such as intersymol interference, non-linearities, etc. And most importantly the noise is purely Gaussian, or AWGN, Additive White Gaussian Noise, and this is a very common assumption in communications. Below we draw two events; one of transmitting a signal of volt peak amplitude and the other a signal of - The figures shows three cases. In each case the nominal voltages are + and - volts ut the variance (the amount of noise) is different. In the first case it is, the second case it is.5 and third case. v. The pictures show the distriution of received voltages for each of the two signals. Proaility of mistaking a for a Proaility of mistaking a for a - Figure - The received voltage of two signals of and - volts are distriuted around the mean value with variance equal to the noise variance of the channel. Even though all three cases have the same mean values, they overlap y a different amount. The amount of overlap is a function of how much noise is present. The larger the noise variance, the larger the overlapping and intuitively that means a larger proaility of error. Computing the Bit Error Proaility How do we figure out the proaility of eing in this overlapped area? Let s take a inary signal, where a is sent y signal of amplitude V and a with amplitude of V. Copyright Charan Langton

9 All Aout Modulation Part II 9 Received amplitudes in this region mean a was sent. Received amplitudes in this region mean a was sent. Proaility of error of mis-decoding a it V T V Proaility of error of mis-decoding a it Figure Two signals with an aritrary threshold s is the signal received.(it is function of time t, ut I am leaving out the index to make it easier to read.) It includes noise, n (also function of time, t.) so that it is really equal to the original amplitude V plus noise. s () t = V + n() t and the noise is given y n = s V Similarly for a it, the received signal is a sum of the original amplitude of V plus noise. s = V + n 3 n= s V We set the decision oundary at an aritrary point etween the two signals, at amplitude T. A decision error is made when noise at the sampling instant exceeds one-half the difference etween the two discriminating amplitudes. It is also ovious we want this difference to e as large as possile to minimize errors. The BER, we will see shortly is a function of this distance etween the signals and we want this distance large. Signals like countries want as few neighors and as far away as possile! This noise level is given y n V V 4 If V = v and V = - v, then the noise level that will cause error is greater than. Two different errors can happen, one for mistaking a for a and vice versa. The proaility of each is the area under the curve starting at the threshold, T and going to infinity in the opposite direction as we can see in Figure. Let s write the expression for this error. We recognize that the received voltages are randomly distriuted with mean equal to the transmit voltage and variance equal to noise power. We use the proaility density function (Eq. 7) of a Gaussian variale as our starting point with mean of the process equal to V and V and s an s eing the received random variales. The variance of the process is σ. ( ) T ( s V ) /σ = 5 σ π Pe e ds This is equal to the shaded area shown on the left in Figure. Similarly, the proaility of error when a is sent is, shown in right shaded in Figure. Copyright Charan Langton

10 All Aout Modulation Part II ( s V ) /σ = 6 σ π Pe ( ) e ds T The total proaility of oth events, using Eq.7 is T ( s V ) /σ T ( s V ) /σ σ π σ π Pe () = e ds+ e ds 7 Without making it any more complicated, let me say that when we take the differential of the aove equations and set it to zero, we otain the following value for the optimum threshold, T. Threshold V + V = 8 Makes sense! This is the average value, assuming that oth s and s are equiproale as they usually are. Now sustitute the value of the threshold T, into the Eq. 7, and using Eq. 9 as a guide, we can write Eq. 7 in at least shorter if not totally friendly form as Pe () V V σ = erfc 9 This can also e expressed in the alternate Q function format, which is equal to the erfc function y erfc( x) = Q( x) Q function is often used in ooks ut since erf and erfc taulations are more readily availale in Excel and Matla, Q function is used less often. Here is the aove BER equation in Q format. Pe () V V σ = Q We can further manipulate this equation y noting that we can write this equation as Pe () where A = erfc σ A = V V By Parsevals Theorem, we determine the power of this difference signal as Copyright Charan Langton

11 All Aout Modulation Part II A = ( V V) dt = ( V) dt + ( V) dt ( VV ) dt = E + E E E σ 3 We note that it energy is defined as the product of it time with the square of the instantaneous amplitude squared. E V dt ( ) = 4 To deal with the last term in Eq, 3, we define correlation factor ρ, as ρ = VV dt 5 EE Now rewrite the Eq. 3 as Pe () = where E+ E EE ρ erfc N 6 N σ = The average it energy of a signal is the average energy of all its individual it energies. In this inary case, we have E E + E avg ( ) = 7 Now set ρ = ecause this will give us the conditions for the lowest BER. E ( R ) Pe () = erfc N 8 where EE R = + E E ρ 9 The value of R is lowest when E = E and ρ =, all these conditions are true for BPSK. Copyright Charan Langton

12 All Aout Modulation Part II So now we get the easy to use (Ha!) BER equation for a inary BPSK signal with unshaped pulses (unlimited andwidth), no distortions and Gaussian noise environment using an optimum receiver. We know that BPSK satisfies the important condition that its ρ = -, since cos wt and sin wt, the two pulses used are exactly anti-podal. Pe () E = erfc N 3 Since erfc is a monotonically decreasing function, we conclude. The it error rate experienced y the link is proportional to the distance or dissimilarity etween the symol.. The most dissimilar signals are BPSK ecause we use a sine and a cosine to represent the symols. Hence BPSK BER forms the limit of lowest BER for a signal. 3. All other modulations have a ρ which is larger than - and so their BERs must also e larger than BPSK. 4. The it error rate is proportional to the variance of the noise. Large variance means larger BER. Bit error of a QPSK signal A QPSK signal can e seen as two independent BPSK signal, on I and Q channels. If P ei is the proaility of error of the I channel (computed y Eq. 3 ) and P eq is the proaility of errors of the Q channels, then we can write the proaility of QPSK symol error (that is either the I or Q or oth channels make an incorrect decision, causing the QPSK symol to e mis-decoded.) Pe () = PeI + PeQ PP ei eq 3 Assuming that these proailities are small, and oth channels have same error rate, we can ignore the last term. Pe () P = P QPSK ei eq This says that the QPSK symol error rate is twice that of BPSK. Now write the error proaility of the QPSK. Es If P() e sym BPSK = erfc N Es then P() e sym QPSK = erfc 3 N Note that although I have een kind of loose with my terms, mentioning its and symols in the same reath, I need now to explicitly state that the error rate we computed for BPSK case was the error rate of its symol and not it. But since for BPSK, a symol is equal to just it, it causes no errors. That s no longer true for QPSK which has its per symol. An error in a symol can mean or errors. So the aove calculation is for a QPSK symol. To compute the it error rate, need to consider Gray coding. Copyright Charan Langton

13 All Aout Modulation Part II 3 Bit and Symol error rates for QPSK Example: Imagine two independent BPSK streams. We transmit its and see two errors in one and three in the other. The it errors rate individually is. and.3 and is.5 on the average. Figure 3 QPSK can e represented y two independent BPSK streams. Its BER is the sum of it errors of oth streams vs. total its. Now imagine the same thing ut as a QPSK signal, which is what it is when you have two BPSK streams together. Figure 4 Each it error represents a symol mistake in QPSK. Each time a it is decoded incorrectly, the whole symol is wrong. Now we have 5 symol errors. (φ = 9) (φ = 9) (φ = 8) QPSK Natural Ordering (φ = ) (φ = 8) QPSK Gray Coding (φ = ) (φ = 7) (φ = 7 ) Figure 5 a On the average if a symol is mistaken, there will e.33 it errors. If we consider only the adjacent symol errors, (ignore the red link) then on the average, there will e.5 errors per symol for a QPSK that is numered according to natural numering and one it per symol for Gray coded QPSK. If Symol is mis-decoded, then we can have on the average ++/3 =.33 its in error per miscoded symol If we ignore the possiility that symol will e mistaken for symol 3 they are 8 degrees apart in phase the average numer of its decoded incorrectly is.5. But we can reduce this BER y utilizing Gray coding, which will reduce average miscoded its down to one per symol. Copyright Charan Langton

14 All Aout Modulation Part II 4 If we organized the symols such that adjacent symols are only it apart then, decoding a symol incorrectly will lead to making only a it error. If adjacent symols are organized in a inary fashion, this is not the case and 5% of the time, a miscoded symol will mean two it errors. Gray coding helps us to organize our signaling such that adjacent symols are different y exactly one it and hence the Bit Error rate for QPSK is one-half the symol error rate. Since there are two its per symol, the BER of a QPSK when it is Gray coded can e written as E Pe () QPSK = erfc 33 N This is exactly the same as BPSK. Gray coding and 8-PSK modulation In 8-PSK modulation we transmit 8 symols. Each one of these symols represents 3 its. How do we decide which symol should represents which it pattern. The first choice is the inary order, called natural order here in Column. We can alternately also numer the symols y a different ordering, Gray coding as in Column 4. Tale I Natural vs. Gray coding of symols Symol Natural Bit Diff. to Gray Bit Diff. to Numer Ordering Next Coding Next Neighor Neighor Gray code ordering Natural ordering Copyright Charan Langton

15 All Aout Modulation Part II 5 Figure 6 - Mapping its to symols, natural and Gray Coding Natural Ordering Notice that symol differs from symol y only 45 degrees of phase. So is it likely that when an error is made, one symol will e mistaken for one that looks similar to it. And that means we may confuse symol for either 3 or symol. It is more unlikely that we will mistake symol for symol 6 as these two differ y 8 degrees. When a symol is misidentified then the it pattern that it stands for is also misdecoded. Symol 4 is mistaken for Symol 5; the it pattern of symol 4, is decoded as. Three it errors are made in the final information signal. Symol 4 is mistaken for Symol 3; the it pattern of symol 4, is decoded as. One it error is made in the final information signal. The average numer of its that are decoded incorrectly for this symol is its. Gray Coding Now examine a different type of mapping where adjacent it mappings differ y just one it. Now when a mistake is made in identifying a symol, then the maximum numer of its that will e wrong is one and not two as was the case for natural ordering. This way of mapping the it groups to the symols is called Gray coding. Gray coding is far more natural then inary system. In nature such as in DNA and chemical structures we find that Mother Nature prefers Gray coding and uses it lierally where errors can e devastating. Gray coding is used for nearly all digital modulations starting with QPSK Relationship of Bit Error Rate with Symol Error Rate If a symol consists of N its, and if it is decoded incorrectly then the net effect is that anywhere from to N its could e decoded incorrectly. Let's take a look at QPSK which has 4 symols. Since only one is correct, the proaility of a it error comes from Symol - it error out of Symol 3 - it error out of Symol 4 - it error out of So now we can write the it error as + P = 3 ( it error) + ( it error) ( it error) There are three symols that have errors, and each adds to the total it errors as shown aove. P = P s 34 3 Copyright Charan Langton

16 All Aout Modulation Part II 6 And if the signal is gray coded this improves y factor of N which is the numer of its per symol. M / P = P s 35 M The following tale gives a list of BER relationships. One of the outstanding things is that BER is a function of only one variale, E/N. However, don t forget that these are theoretical equations under ideal conditions, that is no ISI, no distortions, no filtering effects, no amplitude non-linearities, no fading. In fact when we incorporate all these effects, the required E /N to achieve a certain BER goes up consideraly. What these numers do provide is a aseline from which we can judge improvements and degradations. Inter-symol distance concept Constellation diagrams are very important in developing intuitive understanding of BER. The constellation lets us see the signals as population of ojects from which the receiver must pick out the correct one. How densely packed are the symols? This is oviously the key to how good the it error rate of a signal will e. We want a lot of signals ut we also want them to e different from each other so mistakes are not frequent. Note that the it energy of a modulated signal is equal to E = PT 36 c Where P c is carrier power and T the it time of the information signal. The carrier power is defined as A P c = The symol energy is defined as E = PT 37 s c s The it time and symol time of BPSK is the same, so its it and symol energies are the same. For QPSK, which has two independent I and Q channels, we have E ATs = PT = 38 s c s This energy is divided equally etween the I and Q channels, so we have E ATs = PT = s c s ATs AT E = 4 ATs AT E = 4 39 Bit times for QPSK are related y Copyright Charan Langton

17 All Aout Modulation Part II 7 T = T from which we get that s E = BPSK ( ) E = QPSK ( ) AT AT 4 Now lets examine the constellation of a BPSK signal. Each point is located at an amplitude equal to from the origin. E - E s E s E s Figure 7 The distance etween the two BPSK symols is the largest that is possile and that means that these two symols are least like each other. We define distance as the degree of dissimilarity etween signals. The distance of BPSK symols to the origin is and the distance etween the two points is E d = PT = E 4 c This distance although expressed as a scalar quantity is the property of the signal space. It has no physical meaning ut is very very helpful in understanding how different modulations result in different it error rates. Let's rewrite the BER equation of a BPSK signal y setting E = d Now sustitute d for E, E Pe () = erfc N d = erfc 4N 4 This equation tells us that the BER is directly related to intersymol distance d. Let's see how this applies to other PSK modulations. BER and QPSK inter-symol distance Copyright Charan Langton

18 All Aout Modulation Part II 8 E E The QPSK inter-symol distance is equal to Figure 8 QPSK inter-symol distance is same as BPSK E E 43 when we sustitute this into Eq. 4, we get d Pe () = erfc 4N 4E = erfc 4N = erfc E N 44 This is same as BPSK, again. BER and Inter-symol distance for 6-PSK Now examine a M-PSK signal space where M = 8. The signal space for a M-PSK is always a circle, and each it sequence of N its is represented y one of these M symols. M = N and M = 8, N = 3, for 8-PSK, where three its can e represented y one symol. Each signal is this circular space is located at a distance P sts from the origin. Each symol is 36/M degrees apart. So 8- PSK signals are 36/8 = 45 degrees apart. For small M, the distance d etween symols is given y d π = 4 NE sin 45 M for 8-PSK, M = 8 and N=3, we get d = E Plug this into equation 4, we get BER of 8-PSK, 46 Copyright Charan Langton

19 All Aout Modulation Part II 9 E Pe () = erfc N d = erfc 4N E = N erfc 4N 3 = erfc 3E N 47 Of course, we can extrapolate to this any M-PSK y using consistent values of M and N in equation. BER and Inter-symol distance of 6-QAM Now let's do M-QAM, specifically 6-QAM. We will compute its BER using the concept of the distance. Now we have 6 symols to transmit, with each symol standing for 4 its. Figure 6_QAM has three it sequence. N = 4, and M = 6 Computing its BER is different from 8- Figure 9 6-QAM constellation. Its inter-symol distance is smaller than 6-PSK. In PSK, all signals were at the same distance from its neighor. There was only one inter-symol distance to e computed. Here we have three different types of symols, those that are in the middle and have neighors on all sides, and those that are on the sides so have neighors on only one side and those in the corners. We assume that the inner most symol is at (a, a) volts. First we will compute the signal energy. Energy of a symol is equal to the sum the energy of all the individual symols. Another way to look at symol energy is to see it as the average energy of all different types of symols. In case of PSK, all symols have the same energy (since all the amplitudes are the same), PAM and others where the amplitudes vary would have different energies for the symols. So we write the average energy as E = M av E m M m= 48 Copyright Charan Langton

20 All Aout Modulation Part II The symol energy is just the sum of all the individual energies of the different looking symols. E Eav Eav = = 49 N log M Using a as the distance, we can express the average energy of the symol as E s = 4 = a Now we have [( a + a ) + (9a + a ) + ( a + 9a ) + (9a + 9a )] a =.Es d =.E s 5 And since in this modulation, each symol represents 4 its, we have E s = 4E d =.4 E 5 Compare this to the inter-symol distance of QPSK which is d = E And 6-PSK, which is π d = 6E sin =. 5E 6 5 The distance we just computed for 6-QAM is less than QPSK ut more than 6-PSK. And what this says is that 6-QAM has a BER that is in etween those two other schemes. The BER of a 6-QAM is computed y plugging in this value of inter-symol distance d in Eq 4. And we get, P( e) = erfc E N = erfc = 4 erfc = erfc d 4N.4E N.4E N 53 BER and inter-symol distance of a FSK signal Copyright Charan Langton

21 All Aout Modulation Part II E E E Figure FSK with two signals of two different frequencies In a FSK system, we have symol signals given y u ( t) = T cos π m f t c u ( t) = cos π n fct 54 T These are the FSK asis functions and are harmonics. The symol signals are now given y (after scaling with E ) s( t) = E u( t) s ( t t) = E u ( ) 55 The signals are orthogonal when n and m are integers. Now imagine just a two dimensional system. The signals lie on the two axes and the distance etween these two is easily computed as d = E 56 This distance is larger than 4-PSK ut smaller than 6QAM. Now the BER can e written as P e = erfc d N E Pe = erfc N 57 Relative Inter-symol distances Copyright Charan Langton

22 All Aout Modulation Part II Gray code ordering BPSK QPSK Gray code ordering 8PSK 6-QAM s Figure - Inter-symol distance decreases as numer of symols in a signal space goes up. The aove figure shows that for the same power signal, one with more symols will have smaller intersymol distance. This leads to larger it error rates. This prolem is excerated y other distortions making the power required to operate these modulations non-linearly related to the linear case. Other techniques can e applied to control the distortions ut there is no way to overcome the minimum required E /N for these modulations. Only with coding can we reduce the requirements down ut then we do pay the price of a smaller throughput. No pain, no gain. Copyright Charan Langton

23 All Aout Modulation Part II 3 Bit Error Rate.E-.E-.E-3.E-4 Summary.E-5.E-6.E-7 E/N vs. Bit Error Rate Threoretical Bounds BP SK, QP SK 3QAM 8PSK 6 PSK 3PSK FSK QPR.E-8.E-9.E E/N, db Figure BER curves for various modulations Tale II - BER Equations Modulation M = N, N Pe Bandwidth BPSK, / R E erfc N QPSK 4, / R E erfc / N SQPSK 4, / R E erfc / N MSK 4, / E R / erfc N M-PSK M, N / NE R /N erfc sin π / M N 6-QAM 6, 4 / E R /4 erfc.4 N Copyright Charan Langton

24 All Aout Modulation Part II 4 M-QAM M, N / 3 E erfc N N N QPR L levels / π E erfc 6 N LQPR L levels π / 6 ( log ) erfc L L 4 L M-FSK M, N / M N E erfc N / E N / R /N R /4 R /L M R /N Charan Langton Errors, corrections, comments, please me: mntcastle@earthlink.net Symols Bits AM FM PM A sin( ω t +φ ) A ( ω t +φ ) A sin ω t + φ A sin( ω t +φ) A sin ( ω t +φ) A sin( ω t + φ) 3 A3 sin( ω t +φ) A sin ( ω 3 t +φ ) A sin( ω t + φ3) 4 A4 sin( ω t +φ) A sin ( ω 4 t +φ) A sin( ω t + φ4) sin ( ) Copyright Charan Langton