Power Spectral Density of Digital Modulation Schemes

Transcription

1 Digital Communication, Continuation Course Power Spectral Density of Digital Modulation Schemes Mikael Olofsson Emil Björnson Department of Electrical Engineering ISY) Linköping University, SE Linköping, Sweden January 016

2 Note: his material is used in the course SKS04 Digital Communication, Continuation Course, given by Department of Electrical Engineering ISY) at Linköping University, for engineering students at Master level. Additional Material for Digital Communication, Continuation Course: Power Spectral Density of Digital Modulation Schemes c Mikael Olofsson. Edited by Emil Björnson in 016. Department of Electrical Engineering ISY) Linköping University SE Linköping Sweden his document was prepared using L A EXε via Kile from toolscenter.org).

3 Contents A Power Spectral Density of Digital Modulation Schemes 1 A.1 Multi-Dimensional Pulse-Amplitude Modulation A.1.1 General expressions A.1. Important Special Cases A.1.3 Sine-Shaped Basis Functions A. One-Dimensional Constellations A..1 On-Off Keying OOK) A.. Binary Phase-Shift Keying BPSK) A..3 Amplitude-shift keying ASK) A.3 wo-dimensional Constellations using One Carrier Frequency A.3.1 Non-Binary PSK A.3. Quadrature Amplitude Modulation QAM) i

4 ii Contents

5 Appendix A Power Spectral Density of Digital Modulation Schemes We would like to determine the power spectral density PSD) of a digitally modulated signal. We are interested both in a general expression of the PSD and in explicit expressions for the standard signal constellations that we have considered previously. o determine an expression of the PSD of digital modulation, we need to consider not just one signal interval as in the course SKS01 Digital Communication. Instead we consider the input to be an infinite sequence A[n] of symbols. For the explicit expressions for standard signal constellations, we will assume that those symbols are independent and identically distributed and that the basis functions are orthonormal. A.1 Multi-Dimensional Pulse-Amplitude Modulation Consider a wide-sense stationary process which is an infinite sequence A[n] of symbols from the symbol alphabet used n =,...,+ ). We refer to this process as the information process. his sequence is first mapped onto an infinite sequence S[n] of N- dimensional signal vectors. Each dimension in this vector sequence is a real-valued timediscrete stochastic process S i [n] of its own, which we will refer to as a component process. Since the information process is wide-sense stationary, then the component processes are jointly stationary in the wide sense. hese vectors are then modulated using the N basis functions. For the n-th vector in that sequence, we use the time-shifted real-valued) basis functions{φ i t n Ψ)} N. HereΨisarandomdelay, independent ofa[n]foralln, and uniformly distributed over the signalling interval [0, ). his random delay is included to maintain the wide-sense stationarity, despite the periodic structure that modulated signal 1

6 Appendix A. Power Spectral Density of Digital Modulation Schemes. actually has. he resulting signal is then St) = n S i [n]φ i t n Ψ), A.1) that is, the component processes S i [n] are pulse-amplitude modulated using their basis functions as pulse shapes and then added to form St). Note that we are using the same Ψ in all dimensions to maintain orthogonality among the basis functions. A.1.1 General expressions We will first compute the autocorrelation function ACF) of A.1) and then use it to compute the PSD. We can write the ACF of the resulting signal St) as r S τ) = E { St+τ)S t) } { = E S i [n]φ i t+τ n Ψ) } Sl [m]φ l t m Ψ) n m l=1 { } = E S i [n]sl [m]φ i t+τ n Ψ)φ lt m Ψ). n m We use the linearity of the expectation to rewrite that as r S τ) = n m l=1 E { S i [n]sl [m]φ it+τ n Ψ)φ l t m Ψ)}. l=1 From the assumption that the random delay Ψ and the information process A[n] are independent of each other we find that also the signals S i [n] are independent of the delay. hen we get r S τ) = n m E { S i [n]sl [m]} E { φ i t+τ n Ψ)φ l t m Ψ)}. A.) l=1 We identify the first expectation in A.) as the cross correlation of two component processes. Recall that the component processes are jointly stationary in the wide sense. hus, this cross correlation is only a function of n m. We denote this as E { S i [n]s l [m]} = r Si,S l [n m]. he second expectation A.) is the expectation of a function of Ψ. Since Ψ is uniformly distributed over [0, ), its probability density is 1/ in that interval, and zero elsewhere. hese observations give us r S τ) = n r Si,S l [n m] m l=1 0 φ i t+τ n ψ)φ l t m ψ)1 dψ.

7 A.1. Multi-Dimensional Pulse-Amplitude Modulation. 3 We introduce the new variables k = n m and u = t+τ n ψ. hen we get r S τ) = 1 = 1 k k r Si,S l [k] n l=1 r Si,S l [k] l=1 t n +τ t n +τ φ i u)φ lu τ +k)du φ i u)φ lu τ +k)du. A.3) his is as far as we can get in computing the ACF, without making further assumptions around the correlation between the symbols in the information sequence and around the basis functions. Using the general ACF expression in A.3), the PSD of St) is the Fourier transform of r S τ). We get R S f) = 1 k r Si,S l [k] In the inner integral, we let v = u τ +k and then we have R S f) = 1 = 1 k l=1 r Si,S l [k] l=1 r Si,S l [k]e jπkf l=1 k φ i u)φ lu τ +k)du e jπτf dτ. φ i u)φ l v)e jπk+u v)f dudv φ i u)e jπuf du φ l v)ejπvf dv. A.4) We identify the two integrals at the second line of A.4) as the continuous-time Fourier transform of φ i t) and φ l t), respectively, and the innermost sum as the discrete-time Fourier transform of r Si,S l [k]. Hence, we have R S f) = 1 R Si,S l [f]φ i f)φ l f). Notice that we can also express this formula as the quadratic form R S f) = 1 ) R S1,S 1 [f] R S1,S N [f] Φ 1 f),...,φ N f)..... R SN,S 1 [f] R SN,S N [f] l=1 Φ 1 f). Φ N f) A.5). A.6) his PSD expression does not hold only for digital modulation. It is a general expression for simultaneous pulse-amplitude modulation of several stochastic processes using possibly different pulse shapes for each process. We stress that the basis function does not need to be orthonormal in order for A.5) and A.6) to hold.

8 4 Appendix A. Power Spectral Density of Digital Modulation Schemes. Real-valued signals In most cases we are only interested in real-valued signals. Assuming that all involved processes are real-valued, then their PSDs have to be real as well. Nevertheless, many of the terms in A.5) has non-zero imaginary parts, which means that all the imaginary parts of the terms R Si,S l [f]φ i f)φ l f) have to cancel out in the summation. Actually, they cancel out in pairs, which can be realized by observing that the Fourier transforms of the basis functions satisfy Φ i f)φ l f) = Φ l f)φ if)) and that the cross-correlations of the symbols satisfy R Si,S l [θ] = RS l,s i [θ]. herefore, the imaginary part of the mixed term R Si,S l [f]φ i f)φ l f) cancels out the imaginary part of R S l,s i [f]φ l f)φ i f). he terms on the main diagonal of the matrix in A.6) are all real and positive. his observation will have implications further on. A.1. Important Special Cases Independent symbols As stated in the introduction, we would like to express the PSDs of standard choices of signal constellations and basis functions, for the simple case where the information process A[n] consists of independent and identically distributed variables. his is a realistic assumption when the information process describes data that has been source coded to remove the correlation that might have existed in the original source. Next, we would like to determine all involved cross spectra R Sl,S i [f] for this assumed situation. We should notice that the assumption implies that the information process is stationary in the strict sense. he mapping to the vector sequence S[n] then results in component processes S i [n], i {1,,...,N},thatarejointlystationaryinthestrictsense. Moreover, theindependence of the symbols in the information process results in that S i [n 1 ] and S l [n ] are independent if n 1 n holds regardless of i and l. his means that the power spectral densities and cross spectra of the component processes become particularly simple. Let λ Sl,S i [k] = E{S l [n+k] m Sl )S i [n] m Si ) } denote the cross covariance of the two component processes and let m Sl = E{S l [n]} denote the mean value. hen the cross correlation is r Sl,S i [k] = E{S l [n+k]s i [n] } = λ Sl,S i [k]+m Sl m Si = λ Sl [n],s i [n]δ[k]+m Sl m Si, A.7) where λ Sl [n],s i [n] = λ Sl,S i [0] is the covariance of the stochastic variables S l [n] and S i [n]. By taking the Fourier transform of A.7) we have the cross spectrum R Sl,S i [θ] = λ Sl [n],s i [n] +m Sl m Si δθ m). A.8) m

9 A.1. Multi-Dimensional Pulse-Amplitude Modulation. 5 hus, all we need to do to determine those cross spectra is to analyze the cases k = 0 and k 0 separately. An important observation that we can make is that these cross spectra are all real-valued. We have observed the general relation which gives us r Sl,S i [k] = r Si,S l [ k] R Sl,S i [θ] = R S i,s l [θ]. But as noted, the imaginary parts are zero, so we have R Sl,S i [θ] = R Si,S l [θ]. Uncorrelated dimensions Next, we make another set of assumption namely that the components S l [n+k] and S i [n] are uncorrelated for i l for all k and that they have zero mean E{S i } = 0 for all i. In this special case, the cross correlation is { r Sl,S i [k] = E{S l [n+k]s i [n] r } = Si [k] if i = l, A.9) E{S l [n+k]}e{s i [n] } = 0 if i l. If we apply this property to A.6), then the general expression of the PSD collapses to R S f) = 1 R Si [f] Φi f) since all terms in the matrix then are zero except on the main diagonal, where we actually have the power spectral densities r Si [k] of the component processes. Special basis functions Again we assume that consecutive symbols are independent. Our observations that the imaginary parts of the mixed term R Si,S l [f]φ i f)φ l f) cancels out the imaginary part of R Sl,S i [f]φ l f)φ i f) and that the cross spectra R S l,s i [θ] are real-valued have important implications that help us identify another interesting special case. If Φ i f)φ l f) is purely imaginary for all i l then we have Φ i f)φ l f) = Φ lf)φ i f), A.10) and all mixed terms in A.5) cancel out. Once again, the general PSD expression collapses to R S f) = 1 R Si [f] Φi f). A.11)

10 6 Appendix A. Power Spectral Density of Digital Modulation Schemes. A.1.3 Sine-Shaped Basis Functions Our final goal is to derive explicit expressions for the PSDs of standard signal constellations. Since the general PSD expression in A.5) contains the Fourier transforms of the basis functions, we need to compute them for the standard choices of basis functions; namely sine-shaped basis functions with N =. Same frequency Consider the two basis functions φ 1 t) = cosπf 1t), for 0 t <, φ t) = sinπf 1t), for 0 t <, and assume, as usual, that the f 1 is a positive integer since this guarantees that the two basis functions are orthonormal. By noting that φ 1 t),φ t) are the product of a sinusoidal signal and a rectangular box with support between 0 and, we can use standard relations for the Fourier transform and the basis functions have the Fourier transforms Φ 1 f) = e jπf j f1 sinc f +f 1 ) ) +j f1 sinc f f 1 ) )), A.1) Φ f) = j e jπf j f1 sinc f +f 1 ) ) j f1 sinc f f 1 ) )). A.13) Note that we have utilized that e jπf 1 = e jπ/) f 1 = j f 1 since f 1 is an integer. We can see that we have Φ 1 f)φ f) = j sinc f +f 1 ) ) sinc f f 1 ) )), which obviously is purely imaginary. hus, the condition in A.10) is satisfied and in this two-dimensional situation the simplified PSD expression in A.11) becomes R S f) = 1 R S1 [f] Φ1 f) +R S [f] Φ f) ) for the case of independent symbols. Moreover, we have Φ1 f) = Φ f) = sinc f +f 1 ) ) + 1) f1 sinc f f 1 ) )), sinc f +f 1 ) ) 1) f1 sinc f f 1 ) )). A.14) A.15)

11 A.1. Multi-Dimensional Pulse-Amplitude Modulation. 7 Φ1 f) Φ f) 0 f 1 / Figure A.1: Illustration of the energy spectra Φ1 f) and Φ f) in A.14) and A.15), which are sine-based basis functions with same frequency. Most often the carrier frequency f 1 is much larger than 1/ which is proportional to the signal bandwidth). Consequently, f 1 is a large integer and the mixed terms in A.14) and A.15) will be small compared to the squared terms, andwe can use the approximation Φ1 f) sinc f +f 1 ) ) +sinc f f 1 ) )) Φ f). Based on this observation, we deduce that the bandwidth of a two-dimensional signal constellation using these basis functions is roughly the same as the bandwidth of a onedimensional signal constellation using one of those basis functions. his is illustrated in Figure A.1, which shows the spectra Φ1 f) and Φ f) around the positive carrier frequency f 1. Since these spectra are overlapping, it is very common to count pairs of dimensions instead of just dimensions when counting the bandwidth content. Note that each energy spectrum takes the form of a squared sinc-function, which theoretically has infinite bandwidth. However, 90% of the energy lies within the main lobe, which has a width of /, and this is a common approximation of the bandwidth.

12 8 Appendix A. Power Spectral Density of Digital Modulation Schemes. Different frequencies For basis functions using different frequencies, the situation is a bit more complicated than the above, as we will see next. Let f 1 and f be different positive integers, which again guarantees that the two basis functions φ 1 t) = cosπf 1t), for 0 t <, φ t) = cosπf t), for 0 t <, are orthonormal. Similar to A.1) using standard relations for the Fourier transform), we find that these signals have the Fourier transforms Φ 1 f) = e jπf j f1 sinc f +f 1 ) ) +j f1 sinc f f 1 ) )), Φ f) = e jπf j f sinc f +f ) ) +j f sinc f f ) )). We obviously have Φ 1 f)φ f) = j f 1+f sinc f +f 1 ) ) sinc f +f ) ) +j f 1 f sinc f +f 1 ) ) sinc f f ) ) +j f 1+f sinc f f 1 ) ) sinc f +f ) ) +j f 1 f sinc f f 1 ) ) sinc f f ) ) ). A.16) If both f 1 and f are even or both are odd, then the exponents f 1 +f, f 1 f, f 1 +f and f 1 f are all even. Consequently, all those powers of j are real, and the complete expression of Φ 1 f)φ f) is real. If instead one of f 1 and f is odd and the other even, then all those exponents are odd, and all those powers of j are imaginary and the whole expression of Φ 1 f)φ f) is purely imaginary. hus, still assuming independent symbols, in the first case both even or both odd) we have the general PSD expression R S f) = 1 R S1 [f] Φ1 f) +R S1,S [f]φ 1 f)φ f)+r S [f] Φ f) ) A.17) for this two-dimensional situation. In the second case, though, we have the PSD R S f) = 1 R S1 [f] Φ1 f) +R S [f] Φ f) ) A.18) for this two-dimensional situation according to our observations in Section A.1.. Note that A.18) can also be computed directly from A.17), by observing that Re{Φ 1 f)φ f)} is

13 A.1. Multi-Dimensional Pulse-Amplitude Modulation. 9 zero in the latter case. We can also observe that if f 1 and f are fairly large integers, then the second and third term in A.16) containing sinc f +f 1 ) ) sinc f f ) ) and sinc f f 1 ) ) sinc f +f ) ) are small compared to the two other terms containing sinc f +f 1 ) ) sinc f +f ) ) and sinc f f 1 ) ) sinc f f ) ) in the expression of Φ 1 f)φ f) above, since the offsets between the sinc-functions are smaller. herefore, we can approximate A.16) as Φ 1 f)φ f) j f 1 f sinc f +f 1 ) ) sinc f +f ) ) +j f 1+f sinc f f 1 ) ) sinc f f ) ) ). Furthermore, if the integers f 1 and f are fairly far apart, then we can also claim that those two remaining parts are small and we can even disregard Φ 1 f)φ f) altogether, regardless of which of the two observed cases we have. Similar to the case with same frequency, we have the energy spectra Φ1 f) = Φ f) = sinc f +f 1 ) ) + 1) f1 sinc f f 1 ) )), sinc f +f ) ) + 1) f sinc f f ) )), A.19) A.0) of the basis functions, and again we can use the approximations Φ1 f) sinc f +f 1 ) ) +sinc f f 1 ) )) 4 Φ f) sinc f +f ) ) +sinc f f ) )) 4 if f 1 and f are fairly large integers. o keep the total bandwidth of the signal low, we want the two carrier frequencies to be as close to each other as possible, but still chosen such that f 1 and f are integers to guarantee orthogonality. hus, we could choose them to fulfill f 1 = f ±1, or equally that f 1 = f ±1/). Comparison of Energy Spectra Next, we illustrate and compare the energy spectra Φ 1 f) and Φ f) in A.19) and A.0), respectively. Recall that these represented two cosine basis functions with different frequencies: f 1 and f. Figure A. shows these spectra at positive carrier frequencies, assuming a difference of 1/ between f 1 and f ; that is; f = f 1 + 1/. Note that this

14 10 Appendix A. Power Spectral Density of Digital Modulation Schemes. Φ1 f) Φ f) 0 f 1 f / Figure A.: Illustration of the energy spectra Φ1 f) and Φ f) in A.19) and A.0), which are sine-based basis functions with different frequencies. carrier separation is sufficient for having one cosine and one sine basis function at each of the frequencies and still guarantee orthogonality between all four of them. Each energy spectra in Figure A. has the form of a squared sinc-function. Around 90% of theenergylieswithinthemainlobe, which hasawidthof/ aroundthecarrier frequency. Hence, the approximate bandwidth of the linearly modulated signal sent over one of the basis function is /. Recall that the Nyquist criterion prescribes that a bandwidth of 1/ is sufficient for a pulse-amplitude modulated PAM) passband signal with symbol duration, which is substantially smaller bandwidth than what we obtained here. his is because we use a box-shaped pulse multiplied with a cosine) instead of a sinc pulse. Interestingly, the energy spectra of the two basis functions overlap substantially in Figure A., although the basis functions are orthogonal. In fact, even the main lobes overlap by 50 %. his has the important consequence that if we utilize K cosine basis functions with

15 A.. One-Dimensional Constellations. 11 frequencies f k = f 1 +k 1)/, then the total bandwidth of the signal will not be K /) i.e., adding each separate bandwidth) but rather K + 1)/. his is approximately equal to K/ if K is large and thus essentially the same as achieved by K ideal PAMs with sinc pulses. his principle is used in orthogonal frequency division multiplexing OFDM) to obtain orthogonal basis functions and to have relatively good bandwidth utilization. At the order of K = 1000 carrier frequencies are common in such systems. A. One-Dimensional Constellations For the one-dimensional case, N = 1, the general PSD expression in A.5) simplifies to R S f) = 1 R S 1 [f] Φ1 f), which we recognize as the expression that holds for pulse-amplitude modulation of a timediscrete stochastic process S 1 [n]. his should not be a surprise, since we are using pulseamplitude modulation in each dimension. We assume that f 1 is a positive integer, as before. Here it is not needed to guarantee any orthogonality, since we only have one dimension. Instead we simply demand that, in order to make the calculation of the signal energy simple. We have seen previously that the standard choice of basis function φ 1 t) = cosπf 1t), for 0 t < has Fourier transform Φ 1 f) = e jπf j f1 sinc f +f 1 ) ) +j f1 sinc f f 1 ) )), and energy spectrum Φ 1 f) = sinc f +f 1 ) ) + 1) f1 sinc f f 1 ) )). For an information process that consists of independent and identically distributed variables, we have seen in A.8) that the PSD of a component process is R S1 [θ] = σs 1 +m S 1 δθ m). A.1) If f 1 is even, such that f 1 is also an integer, then Φ1 f) = 0 for all integer values of f, except for f = ±f 1 which are the centers of the sinc-functions. Hence, only two m

16 1 Appendix A. Power Spectral Density of Digital Modulation Schemes. of the impulses in R S1 [f] survive the multiplication with Φ1 f), namely those at ±f 1. All other impulses in R S1 [f] coincide with zeros in Φ 1 f). he resulting PSD is R S f) = σ S 1 sinc f +f 1 ) ) + 1) f1 sinc f f 1 ) )) + m S 1 δ f +f 1 ) ) +δ f f 1 ) )) A.) Notice that 1) f 1 = 1 in this particular case, thus A.) can be further simplified. If on the other hand f 1 is odd, then no impulses in R S1 [f] coincide with zeros in Φ1 f). All impulses in R S1 [f] survive, but will be differently scaled. he dominating impulses are those near ±f 1, just as in the even case. We note that if m S1 = 0, then the impulses in A.1) vanish and we obtain R S f) = σ S 1 sinc f +f 1 ) ) + 1) f1 sinc f f 1 ) )) for both even and odd values of f 1. A.3) In the following, let E max and E avg denote the largest signal energy and the average signal energy, respectively, in the considered signal constellation. In the average case, we assume that all signals are equally probable. A..1 On-Off Keying OOK) Recall that the vectors used in on-off keying are 0) and E max ). Assuming independent and equally probable symbols, we get the mean m S1 = E max / and the variance σ S 1 = E max /4. Here we have E avg = E max /. Since the mean is non-zero, we only want to study the case where f 1 is even, to avoid the problems that we observed above for odd f 1. In this case, we can use A.) and immediately obtain the PSD R S f) = E max sinc f +f 1 ) ) + 1) f1 sinc f f 1 ) )) 8 + E max δ f +f 1 ) ) +δ f f 1 ) )). 8 Expressed in the average signal energy, we have R S f) = E avg sinc f +f 1 ) ) + 1) f1 sinc f f 1 ) )) 4 + E avg 4 δ f +f 1 ) ) +δ f f 1 ) )).

17 A.3. wo-dimensional Constellations using One Carrier Frequency. 13 A.. Binary Phase-Shift Keying BPSK) In BPSK we have the vectors ± E max ) as constellation points. Assuming independent and equally probable symbols, we get the mean m S1 = 0 and the variance E max. Here we have E avg = E max Since the mean is zero, we can allow f 1 to be any positive integer. Plugging the above into the one-dimensional PSD expression in A.3), we get R S f) = E max sinc f +f 1 ) ) + 1) f1 sinc f f 1 ) )). Expressed in the average signal energy, we have R S f) = E avg sinc f +f 1 ) ) + 1) f1 sinc f f 1 ) )). A..3 Amplitude-shift keying ASK) Many different ASK constellations can be considered. Here we focus at 4-ASK, in which we have the vectors ± E max ) and ± E max /3) as constellation points. Assuming independent and equally probable symbols, we get the mean m S1 = 0 and the variance 5 9 E max. Here we have E avg = 5 9 E max Since the mean is zero, we can allow f 1 to be any positive integer. Plugging the above into the one-dimensional PSD expression A.3), we get R S f) = 5E max sinc f +f 1 ) ) + 1) f1 sinc f f 1 ) )). 18 Expressed in the average signal energy, we have R S f) = E avg sinc f +f 1 ) ) + 1) f1 sinc f f 1 ) )). A.3 wo-dimensional Constellations using One Carrier Frequency Next, we consider the two-dimensional case with the two same-frequency basis functions φ 1 t) = cosπf 1t), for 0 t <, φ t) = sinπf 1t), for 0 t <,

18 14 Appendix A. Power Spectral Density of Digital Modulation Schemes. where f 1 is a positive integer. We have seen that the general PSD expression in A.6 simplifies into R S f) = 1 R S1 [f] Φ1 f) +R S [f] Φ f) ) for these basis functions. We have also seen in Section A.1.3 that these basis functions have Fourier transforms Φ 1 f) = e jπf j f1 sinc f +f 1 ) ) +j f1 sinc f f 1 ) )), Φ f) = j e jπf j f1 sinc f +f 1 ) ) j f1 sinc f f 1 ) )), and energy spectra Φ1 f) = Φ f) = sinc f +f 1 ) ) + 1) f1 sinc f f 1 ) )), sinc f +f 1 ) ) 1) f1 sinc f f 1 ) )). For an information process that consists of independent and identically distributed variables, we have seen in A.1) that the PSD of the component processes are R S1 [θ] = σs 1 +m S 1 δθ m), A.4) R S [θ] = σs +m S δθ m). m m A.5) When multiplying these expressions with the Fourier transforms of the basis functions, the situation is the same as in the one-dimensional case when it comes to even and odd f 1. In particular, for even values of f 1 the resulting PSD is R S f) = σ S 1 sinc f +f 1 ) ) + 1) f1 sinc f f 1 ) )) δ f +f 1 ) ) +δ f f 1 ) )) + m S 1 + σ S + m S sinc f +f 1 ) ) 1) f 1 sinc f f 1 ) )) δ f +f 1 ) ) +δ f f 1 ) )). Again, if f 1 is odd, then no impulses in R S1 [f] or R S [f] coincide with zeros in Φ1 f). All impulses in R S1 [f] survive, but are differently scaled depending on how far from the center of sinc-function it is. he dominating impulses are those near ±f 1. We note that if m S1 and m S are zero, then we do not have any of those problems.

19 A.3. wo-dimensional Constellations using One Carrier Frequency. 15 A.3.1 Non-Binary PSK For an M-ary PSK, we have M signal constellation points evenly distributed over a circle of radius E avg withitscenterintheorigin. NotethatE avg isboththemaximumandaverage energy. Both component processes have zero mean, and variance E/. he resulting PSD is R S f) = E avg sinc f +f 1 ) ) + 1) f1 sinc f f 1 ) )) 4 + E avg 4 = E avg sinc f +f 1 ) ) 1) f 1 sinc f f 1 ) )) sinc f +f 1 ) ) +sinc f f 1 ) )). A.3. Quadrature Amplitude Modulation QAM) here are QAM constellations of many different sizes. 4-QAM is identical to 4-PSK, thus we will instead consider 16-QAM in detail. In 16-QAM we have 16 points symmetrically placed on a rectangular grid. In each dimension, we have a 4-ASK modulation. Both component processes have zero mean, and variance 5E max /18. he resulting PSD is R S f) = 5E max sinc f +f 1 ) ) + 1) f1 sinc f f 1 ) )) E max sinc f +f 1 ) ) 1) f1 sinc f f 1 ) )) 36 = 5E max 18 sinc f +f 1 ) ) +sinc f f 1 ) )). Notice that we have E avg = 5 9 E max, just as in the case of 4-ASK. he PSD can also be expressed in terms of the average signal energy: R S f) = E avg sinc f +f 1 ) ) +sinc f f 1 ) )).