What if the bandpass and complex baseband signals are random processes? How are their statistics (autocorrelation, power density) related?

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Antonia Robbins
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1 .3 Bandpass Random Processes [P4.1.4].3-1 What if the bandpass and complex baseband signals are random processes? How are their statistics (autocorrelation, power density) related?.3.1 Complex Random Processes Consider the process t () = xt () + jyt (), where xt () and y() t are real processes characterized by 1 st and nd order statistics. o he mean is [ ] μ () t = E () t =μ () t + jμ y () t, and the ariance is defined x as σ () t = E () t μ () t. For simplicity, make zero mean, so σ () t = E () t. o Also assume xt () and y() t are wide-sense stationary, so Also, [ ] [ ] E x() t x( t τ ) = R ( τ) and E y() t y( t τ ) = R ( τ ) [ ] [ ] x E x() t y( t τ ) = E y() t x( t+τ ) = R () τ = R ( τ ) xy yx y.3-1

2 Normally, we work with circularly complex processes: Ett τ [ ( ) ( )] 0 circularly complex In detail, [ () ( τ )] = ( () + ())( ( τ ) + ( τ) ) Ett E xt jyt xt jyt ( ) = R () τ R () τ + j R () τ + R () τ 0 x y xy yx.3- o So circularly complex means R () τ = R () τ components hae same stats, een autocorr n x y R () τ = R () τ = R ( τ ) cross correlations are odd functions yx xy xy o he autocorrelation function of a complex signal is defined as ( )( ) * R () τ = E () t ( t ) τ = E x() t + jy() t x( t τ) jy( t τ) ( ) = R () τ + R () τ + j R () τ R () τ x y yx xy ( Rx jryx ) ( Ry jrxy ) = ( τ ) + ( τ ) = ( τ ) + ( τ) (Why?) o It s conjugate symmetric * R ( τ ) = R ( τ ), since real part is een and imaginary part is odd. Hence the power spectrum [ ] S ( f) = F R ( τ) is real (as a power spectrum should be)..3-

3 Symmetries and asymmetries: o From Fourier relationships (Appendix A), the power spectrum een if and only if the autocorrelation function makes components xt () and y() t uncorrelated. S ( f ) R () τ is real, which.3-3 is o If the mean frequency is not zero, then speed. t () f a = f S S ( f) df ( f) df has a net tendency to rotate in that direction and his is because discriminator output is proportional to * θ () t A () t = Im () t () t, where t () = Ate () jθ() t. hen relate the sign of E () t A () t θ to f S ( f ) df. An exercise..3-3

4 .3. Statistics of Complex Baseband and Bandpass Signals.3-4 Consider quadrature modulation and demodulation: How are the second order statistics of t () related to those of t ()? Start with the power in a bandpass signal jπ fct jπ fct P = E () t = E Re () t e Re () t e j4π fct = E () t + E Re () t e = E () t = P where the nd line comes from our useful identity and the 3 rd line comes from (a) cases in which the real and imaginary components of (t) are i.i.d. or (b) in all cases by time-aeraging the 3 rd line. In any case, the bandpass power equals the lowpass power. Also note that the lowpass power is the sum of the powers in the real and imaginary parts. If (t) is real, then it s just the power in the real signal..3-4

5 We can extend this approach to autocorrelation functions and power spectra: P E t R ( ) = E ( t ) ( t τ ) * = () τ.3-5 * Rw ( τ ) = E ( t ) w ( t τ ) [ τ ] R ( τ) = Ett ( ) ( ) j fct j fc( t ) E Re π ( t) e Re π τ = ( t τ ) e * jπ fcτ jπ fc( t τ) = E Re ( t) ( t τ) e + E Re ( t) ( t τ) e = 1 Re R ( ) τ e jπ f τ S ( f) = S ( f f ) + S ( f + f ) 1 1 c c c So the complex baseband equialent of the bandpass autocorrelation function is the lowpass autocorrelation function (diided by ), and the bandpass images are each half the height of the baseband image..3-5

6 he point? o Autocorrelation function and power spectrum of bandpass signal are related simply to complex baseband equialents. o Autocorrelation function and power spectrum of complex baseband signals are defined like the equialents for real lowpass signals just garnished with some conjugations and magnitudes..3-6 o So power calculation gies same result at baseband or passband autocorrelation at τ=0 or area under power spectrum. Unless each image of S ( f ) is symmetric about f c, the components and y() t are correlated. xt () So choice of f c can determine whether xt () and y() t are correlated, and whether t () tends to spin in one direction or the other..3-6

7 .3.3 Special Case White Noise.3-7 White noise is a conenient fiction. If the actual noise PSD is flat across the signal band, it will hae the same effect as white noise, which can be specified with just one parameter. Let s get the bandpass and complex lowpass relations right: he bandpass power and the complex lowpass power are equal, at NW, but the real and imaginary parts hae PSD N 0, so power NW 0 /. 0 he real and imaginary components each hae PSD with height N 0, just as for real lowpass systems. An easy link with the systems you analyzed as an undergraduate, where all signals were real..3-7

8 .3.4 Special Case Serial ransmission.3-8 he power spectral density (PSD) of a digital signal is quite different from that of an analog (noise-like) signal. Because digital signals hae more structure, their autocorrelation functions and power spectra must be defined carefully. hey are cyclostationary. Why do we care? Because the PSD gies us the bandwidth of the signal, and bandwidth is one of the two precious resources (the other is transmitter power). his topic can get complicated and take us far astray, so we ll examine only the simplest case a single pulse shape, amplitude modulated with multiple leels (M-PAM, but it also works for PSK and QAM). Shown below for PAM, with RC and with rectangular pulses. Below, we ll obtain the PSD for general pulse shapes..3-8

9 .3-9 he transmitted signal uses a basic pulse shape g() t s() t = A I g( t n) n n where A is an amplitude scale factor, g( t n) is the delayed pulse for symbol number n, and I n is the data alue for symbol n. he data alues are assumed to be independent and identically distributed (iid) and to hae zero mean, and are drawn from a finite set of alues (like 16QAM). his representation of the signal is not well normalized we haen t specified (for example) unit energy pulses or unit ariance data alues but that doesn t matter for this discussion. Let s start with an apparently simple question. What is the expected power of the signal? (Expectation oer the I n ensemble.) Written for complex, you can conert to real. * * P() t = E s() t = A E InImg( t n) g ( t m) n m = A E I I g( t n) g ( t m) * * n m n m * σi δnm n m = A g( t n) g ( t m) σ I = A g( t n) It still depends on time. n.3-9

10 he power is periodic with period. o see this, substitute t+ k for t, where k is any integer, and you get the same result o So power pulsates oer a symbol: 4 P(t) for RC pulses How did we get this graph? P() t t Great we can use this property for synchronizing the receier to the incoming signal. But does it pulsate for rectangular pulses? o If the moments of a process, including power (the ariance), are periodic, then the signal is cyclostationary..3-10

11 .3-11 For SNR calculations, we are interested in the time-aeraged power oer a symbol. hat is, 1 A σ I Pa = P() t dt g( t n) dt = n 0 0 ( n 1) A σ + I A σi = gt () dt = gt () dt n n σ I A = E g Back to our original goal the PSD of the signal. Work through the autocorrelation function. As in the power calculation, we hae Rs(, t t τ) = E s() t s ( t ) A E I I = g( t n) g ( t τ m) * * * τ n m n m * σi n = A g( t n) g ( t τ n) o he autocorrelation function depends on t as well as τ, so is not stationary. R (, t t τ ) is periodic in t, though (test this), so st () is cyclostationary. s.3-11

12 .3-1 o We can get rid of the t dependence by time-aeraging oer one symbol. Use mixed ensemble and time aerages now. ime aerage oer a symbol: ( n+ 1) σi * A σi * A Rs ( τ) = g( t n) g ( t τ n) dt = g( t) g ( ) t τ dt n 0 n n σi * A σi A = gt () g( t τ) dt Rg ( τ) = o It was worth all the work to get this nice, simple result: the autocorrelation function of the transmitted signal is proportional to the pulse autocorrelation function. Now the PSD is easy just Fourier transform: A σi A σi s = F[ s τ ] = F g τ = S ( f) R ( ) R ( ) G( f) Easy it s proportional to the squared magnitude of the pulse transform! In Appendix C, we define some common pulse shapes: rectangular, sinc, raised cosine and root raised cosine..3-1

13 .3-13 As an example, calculate the baseband and passband spectra for square root raised cosine signaling and a typical normalization: unit energy pulses and unit ariance data. o Area under G( f ) is one, so height : I o and σ = 1. o he power P s is the area under A = E s, the energy per symbol. Ss ( f), or A. Symbol rate is 1, so o Use these with spectra: σ I A Ss ( f ) = G( f ) to get the baseband, passband.3-13

Objectives. Presentation Outline. Digital Modulation Lecture 03

Objectives. Presentation Outline. Digital Modulation Lecture 03 Digital Modulation Lecture 03 Inter-Symbol Interference Power Spectral Density Richard Harris Objectives To be able to discuss Inter-Symbol Interference (ISI), its causes and possible remedies. To be able