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# Tele4653 l4

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### Tele4653 l4

1. 1. TELE4653 Digital Modulation & Coding PSD Wei Zhang w.zhang@unsw.edu.au School of Electrical Engineering and Telecommunications The University of New South Wales
2. 2. Outline PSD of Modulated Signals with Memory PSD of Linearly Modulated Signals PSD of CPM Signals TELE4653 - Digital Modulation & Coding - Lecture 4. March 22, 2010. – p.1/1
3. 3. PSD of Mod. Signal with MemoryAssume that the BP modulated signal v(t) with a LP equivalentsignal vl (t) as ∞ vl (t) = sl (t − nT ; In ) (1) n=−∞where sl (t; In ) ∈ {s1l (t), s2l (t), · · · , sM l (t)} is one of the possibleM LP equivalent signals determined by the informationsequence up to time n, denoted by In = (· · · , In−2 , In−1 , In ). Weassume that In is stationary process. TELE4653 - Digital Modulation & Coding - Lecture 4. March 22, 2010. – p.2/1
4. 4. PSD of Mod. Signal with MemoryThe autocorrelation function (ACF) of vl (t) is given byRvl (t + τ, t) = E[vl (t + τ )vl∗ (t)] (2) ∞ ∞ = E[sl (t + τ − nT ; In )s∗ (t − mT ; Im )] l n=−∞ m=−∞It can be seen that vl (t) is a cyclostationary process. Theaverage of Rvl (t + τ, t) over one period T is given by ∞ ∞ T¯ 1Rvl (τ ) = E[sl (t + τ − nT ; In )s∗ (t − mT ; Im )]dt l T n=−∞ m=−∞ 0 ∞ 1 ∞ = E[sl (u + τ − kT ; Ik )s∗ (u; I0 )]du l (3) T k=−∞ −∞ TELE4653 - Digital Modulation & Coding - Lecture 4. March 22, 2010. – p.3/1
5. 5. PSD of Mod. Signal with Memory ∞Let gk (τ ) = E[sl (t + τ ; Ik )s∗ (t; I0 )]dt. l (4) −∞The Fourier transform of gk (τ ) can be calculated as Gk (f ) = E [Sl (f ; Ik )Sl∗ (f ; I0 )] (5)Using (4) in (3) yields ∞ ¯ 1 Rvl (τ ) = gk (τ − kT ) (6) T k=−∞ ¯The Fourier transform of Rvl (τ ), i.e., PSD of vl (t) is given by ∞ 1 Svl (f ) = Gk (f )e−j2πkf T (7) T k=−∞ TELE4653 - Digital Modulation & Coding - Lecture 4. March 22, 2010. – p.4/1
6. 6. PSD of Mod. Signal with MemoryWe further deﬁne Gk (f ) = Gk (f ) − G0 (f ). (8)Eq. (7) can be written as (using G−k (f ) = Gk∗ (f )) ∞ ∞ 1 1Svl (f ) = Gk (f )e−j2πkf T + G0 (f )e−j2πkf T T T k=−∞ k=−∞ ∞ ∞ 2 −j2πkf T 1 k = Gk (f )e + 2 G0 (f )δ(f − ) T T T k=1 k=−∞ (c) (d) Svl (f ) + Svl (f ) (9)where (c) and (d) represent the continuous and the discretecomponents. TELE4653 - Digital Modulation & Coding - Lecture 4. March 22, 2010. – p.5/1
7. 7. PSD of Linearly Mod. SignalsFor linearly modulated signals (ASK, PSK, QAM), the LPequivalent of the modulated signal is of the form ∞ vl (t) = In g(t − nT ) (10) n=−∞where {In } is the stationary information sequence and g(t) is thebasic modulation pulse. Comparing Eq. (10) and (1), we have sl (t; In ) = In g(t) (11) TELE4653 - Digital Modulation & Coding - Lecture 4. March 22, 2010. – p.6/1
8. 8. PSD of Linearly Mod. SignalsUsing (11) in (5) yields Gk (f ) = E[Ik I0 |G(f )|2 ] = RI (k)|G(f )|2 ∗ (12)where RI (k) represents the autocorrelation function of {I n } andG(f ) is the FT of g(t). Therefore, using (7) and (12), the PSD ofvl (t) is ∞ 1 Svl (f ) = |G(f )|2 RI (k)e−j2πkf T (13) T k=−∞ 1 = |G(f )|2 SI (f ) (14) T TELE4653 - Digital Modulation & Coding - Lecture 4. March 22, 2010. – p.7/1
9. 9. PSD of Linearly Mod. Signals As can be seen from (14), the shape of PSD is determined by the shape of the pulse |G(f )| and the PSD of the sequence {In }, i.e., SI (f ). One method to control the PSD of the modulated signal is spectral shaping by precoding through controlling the correlation properties of the information sequence. For instance, a precoding form is Jn = In + αIn−1 . By changing the value of α, we can control the PSD. TELE4653 - Digital Modulation & Coding - Lecture 4. March 22, 2010. – p.8/1
10. 10. PSD of CPMThe CPM is expressed as s(t; I) = A cos[2πfc t + φ(t; I)] (15)where ∞ φ(t; I) = 2πh Ik q(t − kT ) (16) k=−∞The ACF of the LP equivalent vl (t) = ejφ(t;I) is given by ∞Rvl (t + τ ; t) = E exp j2πh Ik [q(t + τ − kT ) − q(t − kT )] k=−∞ ∞ = E exp {j2πhIk [q(t + τ − kT ) − q(t − kT )]} (17) k=−∞ TELE4653 - Digital Modulation & Coding - Lecture 4. March 22, 2010. – p.9/1
11. 11. PSD of CPMAssume the symbols in {Ik } are statistically i.i.d. withprobabilities Pn = Prob{Ik = n}, n = ±1, ±3, · · · , ±(M − 1).Taking expectation of (17) over the symbols {Ik }, we obtain Rvl (t + τ ; t)   ∞ M −1 =  exp{j2πhn[q(t + τ − kT ) − q(t − kT )]} k=−∞ n=−(M −1),n odd (18)Finally, the average ACF is T0 ¯ v (τ ) = 1 Rl Rvl (t + τ ; t)dt (19) T 0 TELE4653 - Digital Modulation & Coding - Lecture 4. March 22, 2010. – p.10/1
12. 12. PSD of CPMDeﬁne ΦI (h) the characteristic function of the random sequence{In } as M −1 ΦI (h) = E[ejπhIn ] = Pn ejπhn (20) n=−(M −1),n oddThen, the PSD of the CPM signal is given by [proof pp. 139-141] ∞Svl (f ) = ¯ Rvl (τ )e−j2πf τ dτ (21) −∞ (L+1)T ¯ LT ¯ Rvl (τ )e−j2πf τ dτ = 2 Rvl (τ )e−j2πf τ dτ + LT 0 1 − ΦI (h)e−j2πf T (22) TELE4653 - Digital Modulation & Coding - Lecture 4. March 22, 2010. – p.11/1
13. 13. PSD of CPFSKFor CPFSK, the pulse shape g(t) is rectangular and zero outsidethe interval [0, T ]. In this case, the PSD may be expressed as M M M 1 2 2Sv (f ) = T An (f ) + 2 Bnm (f )An (f )Am (f ) (23) M M n=1 n=1 m=1where sin π[f T − 1 (2n − 1 − M )h] 2 An (f ) = (24) π[f T − 1 (2n − 1 − M )h] 2 cos(2πf T − αnm ) − Φ cos αnm Bnm (f ) = (25) 1 + Φ2 − 2Φ cos 2πf T αnm = πh(m + n − 1 − M ) (26) sin M πh Φ Φ(h) = (27) M sin πh TELE4653 - Digital Modulation & Coding - Lecture 4. March 22, 2010. – p.12/1
14. 14. PSD of CPFSKThe PSD of CPFSK for M = 2, 4, and 8 is shown in next pagesas a function of f T with modulation index h = 2fd T as aparameter. The origin in the ﬁgures corresponds to the carrier f c . Only half of the bandwidth occupancy is shown. It shows that the PSD of CPFSK is smooth for h < 1, peaked for h = 1, and much broader for h > 1. In system design, to conserve bandwidth we have h < 1. TELE4653 - Digital Modulation & Coding - Lecture 4. March 22, 2010. – p.13/1
15. 15. M=2 from Digital Communications (5th Ed.) – John G. Proakis and Masoud Salehi
16. 16. M=4 from Digital Communications (5th Ed.) – John G. Proakis and Masoud Salehi
17. 17. M=8 from Digital Communications (5th Ed.) – John G. Proakis and Masoud Salehi
18. 18. PSD of MSK and OQPSKAs a special case of CPFSK, MSK has h = 1 . Then, the PSD is 2given by 2 16A2 Tb cos 2πf Tb Sv (f ) = (28) π2 1 − 16f 2 Tb2In contrast, the PSD of Offset QPSK is 2 sin 2πf Tb Sv (f ) = 2A2 Tb (29) 2πf TbThe PSD of the MSK and OQPSK signals are illustrated in theﬁgure on next page. TELE4653 - Digital Modulation & Coding - Lecture 4. March 22, 2010. – p.17/1
19. 19. from Digital Communications (5th Ed.) – John G. Proakis and Masoud Salehi
20. 20. PSD of MSK and OQPSKComparison of spectra: The main lobe of MSK is 50% wider than that for OQPSK. The side lobes of MSK fall off faster. MSK is signiﬁcantly more bandwidth-efﬁcient than OQPSK. TELE4653 - Digital Modulation & Coding - Lecture 4. March 22, 2010. – p.19/1