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. 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. PSD of Mod. Signal with Memory
Assume that the BP modulated signal v(t) with a LP equivalent
signal 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 possible
M LP equivalent signals determined by the information
sequence up to time n, denoted by In = (· · · , In−2 , In−1 , In ). We
assume that In is stationary process.
TELE4653 - Digital Modulation & Coding - Lecture 4. March 22, 2010. – p.2/1

4. PSD of Mod. Signal with Memory
The autocorrelation function (ACF) of vl (t) is given by
Rvl (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. The
average of Rvl (t + τ, t) over one period T is given by
∞ ∞ T
¯ 1
Rvl (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. 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. PSD of Mod. Signal with Memory
We further define
Gk (f ) = Gk (f ) − G0 (f ). (8)
Eq. (7) can be written as (using G−k (f ) = Gk∗ (f ))
∞ ∞
1 1
Svl (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 discrete
components. TELE4653 - Digital Modulation & Coding - Lecture 4. March 22, 2010. – p.5/1

7. PSD of Linearly Mod. Signals
For linearly modulated signals (ASK, PSK, QAM), the LP
equivalent 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 the
basic 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. PSD of Linearly Mod. Signals
Using (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 } and
G(f ) is the FT of g(t). Therefore, using (7) and (12), the PSD of
vl (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. 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. PSD of CPM
The 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. PSD of CPM
Assume the symbols in {Ik } are statistically i.i.d. with
probabilities 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. PSD of CPM
Define ΦI (h) the characteristic function of the random sequence
{In } as
M −1
ΦI (h) = E[ejπhIn ] = Pn ejπhn (20)
n=−(M −1),n odd
Then, 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. PSD of CPFSK
For CPFSK, the pulse shape g(t) is rectangular and zero outside
the interval [0, T ]. In this case, the PSD may be expressed as
M M M
1 2 2
Sv (f ) = T An (f ) + 2 Bnm (f )An (f )Am (f ) (23)
M M
n=1 n=1 m=1
where
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. PSD of CPFSK
The PSD of CPFSK for M = 2, 4, and 8 is shown in next pages
as a function of f T with modulation index h = 2fd T as a
parameter.
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. M=2
from Digital Communications (5th Ed.) – John G. Proakis and Masoud Salehi

16. M=4
from Digital Communications (5th Ed.) – John G. Proakis and Masoud Salehi

17. M=8
from Digital Communications (5th Ed.) – John G. Proakis and Masoud Salehi

18. PSD of MSK and OQPSK
As a special case of CPFSK, MSK has h = 1 . Then, the PSD is
2
given by
2
16A2 Tb cos 2πf Tb
Sv (f ) = (28)
π2 1 − 16f 2 Tb2
In contrast, the PSD of Offset QPSK is
2
sin 2πf Tb
Sv (f ) = 2A2 Tb (29)
2πf Tb
The PSD of the MSK and OQPSK signals are illustrated in the
figure on next page.
TELE4653 - Digital Modulation & Coding - Lecture 4. March 22, 2010. – p.17/1

19. from Digital Communications (5th Ed.) – John G. Proakis and Masoud Salehi

20. PSD of MSK and OQPSK
Comparison 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