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Manufactured signals
7 Manufactured signals Contents 7.1 Introduction ........................................................................................................ 269 7.2 Double side-band amplitude-modulated signal ........................................................ 270 7.3 Pulse-amplitude-modulated signal.......................................................................... 271 7.4 Direct-sequence spread-spectrum signal .................................................................. 277 7.5 Higher-order cyclic spectra of modulated signals ...................................................... 278 7.5.1 PAM signals................................................................................................. 278 7.5.2 QAM signals ................................................................................................ 280 7.5.3 DSB-AM signals ............................................................................................ 281 7.5.4 SSB signals .................................................................................................. 281 7.5.5 ASK signals.................................................................................................. 282 7.6 Cyclic spectral analysis of man-made signals ............................................................ 282 7.7 Proofs ................................................................................................................. 283 Abstract Cyclostationarity properties of man-made communications signals are considered. Secondorder cyclic statistics in both time and frequency domains are derived for the double side-band amplitude-modulated signal, the pulse-amplitude-modulated signal, and the direct-sequence spread-spectrum signal. Higher-order cyclic spectra are derived for PAM, QAM, DSB-AM, SSB, and ASK signals. Keywords Double side-band amplitude modulated, Pulse-amplitude-modulated, Direct-sequence spread spectrum, Higher-order cyclic spectra
7.1 Introduction Periodicities in the multivariate statistical functions of man-made signals used in communications, radar, sonar, and telemetry arise from the construction and/or subsequent processing of the signal, that is, from operations such as modulation, scanning, sampling, multiplexing, and coding. The corresponding cycle frequencies are related to parameters of the processes such as carrier frequency, baud rate, sampling frequency, and code rate. The shapes of the second- and higher-order cyclic statistical functions depend on the shapes of the base pulses and the correlation properties of the modulating sequences (Gardner, 1985, Chap. 12), (Gardner, 1987f), (Gardner et al., 2006, Sec. 5). Cyclostationary Processes and Time Series. https://doi.org/10.1016/B978-0-08-102708-0.00018-2 Copyright © 2020 Elsevier Ltd. All rights reserved.
269
270
Cyclostationary Processes and Time Series
The analytical cyclic spectral analysis of mathematical models of analog and digitally modulated signals was first carried out in (Gardner, 1985) for stochastic processes and in (Gardner, 1987e) and (Gardner et al., 1987) for nonstochastic time series. The effects of multiplexing are considered in (Bukofzer, 1990). Continuous-phase frequency-modulated signals are treated in (Chen, 1989), (Gournay and Viravau, 1998), (Viravau and Gournay, 1998), and (Napolitano and Spooner, 2001). The effects of timing jitter on the cyclostationarity properties of communications signals are addressed in (Gardner, 1987f), (Elmirghani et al., 1995), (Win, 1998). On this general subject, see the general treatments (Gardner, 1985), (Gardner, 1987f), (Brown, 1987) (Gardner, 1990b), (Gardner, 1991c), and also see (Gardner, 1994a), (Dragan and Yavorskii, 1982b), (Leduc, 1990). In this chapter, two fundamental examples of cyclostationary communication signals are considered and their second-order cyclic statistics are derived. Specifically, the double side-band amplitude-modulated signal and the pulse-amplitude modulated signal are considered. The derivations of the second-order cyclic statistics can be accomplished by using the results of Sections 3.2 and 3.3. Then a third example of a more sophisticated communication signal is considered. Higher-order cyclic spectra for several communications signals are derived in Section 7.5. Finally, references are provided where second- and higher-order cyclic statistics of several special cases of communications signals are derived. Several aspects related to the second- and higher-order cyclostationarity properties of communications signals are treated in the blog (Spooner, 2015).
7.2 Double side-band amplitude-modulated signal Let x(t) be the (real-valued) double side-band amplitude-modulated (DSB-AM) signal (Fig. 7.1) x(t) s(t) cos(2πf0 t + φ0 ) .
(7.1)
The cyclic autocorrelation function and cyclic spectrum of x(t) are (Gardner, 1987e) 1 Rxα (τ ) = Rsα (τ ) cos(2πf0 τ ) 2 1 α+2f0 α−2f0 (τ ) e−j 2πf0 τ e−j 2φ0 + Rs (τ ) ej 2πf0 τ ej 2φ0 Rs + 4 1 Sxα (f ) = Ssα (f − f0 ) + Ssα (f + f0 ) 4 α+2f0
+ Ss
(f + f0 ) e−j 2φ0 + Ss
α−2f0
(f − f0 ) ej 2φ0
respectively. If s(t) is a wide-sense stationary signal then Rsα (τ ) = Rs0 (τ ) δα and
(7.2)
(7.3)
Chapter 7 • Manufactured signals
271
FIGURE 7.1 DSB-AM signal x(t).
Rxα (τ ) =
Sxα (f ) =
⎧ ⎪ ⎨ ⎪ ⎩ ⎧ ⎪ ⎨ ⎪ ⎩
1 0 2 Rs (τ ) cos(2πf0 τ ) 1 0 ±j 2πf0 τ e±j 2φ0 4 Rs (τ ) e
0
1 0 0 4 Ss (f − f0 ) + Ss (f 1 0 ±j 2φ0 4 Ss (f ∓ f0 ) e
α=0 α = ±2f0
(7.4)
otherwise + f0 ) α=0 α = ±2f0
(7.5)
otherwise .
0
Thus, x(t) is cyclostationary with period 1/2f0 . In Fig. 7.2A the magnitude of the cyclic autocorrelation function Rxα (τ ), as a function of α and τ , and in Fig. 7.2B the magnitude of the cyclic spectrum Sxα (f ), as a function of α and f , are reported for the DSB-AM signal (7.1) with stationary modulating signal s(t) having triangular autocorrelation function.
7.3 Pulse-amplitude-modulated signal Let x(t) be the complex-valued pulse-amplitude modulated (PAM) signal (Fig. 7.3) x(t)
an q(t − nT0 ) .
(7.6)
n∈Z
In (7.6), q(t) is a complex-valued square integrable pulse and {an }n∈Z , an ∈ C, is an ACS sequence whose cyclostationarity is possibly induced by framing, multiplexing, or coding (Schell, 1995b), with (conjugate) autocorrelation
α αn E an+m an(∗) = (m) ej 2π R aa (∗) α ∈A
(7.7)
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Cyclostationary Processes and Time Series
FIGURE 7.2 DSB-AM signal (7.1). (A) Magnitude of the cyclic autocorrelation function Rxα (τ ), as a function of α and τ , and (B) magnitude of the cyclic spectrum Sxα (f ), as a function of α and f . Source: (Gardner et al., 2006) Copyright of Elsevier.
(conjugate) cyclic autocorrelation N
1 αn E an+m an(∗) e−j 2π N→∞ 2N + 1
α (m) lim R aa (∗)
(7.8)
n=−N
and countable set of (conjugate) cycle frequencies α A α ∈ [−1/2, 1/2) : R (m) = 0 . (∗) aa
(7.9)
The (conjugate) autocorrelation function of x(t) is given by (Section 7.7) E x(t + τ ) x (∗) (t)
α αn = (m) ej 2π q(t + τ − (n + m)T0 ) q (∗) (t − nT0 ) . R aa (∗) m∈Z α ∈A
n∈Z
(7.10)
Chapter 7 • Manufactured signals
273
FIGURE 7.3 PAM signal x(t) with stationary white binary modulating sequence {an } and bit period T0 = 16Ts . (Top) full duty-cycle rectangular pulse; (middle) 50% duty-cycle rectangular pulse; (bottom) raised cosine pulse.
The (conjugate) cyclic autocorrelation function and (conjugate) cyclic spectrum of x(t) are (Section 7.7)
1 α (m) r α (∗) (τ − mT0 ) Raa (∗) α =αT0 qq T0 m∈Z
1 α α Sxx (f ) = (ν) Q(f ) Q(∗) ((−)(α − f )) S (∗) (∗) ν=f T0 , α =αT0 T0 aa
α Rxx (∗) (τ ) =
(7.11) (7.12)
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Cyclostationary Processes and Time Series
respectively, where α Saa (∗) (ν)
α (m) e−j 2πνm R aa (∗)
(7.13)
m∈Z
is the (conjugate) cyclic spectrum of the sequence {an }n∈Z , Q(f ) q(t) e−j 2πf t dt R
(7.14)
and (∗) α (−τ ) ej 2πατ rqq (∗) (τ ) q(τ ) ⊗ q = q(t + τ ) q (∗) (t) e−j 2παt dt . R
(7.15)
From (7.11) and (7.12) it follows that x(t) exhibits (conjugate) cyclostationarity at (con where mod 1 denotes the modulo 1 jugate) cycle frequency α only if (αT0 mod 1) ∈ A, operation with values in [−1/2, 1/2). If the sequence {an }n∈Z is wide-sense stationary, the (conjugate) autocorrelation (7.10) specializes into E x(t + τ ) x (∗) (t)
0 (∗) (m) = q(t + τ − (n + m)T0 ) q (∗) (t − nT0 ) (7.16a) R aa m∈Z
=
n∈Z
0 (∗) (m) repT R 0 aa
q(t + τ − mT0 ) q (∗) (t)
(7.16b)
m∈Z
where repT0 [·] denotes periodic replication with period T0 (Section C.2). Thus, x(t) is cyclostationary with period T0 . If the sequence {an }n∈Z is wide-sense stationary and white, then α 0 (∗) (0) δ( (m) = R R α mod 1) δm . aa (∗) aa
In such a case, the (conjugate) autocorrelation specializes into
0 (∗) (0) repT q(t + τ ) q (∗) (t) E x(t + τ ) x (∗) (t) = R 0 aa
(7.17)
(7.18)
and the (conjugate) cyclic autocorrelation and the (conjugate) cyclic spectrum of x(t) become α Rxx (∗) (τ ) = α Sxx (∗) (f ) =
0 (∗) (0) R aa T0
0 (∗) (0) R aa T0
α δ(αT0 mod 1) rqq (∗) (τ )
(7.19)
δ(αT0 mod 1) Q(f ) Q(∗) ((−)(α − f ))
(7.20)
Chapter 7 • Manufactured signals
275
FIGURE 7.4 PAM signal (7.6) with full duty-cycle rectangular pulse (Fig. 7.3 (top)). (A) Magnitude of the cyclic autoα (τ ), as a function of α and τ , and (B) magnitude of the cyclic spectrum S α (f ), as a function correlation function Rxx ∗ xx ∗ of α and f .
respectively. According with the stated cyclostationarity with period T0 , x(t) exhibits cyclostationarity with cycle frequencies α = k/T0 , k ∈ Z. α (τ ), as a function of α and τ , The magnitude of the cyclic autocorrelation function Rxx ∗ α and the magnitude of the cyclic spectrum Sxx ∗ (f ), as a function of α and f , are reported for the PAM signal (7.6) with stationary white modulating sequence {an }n∈Z , and rectangular pulse q(t) rect((t − Tw /2)/Tw ) with width Tw . In this case, Q(f ) = Tw sinc(f Tw ) e−j πf Tw and (7.15) reduces to α −j πα(Tw −τ ) rqq ∗ (τ ) = e
τ rect 2Tw
|τ | 1− Tw sinc(α(Tw − |τ |)) . Tw
(7.21)
In Fig. 7.4, the case of a full duty-cycle (Tw = T0 ) pulse is represented. For τ = 0 it results α (0) = 0 ∀α = 0. x 2 (t) = 1 (constant ∀t). Thus Rxx ∗ In Fig. 7.5, the case of a 50% duty-cycle (Tw = T0 /2) pulse is represented. α (τ ), as a function In Fig. 7.6A the magnitude of the cyclic autocorrelation function Rxx ∗ α of α and τ , and in Fig. 7.6B the magnitude of the cyclic spectrum Sxx ∗ (f ), as a function of α and f , are reported for the PAM signal (7.6) with stationary white modulating sequence
276
Cyclostationary Processes and Time Series
FIGURE 7.5 PAM signal (7.6) with 50% duty-cycle rectangular pulse (Fig. 7.3 (middle)). (A) Magnitude of the cyclic α (τ ), as a function of α and τ , and (B) magnitude of the cyclic spectrum S α (f ), as a autocorrelation function Rxx ∗ xx ∗ function of α and f .
FIGURE 7.6 PAM signal (7.6) with raised cosine pulse (Fig. 7.3 (bottom)). (A) Magnitude of the cyclic autocorrelation α (τ ), as a function of α and τ , and (B) magnitude of the cyclic spectrum S α (f ), as a function of α and f . function Rxx ∗ xx ∗
Chapter 7 • Manufactured signals
277
{an }n∈Z , and raised cosine pulse ηπt cos t T0 q(t) sinc 2ηt 2 T0 1− T0
(7.22)
where 0 η 1 is the roll-off factor. In this case, ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
T0
T0 πT0 1 − η Q(f ) = 1 + cos |f | − ⎪ 2 η 2T0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 0
|f |
1−η 2T0
1+η 1−η |f | 2T0 2T0 |f |
(7.23)
1+η 2T0
is strictly band limited and the cyclic autocorrelation and cyclic spectrum are nonzero only for α = 0 and α = ±1/T0 .
7.4 Direct-sequence spread-spectrum signal Let x(t) be the short-code direct-sequence spread-spectrum (DS-SS) baseband PAM signal x(t)
ak q(t − kT0 ) ,
(7.24)
k∈Z
where {ak }k∈Z , ak ∈ C, is an ACS sequence, T0 is the symbol period, and q(t)
N
c −1
cn p(t − nTc )
(7.25)
n=0
is the spreading waveform. In (7.25), {c0 , . . . , cNc −1 } is the Nc -length spreading sequence (code) with cn ∈ C, and Tc is the chip period such that T0 = Nc Tc . The (conjugate) cyclic autocorrelation function and (conjugate) cyclic spectrum of x(t) are (Napolitano and Tanda, 2001)
1 α α r α (∗) (τ − mT0 ) ⊗ γcc Raa (∗) (m) (∗) (τ ) α =αT0 pp T0 m∈Z
1 α α Sxx (f ) = (ν) S (∗) (∗) ν=f T0 , α =αT0 T0 aa (∗) α P (f ) P ((−)(α − f ))
(ν) cc(∗)
α Rxx (∗) (τ ) =
ν=f Tc , α =αTc
(7.26)
(7.27)
278
Cyclostationary Processes and Time Series
where α γcc (∗) (τ )
N
c −1 N c −1
n1 =0 n2 =0
cn1 cn(∗) e−j 2παn2 Tc δ(τ − (n1 − n2 )Tc ) 2
(7.28)
and α
(ν) C(ν) C (∗) ((−)( α − ν)) cc(∗)
(7.29)
with C(ν)
N
c −1
cn e−j 2πνn .
(7.30)
n=0
Second-order cyclic spectral analysis of more general classes of DSSS signals is addressed in (Fusco et al., 2006).
7.5 Higher-order cyclic spectra of modulated signals In this section, results of Section 4.4 are exploited to derive higher-order cyclic crossspectra of modulated signals. Higher-order cyclic spectra of single signals are easily obtained by considering xm ≡ x, ∀m in the derived relationships.
7.5.1 PAM signals Let us consider N PAM signals xPAMm (t)
+∞
xm (kTm ) pm (t − kTm ) = xm,δ (t) ⊗ pm (t),
m = 1, ..., N
(7.31)
k=−∞
where the continuous-time signals xm,δ (t) xm (t)
+∞
δ(t − kTm ) = xm (t)
k=−∞
+∞ 1 j 2πrt/Tm e , Tm r=−∞
m = 1, ..., N
(7.32)
are obtained by sampling the continuous-time signals xm (t) with rates 1/Tm (m = 1, ..., N ) and, in the second equality, the Poisson formula (C.8a) is exploited. The RD-CSCMF of the N PAM signals, according to (4.92), can be written as SxαPAM (f ) = PN (α
T
− f 1)
N−1
Pm (fm ) Sxαδ (f )
(7.33)
m=1
where subscripts x PAM ≡ xPAM1 . . . xPAMN , x δ ≡ x1,δ . . . xN,δ , and Pm (f ) denotes the Fourier transform of pm (t). The RD-CSCMF of the signals (7.32) can be expressed in terms of the
Chapter 7 • Manufactured signals
RD-CSCMF of the signals xm (t) (Napolitano, 1995) N
α−r T f 1 α Sx δ (f ) = Sx1 ,...,xNs (f − r ◦ f s ) Tm N m=1
279
(7.34)
r∈Z
where f s [1/T1 , ..., 1/TN ]T , r [r1 , ..., rN ]T , and ◦ denotes the Hadamard matrix product, i.e., r ◦ f s [r1 /T1 , ..., rN−1 /TN−1 ]T . Accounting for the rhs of (7.32), Eq. (7.34) is derived by substituting σm = rm /Tm (with rm any integer number) and cσm = 1/Tm into (4.99). Let xm,d (n) xm (nTs ), n ∈ Z, be the discrete-time signals obtained by uniformly sampling with period Ts the continuous-time signals xm (t). We have that (7.34) with Tm = Ts , m = 1, ..., N , compared with (4.112), yields αf α (ν ) = Ts Sx δ s (f ) |f =ν fs , Sx d
(7.35)
where subscript x d ≡ x1,d · · · xN,d . Let us consider now the N PAM signals xPAMi (t) =
+∞
xi (n) pi (t − nTs )
i = 1, ..., N
(7.36)
n=−∞
whose modulating sequences are N discrete-time signals x1 (n), ..., xN (n) jointly white and wide-sense stationary of order N , i.e., (Dehay et al., 2018b, Definition 4.1) cx δm −1mN Cx (n, m) cum {xi (n + mi ), i = 1, . . . , N } =
(7.37)
cx is the N th-order cross-cumulant of x1 (n), ..., xN (n). where subscript x ≡ x1 · · · xN and The CTCCF of the modulating signals is given by β cx δm −1mN δβ mod 1 Cx (m) =
where
δa mod b
1
a/b ∈ Z,
0
otherwise.
(7.38)
(7.39)
β
x (m ), straightforwardly derivable By Fourier transforming the discrete-time RD-CTCCF C from (7.38), one obtains the CCP xβ (ν ) = cx δβ mod 1 . P
(7.40)
Hence, by substituting (7.40) into the analogous of (7.35) in terms of CCPs and the result into the analogous of (7.33) in terms of CCPs, one has Px PAM (f ) = fs cx PN (β − f T 1) β
N−1 m=1
Pm (fm ) δβ mod fs .
(7.41)
280
Cyclostationary Processes and Time Series
Note that the relations stated in terms of CTCCFs and CCPs for PAM signals modulated by jointly white and wide-sense stationary discrete-time signals cannot be written in terms of RD-CSCMFs by simply substituting CCPs with RD-CSCMFs as in the case of continuous-time modulating signals (see also Remark 4.28). In fact, for the discrete-time TCMF, no simple relation analogous to (7.37) holds.
7.5.2 QAM signals Let us consider N quadrature-carrier amplitude-modulation (QAM) time-series xQAMm (t) xc,m (t) cos(2πfc,m t + φm ) − xs,m (t) sin(2πfc,m t + φm ),
m = 1, ..., N .
(7.42)
These QAM signals can be considered as the outputs of a continuous-time MIMO LAPTV system excited by x(t) [x1 (t), ..., x2N (t)]T , where x2m−1 (t) = xc,m (t)
(7.43)
x2m (t) = xs,m (t) .
(7.44)
The input/output relationship of the system is given by (4.94), where the nonzero elements of the N × 2N matrix C(t) are given by cm,2m−1 (t) = cos(2πfc,m t + φm )
(7.45)
cm,2m (t) = − sin(2πfc,m t + φm ) .
(7.46)
Therefore, from (4.98), letting σmim = ±fc,m , it follows that (Napolitano, 1995) SxαQAM (f ) =
2 1
··· 2N i1 =1
α−f Tcq i
2N
iN =2N−1 q1i1 =±1
,...,iN
· Sxi1 ,...,xi1N
···
qNiN =±1
N
e
j qmim [φm +(im −2m+1)π/2]
m=1
(f − f c ◦ q i1 ,...,iN )
(7.47)
where q i1 ,...,iN [q1i1 , ..., qN iN ]T and f c [fc,1 , ..., fc,N ]T . The CCP of the N QAM signals xQAMi (t)
+∞
xc,i (k) pc,i (t k=−∞ +∞
−
− kTs ) cos(2πfc,i t + φi )
xs,i (k) ps,i (t − kTs ) sin(2πfc,i t + φi ),
i = 1, ..., N
(7.48)
k=−∞
with white sequences xc,i (k) and xs,i (k), can be obtained accounting for (7.41), by substituting
Chapter 7 • Manufactured signals
β−f Tc q i
Pxi1 ,...,xiN1
,...,iN
281
(f − f c ◦ q i1 ,...,iN )
cxi1 ,...,xiN Pcs,iN (β − qN iN fc,N − f T 1) = fs
N−1
Pcs,im (fm − qmim fc,m ) δ(β−f T q
c i1 ,...,iN )
m=1
mod fs
(7.49) into the analogous of (7.47) in terms of CCPs. In (7.49), Pcs,im (f ) is the Fourier transform of pcs,im (t) pc,im (t) δim ,2m−1 + ps,im (t) δim ,2m
(7.50)
and xim (t) = xc,im (t) for im = 2m − 1
xim (t) = xs,im (t) for im = 2m .
7.5.3 DSB-AM signals By letting xs,m (t) ≡ 0 for m = 1, ..., N in (7.42), one obtains double-sideband (DSB) amplitude-modulated (AM) time-series. It results that Sxαi
1
,...,xiN (f
) = Sxαi
1
,...,xiN (f
) δi1 ,1 · · · δiN ,2N−1
= Sxαc,1 ,...,xc,N (f ) δi1 ,1 · · · δiN ,2N−1 .
(7.51)
Therefore, from (7.47) we have (Napolitano, 1995) SxαDSB (f ) =
1
α−q Tf ··· exp{j q Tφ} Sxc,1 ,...,xc c,N (f − q ◦ f c ) N 2 q1 =±1
(7.52)
qN =±1
where φ [φ1 , ..., φN ]T and q [q1 , ..., qN ]T .
7.5.4 SSB signals For Single-Sideband (SSB)-modulated time-series 1 xSSBm (t) xc,m (t) cos[2πfcm t + φm ] − xc,m (t) ⊗ sin[2πfcm t + φm ] πt one has
1 δim ,2m . xim (t) = xc,m (t) ⊗ δ(t) δim ,2m−1 + πt
(7.53)
(7.54)
Then, accounting for (4.92) it follows that (Napolitano, 1995) Sxαi ,...,xi (f ) = Sxαc,1 ,...,xc,N (f ) δiN ,2N−1 − j sign(α − f T 1) δiN ,2N 1
N
·
N−1 m=1
δim ,2m−1 − j sign(fm ) δim ,2m
(7.55)
282
Cyclostationary Processes and Time Series
where sign(·) is the signum function. Therefore, the RD-CSCMF can be obtained by substitution of (7.55) in (7.47).
7.5.5 ASK signals N Amplitude-Shift Keyed (ASK) signals can be viewed as a special case of N QAM signals with xs,i (k) ≡ 0 in (7.48). Thus, their CCP is given by (Napolitano, 1995) Px ASK (f ) = β
1 fs cxc,1 ,...,xc,N ··· exp{j q Tφ} N 2 q1 =±1
qN =±1
Pc,N (β − qN fcN − f T 1)
N−1
Pc,m (fm − qm fc,m ) δ(β−q T f c ) mod fs .
(7.56)
m=1
7.6 Cyclic spectral analysis of man-made signals Second-order cyclostationarity-based spectrum sensing techniques exploit the knowledge of the (conjugate) cycle frequencies, cyclic autocorrelation functions, and cyclic spectra of the signal of interest. They have been derived for the most common analog and digitally modulated signals in (Gardner, 1987e) and (Gardner et al., 1987), respectively. Cyclic statistics have been derived for: • •
• • • • • • •
Long-code direct-sequence spread-spectrum (DSSS) signals (Fusco et al., 2006); Continuous phase modulated (CPM) signals (Gournay and Viravau, 1998), (Viravau and Gournay, 1998), (Cariolaro et al., 2011), (Napolitano and Spooner, 2001); Digital pulse streams modulated by cyclostationary sequences (Oner, 2009); Minimum shift keying (MSK) and offset quadrature phase shift keying (QPSK) (Vu˘ci´c and Obradovi´c, 1999); Multicarrier modulated signals (Zhang et al., 2010); Ultra wide-band (UWB) signals (Oner, 2008a), (Oner, 2008b), (Vucic and Eric, 2009); Orthogonal frequency-division multiplexing (OFDM) signals (Cui et al., 2011); OFDM and single carrier linearly digitally (SCLD) modulated signals (Punchihewa et al., 2010); WiMAX and long-term evolution (LTE) OFDM signals (Al-Habashna et al., 2012);
Chapter 7 • Manufactured signals
283
• Spatial multiplexing OFDM (SM-OFDM) and Alamouti coded OFDM (AL-OFDM) signals (Karami and Dobre, 2015); • Block transmitted SCLD (BT-SCLD) signals (Zhang et al., 2013); • LTE single-carrier frequency-division multiple-access (SC-FDMA) signals (Jerjawi et al., 2015); • LTE advanced signals (Sutton et al., 2014); • Binary offset carrier (BOC) signals (Lohan et al., 2014); • Global position system (GPS) signals (Napolitano and Perna, 2014a); • Universal filtered multi-carrier (UFMC) signal (Zerhouni et al., 2018); • Orthogonal frequency-division multiplexing/offset-QAM (OFDM/OQAM) signals (Vuˇci´c et al., 2017). A symbol constellation having second-order statistics with cyclostationary phase is designed in (Murphy, 2005).
7.7 Proofs In this section, proofs are reported for results of the present chapter. Unless otherwise specified, only formal manipulations are reported, without explicitly justifying inversion of linear operators such as time averages, infinite sums, integrals, and ensemble average (expectation). In general, these manipulations are justified at least in the sense of generalized functions (distributions) (Champeney, 1990), (Zemanian, 1987).
Proof of (7.10) E x(t + τ ) x (∗) (t)
(∗) (∗) =E an1 q(t + τ − n1 T0 ) an2 q (t − n2 T0 ) n1 ∈Z
=
n2 ∈Z
E an1 an(∗) q(t + τ − n1 T0 ) q (∗) (t − n2 T0 ) 2
n1 ∈Z n2 ∈Z
E an+m an(∗) q(t + τ − (n + m)T0 ) q (∗) (t − nT0 ) = n∈Z m∈Z
=
m∈Z α ∈A
α (m) R aa (∗)
n∈Z
αn ej 2π q(t + τ − (n + m)T0 ) q (∗) (t − nT0 )
(7.57)
284
Cyclostationary Processes and Time Series
where in the third equality the variable changes n1 = n + m and n2 = n are made, and in the fourth equality (7.7) is used.
Proof of (7.11) and (7.12) In the right-hand side of (7.57), for every fixed α and m, the function
αn ej 2π q(t + τ − (n + m)T0 ) q (∗) (t − nT0 ) y(t)
(7.58)
n∈Z α n and pulse p(t) is a PAM waveform with deterministic modulating sequence yn ej 2π q(t + τ − mT0 ) q (∗) (t) whose Fourier transform is denoted by P (f ). The results of Section 2.4.1 can be applied to y(t). In particular, we have n α n −j 2π β y β ej 2π e = δ( (7.59) ) mod 1 α −β n
where mod 1 denotes the modulo 1 operation with values in [−1/2, 1/2). Thus, accounting for (2.100), the Fourier coefficient at frequency α of the almost-periodic component of y(t) is given by y α y(t) e−j 2παt
t
=
1 δ( = α −αT0 ) mod 1 T0
1 αT0 y P (α) T 0 q(t + τ − mT0 ) q (∗) (t) e−j 2παt dt R
(7.60)
Accounting for (7.57), one has 1 T
α Rxx (∗) (τ ) lim T →∞
=
m∈Z α ∈A
T /2
−T /2
E x(t + τ ) x (∗) (t) e−j 2παt dt
α (m) R aa (∗)
1 T /2 j 2π e α n q(t + τ − (n + m)T0 ) q (∗) (t − nT0 ) e−j 2παt dt T →∞ T −T /2 n∈Z
α = Raa (∗) (m) lim
m∈Z α ∈A
1 δ( q(t + τ − mT0 ) q (∗) (t) e−j 2παt dt α −αT0 ) mod 1 T0 R 1 αT0 α Raa (∗) (m) rqq = (∗) (τ − mT0 ) T0
(7.61)
m∈Z
where in the third equality (7.60) is used and the fourth equality follows since at most one The function can give nonzero contribution, provided that (αT0 mod 1) ∈ A. value of α∈A α rqq (∗) (τ ) is defined in (7.15).
7.1 Introduction Periodicities in the multivariate statistical functions of man-made signals used in communications, radar, sonar, and telemetry arise from the construction and/or subsequent processing of the signal, that is, from operations such as modulation, scanning, sampling, multiplexing, and coding. The corresponding cycle frequencies are related to parameters of the processes such as carrier frequency, baud rate, sampling frequency, and code rate. The shapes of the second- and higher-order cyclic statistical functions depend on the shapes of the base pulses and the correlation properties of the modulating sequences (Gardner, 1985, Chap. 12), (Gardner, 1987f), (Gardner et al., 2006, Sec. 5). Cyclostationary Processes and Time Series. https://doi.org/10.1016/B978-0-08-102708-0.00018-2 Copyright © 2020 Elsevier Ltd. All rights reserved.
269
270
Cyclostationary Processes and Time Series
The analytical cyclic spectral analysis of mathematical models of analog and digitally modulated signals was first carried out in (Gardner, 1985) for stochastic processes and in (Gardner, 1987e) and (Gardner et al., 1987) for nonstochastic time series. The effects of multiplexing are considered in (Bukofzer, 1990). Continuous-phase frequency-modulated signals are treated in (Chen, 1989), (Gournay and Viravau, 1998), (Viravau and Gournay, 1998), and (Napolitano and Spooner, 2001). The effects of timing jitter on the cyclostationarity properties of communications signals are addressed in (Gardner, 1987f), (Elmirghani et al., 1995), (Win, 1998). On this general subject, see the general treatments (Gardner, 1985), (Gardner, 1987f), (Brown, 1987) (Gardner, 1990b), (Gardner, 1991c), and also see (Gardner, 1994a), (Dragan and Yavorskii, 1982b), (Leduc, 1990). In this chapter, two fundamental examples of cyclostationary communication signals are considered and their second-order cyclic statistics are derived. Specifically, the double side-band amplitude-modulated signal and the pulse-amplitude modulated signal are considered. The derivations of the second-order cyclic statistics can be accomplished by using the results of Sections 3.2 and 3.3. Then a third example of a more sophisticated communication signal is considered. Higher-order cyclic spectra for several communications signals are derived in Section 7.5. Finally, references are provided where second- and higher-order cyclic statistics of several special cases of communications signals are derived. Several aspects related to the second- and higher-order cyclostationarity properties of communications signals are treated in the blog (Spooner, 2015).
7.2 Double side-band amplitude-modulated signal Let x(t) be the (real-valued) double side-band amplitude-modulated (DSB-AM) signal (Fig. 7.1) x(t) s(t) cos(2πf0 t + φ0 ) .
(7.1)
The cyclic autocorrelation function and cyclic spectrum of x(t) are (Gardner, 1987e) 1 Rxα (τ ) = Rsα (τ ) cos(2πf0 τ ) 2 1 α+2f0 α−2f0 (τ ) e−j 2πf0 τ e−j 2φ0 + Rs (τ ) ej 2πf0 τ ej 2φ0 Rs + 4 1 Sxα (f ) = Ssα (f − f0 ) + Ssα (f + f0 ) 4 α+2f0
+ Ss
(f + f0 ) e−j 2φ0 + Ss
α−2f0
(f − f0 ) ej 2φ0
respectively. If s(t) is a wide-sense stationary signal then Rsα (τ ) = Rs0 (τ ) δα and
(7.2)
(7.3)
Chapter 7 • Manufactured signals
271
FIGURE 7.1 DSB-AM signal x(t).
Rxα (τ ) =
Sxα (f ) =
⎧ ⎪ ⎨ ⎪ ⎩ ⎧ ⎪ ⎨ ⎪ ⎩
1 0 2 Rs (τ ) cos(2πf0 τ ) 1 0 ±j 2πf0 τ e±j 2φ0 4 Rs (τ ) e
0
1 0 0 4 Ss (f − f0 ) + Ss (f 1 0 ±j 2φ0 4 Ss (f ∓ f0 ) e
α=0 α = ±2f0
(7.4)
otherwise + f0 ) α=0 α = ±2f0
(7.5)
otherwise .
0
Thus, x(t) is cyclostationary with period 1/2f0 . In Fig. 7.2A the magnitude of the cyclic autocorrelation function Rxα (τ ), as a function of α and τ , and in Fig. 7.2B the magnitude of the cyclic spectrum Sxα (f ), as a function of α and f , are reported for the DSB-AM signal (7.1) with stationary modulating signal s(t) having triangular autocorrelation function.
7.3 Pulse-amplitude-modulated signal Let x(t) be the complex-valued pulse-amplitude modulated (PAM) signal (Fig. 7.3) x(t)
an q(t − nT0 ) .
(7.6)
n∈Z
In (7.6), q(t) is a complex-valued square integrable pulse and {an }n∈Z , an ∈ C, is an ACS sequence whose cyclostationarity is possibly induced by framing, multiplexing, or coding (Schell, 1995b), with (conjugate) autocorrelation
α αn E an+m an(∗) = (m) ej 2π R aa (∗) α ∈A
(7.7)
272
Cyclostationary Processes and Time Series
FIGURE 7.2 DSB-AM signal (7.1). (A) Magnitude of the cyclic autocorrelation function Rxα (τ ), as a function of α and τ , and (B) magnitude of the cyclic spectrum Sxα (f ), as a function of α and f . Source: (Gardner et al., 2006) Copyright of Elsevier.
(conjugate) cyclic autocorrelation N
1 αn E an+m an(∗) e−j 2π N→∞ 2N + 1
α (m) lim R aa (∗)
(7.8)
n=−N
and countable set of (conjugate) cycle frequencies α A α ∈ [−1/2, 1/2) : R (m) = 0 . (∗) aa
(7.9)
The (conjugate) autocorrelation function of x(t) is given by (Section 7.7) E x(t + τ ) x (∗) (t)
α αn = (m) ej 2π q(t + τ − (n + m)T0 ) q (∗) (t − nT0 ) . R aa (∗) m∈Z α ∈A
n∈Z
(7.10)
Chapter 7 • Manufactured signals
273
FIGURE 7.3 PAM signal x(t) with stationary white binary modulating sequence {an } and bit period T0 = 16Ts . (Top) full duty-cycle rectangular pulse; (middle) 50% duty-cycle rectangular pulse; (bottom) raised cosine pulse.
The (conjugate) cyclic autocorrelation function and (conjugate) cyclic spectrum of x(t) are (Section 7.7)
1 α (m) r α (∗) (τ − mT0 ) Raa (∗) α =αT0 qq T0 m∈Z
1 α α Sxx (f ) = (ν) Q(f ) Q(∗) ((−)(α − f )) S (∗) (∗) ν=f T0 , α =αT0 T0 aa
α Rxx (∗) (τ ) =
(7.11) (7.12)
274
Cyclostationary Processes and Time Series
respectively, where α Saa (∗) (ν)
α (m) e−j 2πνm R aa (∗)
(7.13)
m∈Z
is the (conjugate) cyclic spectrum of the sequence {an }n∈Z , Q(f ) q(t) e−j 2πf t dt R
(7.14)
and (∗) α (−τ ) ej 2πατ rqq (∗) (τ ) q(τ ) ⊗ q = q(t + τ ) q (∗) (t) e−j 2παt dt . R
(7.15)
From (7.11) and (7.12) it follows that x(t) exhibits (conjugate) cyclostationarity at (con where mod 1 denotes the modulo 1 jugate) cycle frequency α only if (αT0 mod 1) ∈ A, operation with values in [−1/2, 1/2). If the sequence {an }n∈Z is wide-sense stationary, the (conjugate) autocorrelation (7.10) specializes into E x(t + τ ) x (∗) (t)
0 (∗) (m) = q(t + τ − (n + m)T0 ) q (∗) (t − nT0 ) (7.16a) R aa m∈Z
=
n∈Z
0 (∗) (m) repT R 0 aa
q(t + τ − mT0 ) q (∗) (t)
(7.16b)
m∈Z
where repT0 [·] denotes periodic replication with period T0 (Section C.2). Thus, x(t) is cyclostationary with period T0 . If the sequence {an }n∈Z is wide-sense stationary and white, then α 0 (∗) (0) δ( (m) = R R α mod 1) δm . aa (∗) aa
In such a case, the (conjugate) autocorrelation specializes into
0 (∗) (0) repT q(t + τ ) q (∗) (t) E x(t + τ ) x (∗) (t) = R 0 aa
(7.17)
(7.18)
and the (conjugate) cyclic autocorrelation and the (conjugate) cyclic spectrum of x(t) become α Rxx (∗) (τ ) = α Sxx (∗) (f ) =
0 (∗) (0) R aa T0
0 (∗) (0) R aa T0
α δ(αT0 mod 1) rqq (∗) (τ )
(7.19)
δ(αT0 mod 1) Q(f ) Q(∗) ((−)(α − f ))
(7.20)
Chapter 7 • Manufactured signals
275
FIGURE 7.4 PAM signal (7.6) with full duty-cycle rectangular pulse (Fig. 7.3 (top)). (A) Magnitude of the cyclic autoα (τ ), as a function of α and τ , and (B) magnitude of the cyclic spectrum S α (f ), as a function correlation function Rxx ∗ xx ∗ of α and f .
respectively. According with the stated cyclostationarity with period T0 , x(t) exhibits cyclostationarity with cycle frequencies α = k/T0 , k ∈ Z. α (τ ), as a function of α and τ , The magnitude of the cyclic autocorrelation function Rxx ∗ α and the magnitude of the cyclic spectrum Sxx ∗ (f ), as a function of α and f , are reported for the PAM signal (7.6) with stationary white modulating sequence {an }n∈Z , and rectangular pulse q(t) rect((t − Tw /2)/Tw ) with width Tw . In this case, Q(f ) = Tw sinc(f Tw ) e−j πf Tw and (7.15) reduces to α −j πα(Tw −τ ) rqq ∗ (τ ) = e
τ rect 2Tw
|τ | 1− Tw sinc(α(Tw − |τ |)) . Tw
(7.21)
In Fig. 7.4, the case of a full duty-cycle (Tw = T0 ) pulse is represented. For τ = 0 it results α (0) = 0 ∀α = 0. x 2 (t) = 1 (constant ∀t). Thus Rxx ∗ In Fig. 7.5, the case of a 50% duty-cycle (Tw = T0 /2) pulse is represented. α (τ ), as a function In Fig. 7.6A the magnitude of the cyclic autocorrelation function Rxx ∗ α of α and τ , and in Fig. 7.6B the magnitude of the cyclic spectrum Sxx ∗ (f ), as a function of α and f , are reported for the PAM signal (7.6) with stationary white modulating sequence
276
Cyclostationary Processes and Time Series
FIGURE 7.5 PAM signal (7.6) with 50% duty-cycle rectangular pulse (Fig. 7.3 (middle)). (A) Magnitude of the cyclic α (τ ), as a function of α and τ , and (B) magnitude of the cyclic spectrum S α (f ), as a autocorrelation function Rxx ∗ xx ∗ function of α and f .
FIGURE 7.6 PAM signal (7.6) with raised cosine pulse (Fig. 7.3 (bottom)). (A) Magnitude of the cyclic autocorrelation α (τ ), as a function of α and τ , and (B) magnitude of the cyclic spectrum S α (f ), as a function of α and f . function Rxx ∗ xx ∗
Chapter 7 • Manufactured signals
277
{an }n∈Z , and raised cosine pulse ηπt cos t T0 q(t) sinc 2ηt 2 T0 1− T0
(7.22)
where 0 η 1 is the roll-off factor. In this case, ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
T0
T0 πT0 1 − η Q(f ) = 1 + cos |f | − ⎪ 2 η 2T0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 0
|f |
1−η 2T0
1+η 1−η |f | 2T0 2T0 |f |
(7.23)
1+η 2T0
is strictly band limited and the cyclic autocorrelation and cyclic spectrum are nonzero only for α = 0 and α = ±1/T0 .
7.4 Direct-sequence spread-spectrum signal Let x(t) be the short-code direct-sequence spread-spectrum (DS-SS) baseband PAM signal x(t)
ak q(t − kT0 ) ,
(7.24)
k∈Z
where {ak }k∈Z , ak ∈ C, is an ACS sequence, T0 is the symbol period, and q(t)
N
c −1
cn p(t − nTc )
(7.25)
n=0
is the spreading waveform. In (7.25), {c0 , . . . , cNc −1 } is the Nc -length spreading sequence (code) with cn ∈ C, and Tc is the chip period such that T0 = Nc Tc . The (conjugate) cyclic autocorrelation function and (conjugate) cyclic spectrum of x(t) are (Napolitano and Tanda, 2001)
1 α α r α (∗) (τ − mT0 ) ⊗ γcc Raa (∗) (m) (∗) (τ ) α =αT0 pp T0 m∈Z
1 α α Sxx (f ) = (ν) S (∗) (∗) ν=f T0 , α =αT0 T0 aa (∗) α P (f ) P ((−)(α − f ))
(ν) cc(∗)
α Rxx (∗) (τ ) =
ν=f Tc , α =αTc
(7.26)
(7.27)
278
Cyclostationary Processes and Time Series
where α γcc (∗) (τ )
N
c −1 N c −1
n1 =0 n2 =0
cn1 cn(∗) e−j 2παn2 Tc δ(τ − (n1 − n2 )Tc ) 2
(7.28)
and α
(ν) C(ν) C (∗) ((−)( α − ν)) cc(∗)
(7.29)
with C(ν)
N
c −1
cn e−j 2πνn .
(7.30)
n=0
Second-order cyclic spectral analysis of more general classes of DSSS signals is addressed in (Fusco et al., 2006).
7.5 Higher-order cyclic spectra of modulated signals In this section, results of Section 4.4 are exploited to derive higher-order cyclic crossspectra of modulated signals. Higher-order cyclic spectra of single signals are easily obtained by considering xm ≡ x, ∀m in the derived relationships.
7.5.1 PAM signals Let us consider N PAM signals xPAMm (t)
+∞
xm (kTm ) pm (t − kTm ) = xm,δ (t) ⊗ pm (t),
m = 1, ..., N
(7.31)
k=−∞
where the continuous-time signals xm,δ (t) xm (t)
+∞
δ(t − kTm ) = xm (t)
k=−∞
+∞ 1 j 2πrt/Tm e , Tm r=−∞
m = 1, ..., N
(7.32)
are obtained by sampling the continuous-time signals xm (t) with rates 1/Tm (m = 1, ..., N ) and, in the second equality, the Poisson formula (C.8a) is exploited. The RD-CSCMF of the N PAM signals, according to (4.92), can be written as SxαPAM (f ) = PN (α
T
− f 1)
N−1
Pm (fm ) Sxαδ (f )
(7.33)
m=1
where subscripts x PAM ≡ xPAM1 . . . xPAMN , x δ ≡ x1,δ . . . xN,δ , and Pm (f ) denotes the Fourier transform of pm (t). The RD-CSCMF of the signals (7.32) can be expressed in terms of the
Chapter 7 • Manufactured signals
RD-CSCMF of the signals xm (t) (Napolitano, 1995) N
α−r T f 1 α Sx δ (f ) = Sx1 ,...,xNs (f − r ◦ f s ) Tm N m=1
279
(7.34)
r∈Z
where f s [1/T1 , ..., 1/TN ]T , r [r1 , ..., rN ]T , and ◦ denotes the Hadamard matrix product, i.e., r ◦ f s [r1 /T1 , ..., rN−1 /TN−1 ]T . Accounting for the rhs of (7.32), Eq. (7.34) is derived by substituting σm = rm /Tm (with rm any integer number) and cσm = 1/Tm into (4.99). Let xm,d (n) xm (nTs ), n ∈ Z, be the discrete-time signals obtained by uniformly sampling with period Ts the continuous-time signals xm (t). We have that (7.34) with Tm = Ts , m = 1, ..., N , compared with (4.112), yields αf α (ν ) = Ts Sx δ s (f ) |f =ν fs , Sx d
(7.35)
where subscript x d ≡ x1,d · · · xN,d . Let us consider now the N PAM signals xPAMi (t) =
+∞
xi (n) pi (t − nTs )
i = 1, ..., N
(7.36)
n=−∞
whose modulating sequences are N discrete-time signals x1 (n), ..., xN (n) jointly white and wide-sense stationary of order N , i.e., (Dehay et al., 2018b, Definition 4.1) cx δm −1mN Cx (n, m) cum {xi (n + mi ), i = 1, . . . , N } =
(7.37)
cx is the N th-order cross-cumulant of x1 (n), ..., xN (n). where subscript x ≡ x1 · · · xN and The CTCCF of the modulating signals is given by β cx δm −1mN δβ mod 1 Cx (m) =
where
δa mod b
1
a/b ∈ Z,
0
otherwise.
(7.38)
(7.39)
β
x (m ), straightforwardly derivable By Fourier transforming the discrete-time RD-CTCCF C from (7.38), one obtains the CCP xβ (ν ) = cx δβ mod 1 . P
(7.40)
Hence, by substituting (7.40) into the analogous of (7.35) in terms of CCPs and the result into the analogous of (7.33) in terms of CCPs, one has Px PAM (f ) = fs cx PN (β − f T 1) β
N−1 m=1
Pm (fm ) δβ mod fs .
(7.41)
280
Cyclostationary Processes and Time Series
Note that the relations stated in terms of CTCCFs and CCPs for PAM signals modulated by jointly white and wide-sense stationary discrete-time signals cannot be written in terms of RD-CSCMFs by simply substituting CCPs with RD-CSCMFs as in the case of continuous-time modulating signals (see also Remark 4.28). In fact, for the discrete-time TCMF, no simple relation analogous to (7.37) holds.
7.5.2 QAM signals Let us consider N quadrature-carrier amplitude-modulation (QAM) time-series xQAMm (t) xc,m (t) cos(2πfc,m t + φm ) − xs,m (t) sin(2πfc,m t + φm ),
m = 1, ..., N .
(7.42)
These QAM signals can be considered as the outputs of a continuous-time MIMO LAPTV system excited by x(t) [x1 (t), ..., x2N (t)]T , where x2m−1 (t) = xc,m (t)
(7.43)
x2m (t) = xs,m (t) .
(7.44)
The input/output relationship of the system is given by (4.94), where the nonzero elements of the N × 2N matrix C(t) are given by cm,2m−1 (t) = cos(2πfc,m t + φm )
(7.45)
cm,2m (t) = − sin(2πfc,m t + φm ) .
(7.46)
Therefore, from (4.98), letting σmim = ±fc,m , it follows that (Napolitano, 1995) SxαQAM (f ) =
2 1
··· 2N i1 =1
α−f Tcq i
2N
iN =2N−1 q1i1 =±1
,...,iN
· Sxi1 ,...,xi1N
···
qNiN =±1
N
e
j qmim [φm +(im −2m+1)π/2]
m=1
(f − f c ◦ q i1 ,...,iN )
(7.47)
where q i1 ,...,iN [q1i1 , ..., qN iN ]T and f c [fc,1 , ..., fc,N ]T . The CCP of the N QAM signals xQAMi (t)
+∞
xc,i (k) pc,i (t k=−∞ +∞
−
− kTs ) cos(2πfc,i t + φi )
xs,i (k) ps,i (t − kTs ) sin(2πfc,i t + φi ),
i = 1, ..., N
(7.48)
k=−∞
with white sequences xc,i (k) and xs,i (k), can be obtained accounting for (7.41), by substituting
Chapter 7 • Manufactured signals
β−f Tc q i
Pxi1 ,...,xiN1
,...,iN
281
(f − f c ◦ q i1 ,...,iN )
cxi1 ,...,xiN Pcs,iN (β − qN iN fc,N − f T 1) = fs
N−1
Pcs,im (fm − qmim fc,m ) δ(β−f T q
c i1 ,...,iN )
m=1
mod fs
(7.49) into the analogous of (7.47) in terms of CCPs. In (7.49), Pcs,im (f ) is the Fourier transform of pcs,im (t) pc,im (t) δim ,2m−1 + ps,im (t) δim ,2m
(7.50)
and xim (t) = xc,im (t) for im = 2m − 1
xim (t) = xs,im (t) for im = 2m .
7.5.3 DSB-AM signals By letting xs,m (t) ≡ 0 for m = 1, ..., N in (7.42), one obtains double-sideband (DSB) amplitude-modulated (AM) time-series. It results that Sxαi
1
,...,xiN (f
) = Sxαi
1
,...,xiN (f
) δi1 ,1 · · · δiN ,2N−1
= Sxαc,1 ,...,xc,N (f ) δi1 ,1 · · · δiN ,2N−1 .
(7.51)
Therefore, from (7.47) we have (Napolitano, 1995) SxαDSB (f ) =
1
α−q Tf ··· exp{j q Tφ} Sxc,1 ,...,xc c,N (f − q ◦ f c ) N 2 q1 =±1
(7.52)
qN =±1
where φ [φ1 , ..., φN ]T and q [q1 , ..., qN ]T .
7.5.4 SSB signals For Single-Sideband (SSB)-modulated time-series 1 xSSBm (t) xc,m (t) cos[2πfcm t + φm ] − xc,m (t) ⊗ sin[2πfcm t + φm ] πt one has
1 δim ,2m . xim (t) = xc,m (t) ⊗ δ(t) δim ,2m−1 + πt
(7.53)
(7.54)
Then, accounting for (4.92) it follows that (Napolitano, 1995) Sxαi ,...,xi (f ) = Sxαc,1 ,...,xc,N (f ) δiN ,2N−1 − j sign(α − f T 1) δiN ,2N 1
N
·
N−1 m=1
δim ,2m−1 − j sign(fm ) δim ,2m
(7.55)
282
Cyclostationary Processes and Time Series
where sign(·) is the signum function. Therefore, the RD-CSCMF can be obtained by substitution of (7.55) in (7.47).
7.5.5 ASK signals N Amplitude-Shift Keyed (ASK) signals can be viewed as a special case of N QAM signals with xs,i (k) ≡ 0 in (7.48). Thus, their CCP is given by (Napolitano, 1995) Px ASK (f ) = β
1 fs cxc,1 ,...,xc,N ··· exp{j q Tφ} N 2 q1 =±1
qN =±1
Pc,N (β − qN fcN − f T 1)
N−1
Pc,m (fm − qm fc,m ) δ(β−q T f c ) mod fs .
(7.56)
m=1
7.6 Cyclic spectral analysis of man-made signals Second-order cyclostationarity-based spectrum sensing techniques exploit the knowledge of the (conjugate) cycle frequencies, cyclic autocorrelation functions, and cyclic spectra of the signal of interest. They have been derived for the most common analog and digitally modulated signals in (Gardner, 1987e) and (Gardner et al., 1987), respectively. Cyclic statistics have been derived for: • •
• • • • • • •
Long-code direct-sequence spread-spectrum (DSSS) signals (Fusco et al., 2006); Continuous phase modulated (CPM) signals (Gournay and Viravau, 1998), (Viravau and Gournay, 1998), (Cariolaro et al., 2011), (Napolitano and Spooner, 2001); Digital pulse streams modulated by cyclostationary sequences (Oner, 2009); Minimum shift keying (MSK) and offset quadrature phase shift keying (QPSK) (Vu˘ci´c and Obradovi´c, 1999); Multicarrier modulated signals (Zhang et al., 2010); Ultra wide-band (UWB) signals (Oner, 2008a), (Oner, 2008b), (Vucic and Eric, 2009); Orthogonal frequency-division multiplexing (OFDM) signals (Cui et al., 2011); OFDM and single carrier linearly digitally (SCLD) modulated signals (Punchihewa et al., 2010); WiMAX and long-term evolution (LTE) OFDM signals (Al-Habashna et al., 2012);
Chapter 7 • Manufactured signals
283
• Spatial multiplexing OFDM (SM-OFDM) and Alamouti coded OFDM (AL-OFDM) signals (Karami and Dobre, 2015); • Block transmitted SCLD (BT-SCLD) signals (Zhang et al., 2013); • LTE single-carrier frequency-division multiple-access (SC-FDMA) signals (Jerjawi et al., 2015); • LTE advanced signals (Sutton et al., 2014); • Binary offset carrier (BOC) signals (Lohan et al., 2014); • Global position system (GPS) signals (Napolitano and Perna, 2014a); • Universal filtered multi-carrier (UFMC) signal (Zerhouni et al., 2018); • Orthogonal frequency-division multiplexing/offset-QAM (OFDM/OQAM) signals (Vuˇci´c et al., 2017). A symbol constellation having second-order statistics with cyclostationary phase is designed in (Murphy, 2005).
7.7 Proofs In this section, proofs are reported for results of the present chapter. Unless otherwise specified, only formal manipulations are reported, without explicitly justifying inversion of linear operators such as time averages, infinite sums, integrals, and ensemble average (expectation). In general, these manipulations are justified at least in the sense of generalized functions (distributions) (Champeney, 1990), (Zemanian, 1987).
Proof of (7.10) E x(t + τ ) x (∗) (t)
(∗) (∗) =E an1 q(t + τ − n1 T0 ) an2 q (t − n2 T0 ) n1 ∈Z
=
n2 ∈Z
E an1 an(∗) q(t + τ − n1 T0 ) q (∗) (t − n2 T0 ) 2
n1 ∈Z n2 ∈Z
E an+m an(∗) q(t + τ − (n + m)T0 ) q (∗) (t − nT0 ) = n∈Z m∈Z
=
m∈Z α ∈A
α (m) R aa (∗)
n∈Z
αn ej 2π q(t + τ − (n + m)T0 ) q (∗) (t − nT0 )
(7.57)
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Cyclostationary Processes and Time Series
where in the third equality the variable changes n1 = n + m and n2 = n are made, and in the fourth equality (7.7) is used.
Proof of (7.11) and (7.12) In the right-hand side of (7.57), for every fixed α and m, the function
αn ej 2π q(t + τ − (n + m)T0 ) q (∗) (t − nT0 ) y(t)
(7.58)
n∈Z α n and pulse p(t) is a PAM waveform with deterministic modulating sequence yn ej 2π q(t + τ − mT0 ) q (∗) (t) whose Fourier transform is denoted by P (f ). The results of Section 2.4.1 can be applied to y(t). In particular, we have n α n −j 2π β y β ej 2π e = δ( (7.59) ) mod 1 α −β n
where mod 1 denotes the modulo 1 operation with values in [−1/2, 1/2). Thus, accounting for (2.100), the Fourier coefficient at frequency α of the almost-periodic component of y(t) is given by y α y(t) e−j 2παt
t
=
1 δ( = α −αT0 ) mod 1 T0
1 αT0 y P (α) T 0 q(t + τ − mT0 ) q (∗) (t) e−j 2παt dt R
(7.60)
Accounting for (7.57), one has 1 T
α Rxx (∗) (τ ) lim T →∞
=
m∈Z α ∈A
T /2
−T /2
E x(t + τ ) x (∗) (t) e−j 2παt dt
α (m) R aa (∗)
1 T /2 j 2π e α n q(t + τ − (n + m)T0 ) q (∗) (t − nT0 ) e−j 2παt dt T →∞ T −T /2 n∈Z
α = Raa (∗) (m) lim
m∈Z α ∈A
1 δ( q(t + τ − mT0 ) q (∗) (t) e−j 2παt dt α −αT0 ) mod 1 T0 R 1 αT0 α Raa (∗) (m) rqq = (∗) (τ − mT0 ) T0
(7.61)
m∈Z
where in the third equality (7.60) is used and the fourth equality follows since at most one The function can give nonzero contribution, provided that (αT0 mod 1) ∈ A. value of α∈A α rqq (∗) (τ ) is defined in (7.15).