INFORMATION AND COMMUNICATION TECHNOLOGIES AND SERVICES VOLUME: 16 |NUMBER: 3 |2018 |SEPTEMBER
Time-F equency Rep ese a ion o Rada Signals
Using Dopple -Lag Block Sea ching Wigne -Ville
Dis ibu ion
Muhammad Noo Muhammad HAMDI, Ahmad Zu i SHA’AMERI
Depa men o Elec onic and Compu e Enginee ing, School o Elec ical Enginee ing,
Uni e si i Teknologi Malaysia, Joho Bah u, Joho , Malaysia
muhdno[email p o ec ed], zu i@ ke.u m.my
DOI: 10.15598/aeee. 16i3.2633
Abs ac . Rada signals a e ime- a ying signals
whe e he signal pa ame e s change o e ime. Fo
hese signals, Quad a ic Time-F equency Dis ibu ion
(QTFD) o e s ad an ages o e classical spec um es-
ima ion in e ms o equency and ime esolu ion
bu i su e s hea ily om c oss- e ms. In gene a -
ing accu a e Time-F equency Rep esen a ion (TFR),
a ke nel unc ion mus be able o supp ess c oss-
e ms while main aining au o- e ms ene gy especially
in a non-coope a i e en i onmen whe e he pa ame e s
o he ac ual signal a e unknown. Thus, a new signal-
dependen QTFD is p oposed ha adap i ely es ima es
he ke nel pa ame e s o a wide class o ada signals.
The adap i e p ocedu e, Dopple -Lag Block Sea ching
(DLBS) ke nel es ima ion was de eloped o se e his
pu pose. Accu a e TFRs p oduced o all simula ed
ada signals wi h Ins an aneous F equency (IF) es-
ima ion pe o mance a e e i ied using Mon e Ca lo
simula ion mee ing he equi emen s o he C ame -
Rao Lowe Bound (CRLB) a SNR > 6 dB.
Keywo ds
Adap i e p ocedu e, au o- e ms, C ame -Rao
lowe bound, c oss- e ms, ke nel unc ion,
quad a ic ime- equency dis ibu ion.
1. In oduc ion
Rada is widely used bo h in mili a y and non-mili a y
applica ions such as acking missiles, ships, land ehi-
cles and ai c a , ligh con ol sys em, ocean su eil-
lance sys em and geological obse a ions. By de ini-
ion, Low P obabili y o In e cep (LPI) ada s u i-
lize special emi ed wa e o m ha has been speci ically
designed o a oid de ec ion o in e cep ion by non-
coope a i e in e cep ecei e [1]. This is achie ed by
in eg a ing addi ional p ope ies such as ul a-low side-
lobe, Ad anced Mul i unc ion Radio F equency Con-
cep (AMRFC), wideband equency and minimum
ansmi ed ene gy. The idea o LPI ada is o see
and no be seen, meaning i mus ha e he capabili y
o de ec a ge s like any ada while s aying in isible
o elec onic econnaissance equipmen .
In e cep ing LPI signals is no easy bu is no o ally
impossible. Some o he impo an p ope ies equi ed
in he mode n in e cep ecei e s in in e cep ing LPI
signals a e channelized ecei e , u iliza ion o supe -
he e odyne ecei e and sidelobe de ec ion capabili y
[2] and [3]. Signal p ocessing algo i hms a e he im-
po an componen s o mode n in e cep ecei e ha
imp o e he de ec ion and analysis o LPI ada sig-
nals. Example o me hods used o de ec ing and an-
alyzing LPI ada signals a e adap i e ma ch il e -
ing, pa allel il e a ays wi h highe o de s a is ics,
Wigne -Ville Dis ibu ion (WVD), quad a u e mi o
il e bank, and cyclos a iona y p ocessing [3].
A ime- a ying signal such as LPI ada signals, he
spec al desc ip ion o which depends on ime is bes
analyzed wi h Time-F equency Dis ibu ion (TFD).
Among TFD classes, Quad a ic TFD (QTFD) is ap-
p op ia ely used because i p o ides high- esolu ion
ep esen a ion bo h in ime and equency [4]. C oss-
e ms a e in oduced in QTFD due o he quad a ic
na u e o he algo i hm, which makes i di icul o
in e p e he ue signal cha ac e is ics and also ex-
agge a es he e ec o noise [5]. Some o he ech-
niques p oposed o p oduce accu a e TFR a e educed
in e e ence dis ibu ion, ac ional Fou ie ans o m,
Radon-Wigne dis ibu ion, adap i e c oss WVD and
modi ied B-dis ibu ion [6] and [7].
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The applica ion o he F ac ional Fou ie T ans-
o m (F FT) is p o en e ec i e in ep esen ing and
p ese ing signal componen s in he Time-F equency
(TF) plane such as Pulse Linea F equency Modula ion
(PLFM) [8]. Besides ha , he echo o he mo ing a -
ge o ai bo ne Syn he ic Ape u e Rada (SAR) can
also be conside ed as a LFM signal. Thus, he F FT
can be used o ep esen and analyze such signals. In
he li e a u e, i was men ioned ha he F FT is equi -
alen o a o a ion o he signal ei he in TF o ambi-
gui y domain. The deg ee o he TF o a ion depends
on he ac ional powe o F FT [9]. The applica ion o
F FT can be u he expanded om his de ini ion o
signal pa ame e s es ima ion and c oss- e m elimina-
ion [10] and [11]. A echnique whe e a combina ion o
a o a ed TF plane WVD wi h a sui able TF il e ing
is compa ed wi h he DLBS-WVD.
The es o he pape is o ganized as ollows. Sec-
ion 2. p esen s he signal model and p oblem de -
ini ion. The signal cha ac e is ics in he ime-lag and
Dopple -lag domain a e discussed in Sec. 3. The
ela ionship be ween F FT and TFR is also explained
in his sec ion. The simula ion esul and discussion
a e p esen ed in Sec. 4. while he ield ials esul s
a e desc ibed in Sec. 5.
2. Signal Model and P oblem
De ini ion
The ou commonly used ada signal ypes u ilized
o e i y he accu acy o ime- equency ep esen a ion
p oduced by he Dopple -Lag Block Sea ching Wigne -
Ville Dis ibu ion (DLBS-WVD) a e: Simple Pulse sig-
nal (SP), 4 Cos as Coded pulse (CC4) signals, PLFM,
and Con inuous Wa e Linea F equency Modula ion
(CW-LFM). The signal pa ame e s a e desc ibed in
Tab. 1. Excep o he SP signal, all he o he signals
can be ca ego ized as LPI ada signal wa e o ms [1].
The signals a e assumed ha hey ha e been downcon-
e ed om adio equency o in e media e equency
whe e hey a e sampled a he Nyquis a e (sampling
equency, s= 40 MHz).
The use o sho pulse epe i ion pe iod, Tbe ween
5 o 20 µs is o simpli y he de elopmen o he ke nel
es ima ion p ocedu e. Howe e , he ac ual ada sig-
nals pa ame e s may a y acco ding o he applica ions
and he de ec ion ange [12], [13] and [14].
Typically, Elec onic Suppo (ES) deals wi h a non-
coope a i e en i onmen whe e p io knowledge o he
ue signal cha ac e is ics – pulse epe i ion pe iod, e-
quency agili ies, modula ion echniques, pulse wid h,
pulse ampli ude – a e unknown. Adap i e ke nel im-
p o es he TFR by es ima ing he ke nel pa ame e s
acco ding o he pa e n o c oss- e ms which a y ac-
Tab. 1: Signal Pa ame e s. Pulse epe i ion pe iod (T), pulse
wid h (Tp), lowes equency ( min), highes equency
( max), bandwid h (BW ).
Signal F equency
Pa ame e s
Time
Pa ame e s
Simple Pulse
(SP) = 10 MHz T= 5 µs
Tp= 1 µs
4 Cos as Coded
pulse (CC4)
4 sub-pulse equencies
b1= 4 MHz
b2= 8 MHz
b3=16 MHz
b4= 12 MHz
T= 16 µs
Tp= 4 µs
Pulse Linea FM
(PLFM)
min = 2 MHz
max = 17 MHz
BW = 15 MHz
T= 9 µs
Tp= 4 µs
Con inuous Wa e
Linea FM
(CW-LFM)
min = 2 MHz
max = 10 MHz
BW = 8 MHz
T= 20 µs
Tp= 10 µs
co ding o he signal. The QTFD wi h he adap i e
ke nel ensu es an accu a e TFR o a b oad class o
signals.
3. Quad a ic Time-F equency
Dis ibu ion
The QTFD p oduces an ene gy ep esen a ion join ly
o e he ime- equency plane. I is also conside ed as
a ela ed class o il e ed WVDs wi h a speci ic ime–lag
ke nel unc ion [15]. When exp essed wi h espec o
he ime- equency ke nel and he WVD, he QTFD is
w i en as [5]
ρz( , ) = γ( , )∗
∗
Wz( , ),(1)
whe e γ( , )is a ime- equency ke nel and Wz( , )
is he WVD. The WVD can be de ined as
Wz( , ) =
∞
Z
−∞
Kz( , τ)e−j2π τ dτ, (2)
whe e Kz( , τ)is he bilinea p oduc o Ins an aneous
Au oco ela ion Func ion (IAF). The bilinea p oduc
can be w i en as
Kz( , τ) = z +τ
2z∗ −τ
2,(3)
whe e z( )is he analy ical o m o he signal.
The o mula ion o he QTFD wi h he ime-lag ke -
nel is gi en as
ρz( , ) =
∞
Z
−∞
G( , τ)∗
Kz( , τ)e−j2π τ dτ, (4)
whe e G( , τ)is he ime-lag ke nel.
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A sepa able ke nel o e s independen con ol o
ime-smoo hing and equency-smoo hing o TFD
which can be de ined in ime-lag unc ion as
G( , τ) = g1( )g2(τ),(5)
whe e g1( )is he smoo hing unc ion in ime and g2(τ)
is he smoo hing unc ion in lag. By using a sepa able
ke nel, he QTFD in Eq. (1) can be desc ibed as
ρz( , ) =
∞
Z
−∞
g1( )∗
( )Kz( , τ)g2(τ)e−j2π τ dτ. (6)
3.1. Gene al Signal Cha ac e is ics
in Time-Lag Domain
The IAF de ined in Eq. (4) can be iewed as he co -
ela ion o signal i sel o e successi e ime in e als.
C oss- e ms a e in oduced due o he quad a ic na-
u e o he IAF which can cause di icul y in he in-
e p e a ion o he ue signal cha ac e is ics [16] and
[17]. Ke nel unc ion as shown in Eq. (4) sol ed his
p oblem.
Va ious ke nel unc ions ga e ise o a ange o di -
e en TFDs [6] and [15]. The gene al IAF de ini ion
is p oduced by mul iplying he signal wi h i s conju-
ga e as shown in Eq. (7). Each ype o inpu signal,
z( ) esul s in di e en IAF de ini ion. Fo example,
wo pulses o SP signal p oduce wo au o- e ms and
wo c oss- e ms while wo pulses o CW-LFM p oduce
ou au o- e ms and wel e c oss- e ms. The IAF ob-
ained by subs i u ing he signal de ini ion in Tab. 1
in o Eq. (4) can be exp essed in he ollowing o m:
Kz( , τ) = hz1 +τ
2+z2 +τ
2−Ti·
·hz∗
1 −τ
2+z∗
2 −τ
2−Ti
=Kz,11 ( , τ) + Kz,22 ( −T, τ)
| {z }
au o− e ms
+
Kz,12  −T
2, τ +T+Kz,21  −T
2, τ −T
| {z }
c oss− e ms
,
(7)
whe e z1( )is he i s pulse and z2( )is he second
pulse o he signal. The accu acy o TFR depends on
he capabili y o he algo i hm o p ese e he au o-
e ms ene gy. C oss- e ms can be di ided in o wo
ca ego ies: in a-pulse and in e -pulses. In a-pulse
c oss- e ms occu as a esul o he co ela ion o signal
be ween he sub-pulse o he signal while in e -pulse
c oss- e ms a e he p oduc o he co ela ion be ween
di e en pulses o he signal.
Figu e 1 shows he ime-lag domain o CW-LFM sig-
nal and p oduced by eplacing z1( )and z2( )in Eq. (7)
1,1 2,2 3,3 4,4
2,1 3,2
1,4
2,41,3
3,42,31,2
4,3
3,1 4,2
4,1
0
Tb
2Tb
-Tb
-2Tb
-3Tb
-4Tb
3Tb
4Tb
τ
4Tb
3Tb
2Tb
Tb
Fig. 1: The ime-lag domain wo pulses o CW-LFM signal.
wi h wo pulses o iangula CW-LFM signal whe e Tb
is he sub-pulse du a ion. Fo iangula CW-LFM, he
Tbis exac ly hal o he signal pulse. The do ed dia-
mond shapes ep esen he in a-pulse c oss- e ms and
he shaded diamond shapes ep esen he in e -pulse
c oss- e ms. The emaining diamond shapes ep esen
he au o- e ms o he signal.
Gene ally, a ke nel ha is able o supp ess he c oss-
e ms a |τ|> Tbis accep able excep o CW-LFM
signal. Figu e 1 shows ha he e a e some c oss- e ms
loca ed be ween 0≤τ≤Tbwhich could lead o in-
accu a e TFR o he signal. Fo comple e c oss- e ms
supp ession, he applied window o ke nel mus ma ch
pe ec ly wi h he au o- e ms. To comple ely sepa a e
he au o- e ms om he c oss- e ms in he ime-lag
domain, al e na i e c oss- e ms supp ession echniques
a e p oposed in he Dopple -lag domain [18].
3.2. Rada Signal Cha ac e is ics in
Ambigui y Domain
The AF is ela ed o he IAF by he Fou ie ans o m
wi h espec o ime as shown in he ollowing equa ion:
Az( , τ) = FT
→ [Kz( , τ)] =
∞
Z
−∞
Kz( , τ)e−j2π d . (8)
The comple e signal equa ions o SP, CC4, PLFM
and CW-LFM signals in he ambigui y domain can be
ound in [19]. The AF o he CW-LFM signal is illus-
a ed in Fig. 2.
The AF ep esen s wo pulses CW-LFM signal whe e
he au o- e m a e ep esen ed by he g ey shaded
shapes, in e -pulse c oss- e ms by he blue shaded
shapes and in a-pulse c oss- e ms by he o ange
shaded shapes. Simila colo codes a e used in Fig. 3,
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0
Tp
T
-Tp
-T
-T-Tp
T+Tp
τ
`
Δ NTΔ NT
T-Tp
G2( )
G2(- )
g1(-τ )
g1(τ )
(1,1),(2,2),
(3,3),(4,4)
(1,3),(2,3),
(1,4),(2,4)
(3,1),(3,2),
(4,1),(4,2)
(4,3),(2,1)
(3,4),(1,2)
Fig. 2: The AF o wo pulses o CW-LFM.
Fig. 4 and Fig. 5. F om Fig. 1, he au o- e ms a e
loca ed a τ≤ |Tp|while mos o he in a-pulse c oss-
e ms a e posi ioned e y close o he au o- e ms. On
he o he hand, he in e -pulse c oss- e ms a e loca ed
be ween T−Tp≤τ≤T+Tp. I is wo h men ioning
he e ha he ene gy o he in a-pulse c oss- e ms o
CW-LFM is signi ican ly lowe compa ed o i s au o-
e ms. Thus, a small po ion o he in a-pulse c oss-
e ms in he AF does no cause a majo deg ada ion
in he TFR.
The p ocedu e o supp essing in e -pulse c oss-
e ms is much simple in he Dopple -lag domain com-
pa ed o he ime-lag domain because he loca ion be-
ween he in e -pulse c oss- e ms and au o- e ms a e
well sepa a ed. Thus, se ing a lag window, g1(τ)a
Tpis su icien o sepa a ing he au o- e ms om he
in e -pulse c oss- e ms.
Figu e 3 shows he AF o wo pulses o PLFM sig-
nal. The posi ion o he au o- e ms is a τ≤ |Tp|and
he in e -pulse c oss- e ms a e loca ed a T−Tp≤τ≤
T+Tp. The cha ac e is ics o PLFM a he AF a e
simila wi h he CW-LFM excep he e a e no in a-
pulse c oss- e ms o PLFM signal. Due o his, he
same c oss- e ms supp ession p ocedu e in CW-LFM
signal is applicable o PLFM signal.
Figu e 4 shows he AF o wo pulses o SP signal.
No e ha he au o- e m and in e -pulse c oss- e ms
loca ion a e simila wi h he PLFM and CW-LFM sig-
nals which a e a τ≤ |Tp|and T−Tp≤τ≤T+Tp
espec i ely. The only di e ence is he shape o he
au o- e ms and he c oss- e ms. Thus, a simila p oce-
0
Tp
T
-Tp
-T
-T-Tp
T+Tp
τ
∆ NT
`
-∆ NT
T-Tp
G2( )
G2(- )
g1(-τ )
g1(τ )
(1,1),(2,2)
(2,1)
(1,2)
Fig. 3: The AF o wo pulses o PLFM signal.
0
Tp
T
-Tp
-T
-T-Tp
T+Tp
τ
1/Tp
-1/Tp
T-Tpg1(τ )
g1(-τ )
G2( )
G2(- )
Fig. 4: The AF o wo pulses o SP signal.
du e o a enua e he c oss- e ms o he PLFM signal
can be applied o he SP signal.
The AF o one pulse o he CC4 signal is shown in
Fig. 5 whe e he au o- e ms o he signal a e loca ed a
τ≤ |Tb|while he in a-pulse c oss- e ms a e sca e ed
on he ambigui y plane a −4Tb≤τ≤4Tb. Se ing
a lag window a τ≤ |Tb|is enough o ex ac he au o-
e ms om he c oss- e ms in he ambigui y domain.
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0
τ
Tb
-Tb
2Tb
3Tb
4Tb
-4Tb
-3Tb
-2Tb
Δ N*T Δ 2N*T Δ 3N*T
-Δ N*T
-Δ 2N*T-Δ 3N*T
G2( )
G2(- )
g1(-τ )
g1(τ )
(1,1)
(3,4) (2,1) (4,2)
(3,1) (4,2)
(4,1)
(4,3)(1,2)
(1,3)
(2,4)
(1,4)
(2,4)
Fig. 5: The AF o one pulse o he CC4 signal.
3.3. Es ima ion o Ke nel
Pa ame e s
Many s udies we e conduc ed o es ima e he sui able
ke nel pa ame e s o educing he in e e ence in he
TFR. Adap i e op imal ke nel TFR (AOK-TFR) [20] is
one o he ea lies echniques ha inco po a es window
in he AF and is able o p oduce accu a e TFR e en a
SNR o 0 dB. Howe e , he capabili y o he me hod is
limi ed o mul i-componen LFM signals.
Adap i e Op imal Ke nel Smoo h-Windowed
Wigne -Ville Dis ibu ion (AOK-SWWVD) [17] and
Adap i e Smoo hed Windowed c oss WVD (ASW-
WVD) [15] a e also able o p oduce accu a e TFR a
a low SNR bu hese solu ions a e limi ed o digi ally
modula ed signals such as FSK and PSK signals. All
he me hods men ioned abo e despi e p o en eliable
o signal ep esen a ion, hey lacked he capabili y o
co e a b oade class o signals. Thus, he Dopple -Lag
Block Sea ching (DLBS) p ocedu e is in oduced in
his sec ion o p oduce an accu a e TFR a low SNR
while co e ing a wide class o signals.
Be o e desc ibing he adap i e p ocedu es, i is c u-
cial o i s discuss he ou ambigui y unc ion quad-
an s as shown in Fig. 6. In he DLBS app oach, i is
su icien o es ima e he ke nel pa ame e s only in he
Q1quad an due o he symme ical p ope ies o he
ambigui y domain. One o he AF p ope ies exploi ed
is he maximum ene gy ha occu s a he o igin o he
ambigui y domain.
As de ined in [21], he AF is highes a he o igin in
compa ison o he o he pa s o he ambigui y domain
acco ding o he ollowing inequali y:
|Az( , τ)|≤|Az(0,0)|.(9)
Thus, he DLBS ini ia es he sea ch a he o igin and
checks o a signi ican d op in ene gy in Dopple and
lag. This is pe o med by ma ching he e e ence block,
0
τ
Q1
Q2
Q3
Q4
Fig. 6: The quad an di ision in he ambigui y domain.
Az(0,0) and he analyzed blocks, Az(λ1, λ2)in he AF
domain o ob ain he ke nel pa ame e s.
The DLBS algo i hm can be exp essed as ollows:
∆Az=|Az(0,0) −Az(λ1, λ2)|,(10)
whe e ∆Azis he ene gy di e ence be ween Az(0,0)
and Az(λ1, λ2)which can be de ined as
Az(0,0) =
∞
Z
−∞
∞
Z
−∞
wa( , τ)Az( , τ)d dτ.
Az(λ1, λ2) =
∞
Z
−∞
∞
Z
−∞
wa( −λ1, τ −λ2)Az( , τ)d dτ,
(11)
0≤λ1<∞,0≤λ2<∞,
whe e wa( , τ)is he analysis window, λ1is he ins an
Dopple and λ2is he ins an lag. The analysis window
can be desc ibed as [16]
wa( , τ) = a
τa
,(12)
0≤ ≤ a<1
Tp
,0≤τ≤τa< Tp,
whe e τaand aa e he analysis window size in e ms
o lag and Dopple espec i ely, Tpis he signal pulse
wid h. In o de o DLBS o accu a ely es ima e he
signal pulse wid h, he analysis window mus be se
smalle han he expec ed signal pulse wid h.
All he analyzed blocks ha ha e ene gy di e ence,
∆zabo e he h eshold alue, Az, hd a e conside ed as
a block wi h a minimum numbe o au o- e ms and can
be excluded om he gene a ion o he TFR. Figu e 7
shows he Q1quad an o PLFM signal and used as
an example o DLBS p ocedu e. The DLBS sea ch
p ocedu es a e as ollows:
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τ
1 2 3 4
6 7 8 9
11 12 13 14
16 17 18 19
Tp
1/4Tp
1/2Tp
3/4Tp
5/4Tp
5
10
15
20
21 22 23 24 25
1/Tp
1/0.5Tp
Figu e 7
Fig. 7: The Q1 o PLFM in DLBS me hod.
•Fo m he Dopple -lag unc ion, Az( , τ) o he Q1
quad an using Eq. (8).
•Compu e he o al ene gy o block 1 as a e e ence
block, Az(0,0) and se a h eshold alue, Az, hd.
•Compu e he ene gy di e ence o block 2,
Az(1/0.5Tp,1/4Tp). Since he ene gy di e ence
is below he h eshold alue con inue o he nex
block on he igh .
•The ene gy di e ence o block 3,
Az(1/0.75Tp,1/4Tp)is ound o be abo e he
h eshold alue which means a his pa icula
block he au o- e ms ene gy is low. Hence, he
e alua ion o ow (1/4)Tpis s opped and he
e alua ion mo es o he nex ow, (1/2)Tp.
•The e alua ion o new ow always s a s a one
column be o e he las e alua ed column because
i will ensu e he e alua ion s a whe e he au o-
e ms ene gy is highe . A he ow (1/2)Tp he
e alua ion s a s a block 7.
•The e alua ion o ow (1/2)Tps a s om block
7 o he igh di ec ion un il i eaches block 9.
Since he ene gy di e ence o block 9 is abo e
he h eshold alue, he sea ch block mo es o he
nex ow, (3/4)Tpand s a a block 13.
•S eps om 3 o 5 a e epea ed o he o he ows
un il eached ow (5/4)Tp. The ene gy di e ence
o all he blocks in his ow a e ound o be abo e
he h eshold alue and indica e ha he e alua-
ion o PLFM signal is comple ed.
•The ke nel pa ame e is es ima ed by choosing he
loca ion o las e alua ed block ha has ene gy
di e ence below he h eshold alue. Fo his ex-
ample, he las e alua ed block is block 19 and i s
loca ion ep esen he Dopple and lag pa ame e s
which is 1/Tpand Tp espec i ely.
The same p ocedu e is applicable o he o he sig-
nals as hei p ope ies in he AF a e simila wi h he
PLFM signal. The h eshold alue o 0.3is chosen
because i p o ides he bes Dopple and lag window
wid h o mos o he signal [19].
3.4. F ac ional Fou ie T ans o m in
C oss-Te ms Reduc ion
The F FT is ac ually a gene alized o m o Fou ie
T ans o m (FT) wi h he α- h o de o ac ional powe
[22] and is bes exp essed wi h he help o ans o ma-
ion ke nel. I x( )is he signal, hen he F FT o x( )
is gi en as [9]
Xα(u) =
∞
Z
−∞
x( )Kα( , u)d , (13)
whe e Kα( , u)is he ans o ma ion ke nel and can be
exp essed as
Kα( , u)= 1−jco α
2πexpj 2+u2
2co α− u csc α
.(14)
Equa ion (14) is alid i αis no mul iple o π. How-
e e , i αis a mul iple o 2π, he ke nel becomes δ( −u).
Fo (α+π)a mul iple o 2π, he ke nel becomes δ( +u).
The ex ension o he F FT o he TFR o he signal is
he o a ed e sion o he WVD o he o iginal signal
by he angle θwhich is gi en as
ρxα( , ) = R−θ{ρx( , )},(15)
whe e Rθ{} is he ope a o which o a es he TF plane
clockwise. I he ime- a ying signal is linea ly sepa-
able in he ime- equency plane, he c oss- e ms can
be sepa a ed wi h he app op ia e o a ion angle and
sui able TF il e ing echnique [5].
3.5. IF Es ima e
Ins an aneous F equency (IF) es ima ion can be used
o desc ibe he equency cha ac e is ics o he signal.
No mally, a good es ima o has o be consis en while
s a is ically and compu a ionally e icien [23]. Rao
and Taylo p o ed ha WVD peak based IF es ima ion
is op imal o linea FM signals o mode a e and high
SNR al hough he es ima o pe o mance deg ades sig-
ni ican ly a low SNR [24]. Peak based IF es ima o
can be exp essed as
a g max
ρ( , ),(16)
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whe e i( )is he IF and ρ( , )is he TFR. Peak based
IF es ima o is able o p oduce a decen es ima ion as
i s capabili y o localize ene gy along he IF law [23].
Fo he pu pose o measu ing he pe o mance o any
unbiased pa ame e es ima o , he C ame –Rao Lowe
Bound (CRLB) is equen ly used because i can p o-
ide he heo e ical limi o he a iance o he es ima-
o . The mos e icien es ima o is he one ha can
achie e he lowe bound on he a iance [25].
The gene al o mula ion o CRLB o IF es ima e is
[15]
a ˆ
l≥24
(2π)2γN (N2−1),(17)
whe e Nis he a e age window wid h, and γis he
SNR. Wi h he assump ion ha he ac ual IF o he
signal is known, and he signal is in disc e e o m, he
a iance o he IF es ima ed om measu emen can be
exp essed as [15]
a ˆ
i=1
N
N−1
X
n=0 ˆ
i(n)− i(n)2,(18)
whe e Nis he o al numbe o samples, ˆ
i(n)is he
ac ual IF and i(n)is he ac ual IF.
4. Resul s and Discussion
This sec ion desc ibes he TFRs and IF es ima es o
SP, CC4, LFM and CW-LFM signals using he DLBS-
WVD and F FT ollowed by he pe o mance o he
IF es ima o benchma ked wi h he CRLB. The pe -
o mance o DLBS-WVD is p esen ed a SNR o 5dB.
This alue is chosen because SNR abo e 10 dB is con-
side ed as high SNR om p e ious wo k on IF es ima-
ion [15].
4.1. TFR and IF Es ima e
Pe o mance
By using he DLBS-WVD, he TFR plo and IF es i-
ma e o he signals using he pa ame e s p esen ed in
Tab. 1 is p oduced. The ene gy o au o- e ms compo-
nen s is highes a he o igin and dec ease posi ioned
away om he o igin. Due o his, DLBS-WVD has
a di icul y o p ese e he en i e ene gy o he signal
componen s especially when he au o- e ms a e loca ed
a om he o igin. This p oblem becomes ob ious o
PLFM and CW-LFM signals compa ed o SP and CC4
signals as discussed in [19]. A small po ion o he
au o- e ms componen s a e supp essed o PLFM and
CW-LFM signals which p oduced mino e o s in he
TFR and IF es ima e.
(b) IF using DLBS
Time (ms)
(a) TFR using DLBS
Time-F equency Rep esen a ion
F equency (Hz)
00.002 0.004 0.006 0.008 0.01 0.012
0
5
10
15
x 106
00.002 0.004 0.006 0.008 0.01 0.012
0
2
4
6
8
10
12 x 10
6
Ins an aneous F eqeuncy
Time (ms)
F equency (Hz)
(a) TFR using DLBS.
(b) IF using DLBS
Time (ms)
(a) TFR using DLBS
Time-F equency Rep esen a ion
F equency (Hz)
00.002 0.004 0.006 0.008 0.01 0.012
0
5
10
15
x 106
00.002 0.004 0.006 0.008 0.01 0.012
0
2
4
6
8
10
12 x 10
6Ins an aneous F equency
Time (ms)
F equency (Hz)
(b) IF using DLBS.
Fig. 8: TFR plo and IF es ima e o wo pulses SP signal.
Figu e 8 shows he TFR plo and IF es ima e o
a wo pulses SP signal. The IF o he signal is accu-
a ely es ima ed a 10 MHz. This is con ibu ed om
a clean TFR o he signal due o he success ul sup-
p ession o he c oss- e ms in he AF domain.
The TFR and IF es ima e o a wo pulses CC4 signal
is shown in Fig. 9. The equency componen s o he
signal a e co ec ly es ima ed gi en ha he Cos as
sequence o he signal is [1 2 4 3]. In his pape , he
CC4 signal is used o illus a e he unc ionali y o he
DLBS-WVD on he Cos as coded class o signals.
Figu e 10 shows he analysis esul s o a wo pulses
PLFM signal. The equency componen o he signal
is es ima ed om 2.27–16.75 MHz while Tand Tpa e
13 ms and 4ms espec i ely. The ime componen s o
he signal a e accu a ely es ima ed bu he equency
componen o he signal is es ima ed wi h 1.6% e -
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(b) IF using DLBS
(a) TFR using DLBS
Time-F equency Rep esen a ion
F equenc y (Hz)
Time (ms)
00.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04
0
5
10
15
x 10
6
00.005 0.01 0.015 0.02 0.025 0.03
2
4
6
8
10
12
14
16
x 10
6
Ins an aneous F eqeuncy
Time (ms)
F equency (Hz)
(a) TFR using DLBS.
(b) IF using DLBS
Time-F equency Rep esen a ion
F equenc y (Hz)
Time (ms)
00.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04
0
5
10
15
x 10
6
00.005 0.01 0.015 0.02 0.025 0.03
2
4
6
8
10
12
14
16
x 10
6
(a) TFR using DLBS
Ins an aneous F equency
Time (ms)
F equency (Hz)
(b) IF using DLBS.
Fig. 9: TFR plo and IF es ima e o wo pulses CC4 signal.
o . This e o is due o ine i able supp ession o he
small pa o signal au o- e ms in he ambigui y do-
main. Howe e , he es ima ion e o is oo small and
can be conside ed insigni ican .
The TFR and IF es ima e is shown in Fig. 11 o
a wo pulses CW-LFM signal. The minimum and max-
imum equency o he signal is es ima ed a 2.19 MHz
and 5.83 MHz espec i ely. The e is abou 4.2% e o
in he equency componen es ima ion which is sligh ly
highe han he LFM signal. This is due o he in a-
pulse c oss- e ms in he CW-LFM signal which educes
he ene gy concen a ion o he signal componen s and
p oduce e o s in he IF es ima ion.
Time (ms)
(b) IF using DLBS
Time (ms)
(a) TFR using DLBS
Time-F equency Rep esen a ion
F equenc y (Hz)
00.005 0.01 0.015 0.02 0.025
0
5
10
15
x 10
6
0 5 10 15
x 10
-3
5
10
15
x 10
6
Ins an aneous F eqeuncy
F equency (Hz)
(a) TFR using DLBS.
Time (ms)
(b) IF using DLBS
Time (ms)
(a) TFR using DLBS
Time-F equency Rep esen a ion
F equenc y (Hz)
00.005 0.01 0.015 0.02 0.025
0
5
10
15
x 10
6
0 5 10 15
x 10
-3
5
10
15
x 10
6Ins an aneous F equency
F equency (Hz)
(b) IF using DLBS.
Fig. 10: TFR plo and IF es ima e o wo pulses PLFM signal.
4.2. TFR o WVD Using F FT
Using he same signal as in he p e ious sec ion, he
pe o mance o F FT in p oducing an accu a e TFR is
e alua ed. The IF es ima e o he signal using F FT
is simila o he esul om he p e ious sec ion gi en
ha he c oss- e ms a e success ully emo ed. I is im-
po an o pe o m a TF o a ion in such a way ha
he ac ional equency spec um is mos compac be-
o e applying any TF il e ing p ocedu e. The ech-
nique used in de e mining he co ec o de o F FT is
sea ching scheme [26]. This me hod de e mines he op-
imal ac ional powe by e alua ing he compac ness
o he ac ional equency spec um. The ac ional
powe ha p o ides he mos compac ac ional spec-
um will be used o TF il e ing.
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Time (ms)
(b) IF using DLBS
(a) TFR using DLBS
Time-F equency Rep esen a ion
F equenc y (Hz)
Time (ms)
00.005 0.01 0.015 0.02 0.025 0.03 0.035
2
4
6
8
x 10
6
00.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04
1
2
3
4
5
6x 10
6
Ins an aneous F eqeuncy
F equency (Hz)
(a) TFR using DLBS.
Time (ms)
(b) IF using DLBS
Time-F equency Rep esen a ion
F equenc y (Hz)
Time (ms)
00.005 0.01 0.015 0.02 0.025 0.03 0.035
2
4
6
8
x 10
6
00.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04
1
2
3
4
5
6x 10
6
(a) TFR using DLBS
Ins an aneous F equency
F equency (Hz)
(b) IF using DLBS.
Fig. 11: TFR plo and IF es ima e o wo pulses CW-LFM
signal.
The SP signal does no equi e any TF o a ion since
he equency spec um is mos compac a ac ional
powe ze o. The c oss- e ms ha a e loca ed be ween
he signal componen s can be easily emo ed by apply-
ing a TF il e ing as shown in Fig. 12(a) wi h a ed
do ed line esul ing in as shown in Fig. 12(b) a c oss-
e ms ee TFR o he SP signal.
Figu e 13 shows a TFR plo o wo pulses o CC4
signal which is hea ily co up ed by c oss- e ms. Fo
e e y pulse o CC4 signal, he e a e ou signal com-
ponen s ep esen ing ou di e en equencies as indi-
ca ed by ed do ed boxes. The localiza ion o c oss-
e ms ha a e e y close o he au o- e ms esul in
s ong deg ada ion o signal powe . The e is no sui -
able o a ion angle ha can be used o comple ely e-
mo e all he c oss- e ms. Fu he mo e, he smea ing
e ec om he c oss- e ms u he educe he quali y
o he signal componen s especially a a equency o
TFR o O iginal Signal
F equency (Hz)
Time (ms)
00.002 0.004 0.006 0.008 0.01 0.012
0
2
4
6
8
10
12
14
16
18
x 10
6
(a)
(b)
F equency (Hz)
Time (ms)
TFR A e Fil e a ion
00.002 0.004 0.006 0.008 0.01 0.012
0
2
4
6
8
10
12
14
16
18
x 10
6
(a)
TFR o O iginal Signal
F equency (Hz)
Time (ms)
00.002 0.004 0.006 0.008 0.01 0.012
0
2
4
6
8
10
12
14
16
18
x 10
6
(a)
(b)
F equency (Hz)
Time (ms)
TFR A e Fil e a ion
00.002 0.004 0.006 0.008 0.01 0.012
0
2
4
6
8
10
12
14
16
18
x 10
6
(b)
Fig. 12: TFR plo o wo pulses SP signal.
TFR o O iginal Signal
F equency (Hz)
Time (ms)
00.005 0.01 0.015 0.02 0.025 0.03
0
2
4
6
8
10
12
14
16
18
x 10
6
Fig. 13: TFR plo o wo pulses CC4 signal.
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