Sistem Komunikasi II(Digital Communication Systems)
Lecture #3: Demodulasi / Deteksi Baseband(Baseband Demodulation / Detection)
- PART I –
Topik:3.1 Pendahuluan.
3.2 Representasi Geometris dari Sinyal.
3.3 Optimal Detection: “Maximum Likelihood Detection”.
3.4 Energy/Symbol, Energy/Bit, and Minimum Distance.
3.5 Probabilitas Error untuk Transmisi Binary PAM dengan (Optimal) Maximum Lkelihood Detection.
3.6 Optimal Filter: “Matched Filter” or “Correlator
3.1. Introduction
Encoder ModulatorRF
Modulator
DecoderDemodulator& Detector
RF Demodulator
100101…
100101…
10101…
1011…
Filtering <Mapping <
Detection <
Block Diagram dari Sistem Komunikasi Digital:Transmitter
Receiver
noise
Kanal
3.1. Pendahuluan – cont.
mi adalah simbol digit yang me-representasikan informasi digital (message).
1 2[ , ,..., ]i M Alfabet simbolm m m m∈ ←
Transmiter Receiver(Channel)mi Si (t) x(t)
Simbol Digit
n(t)
Estimasi of mi
Contoh:1. Binary PAM: m1 = 0, m2 = 12. 4-ary PAM: m1 = 00, m2 = 01 , m3 = 10 , m4 = 11
ˆ im10101… 10101…
Sistem Komunikasi Digital (Baseband):
Simbol Waveform
3.1. Pendahuluan – cont.
Transmiter Receiver
Kanal AWGNmi Si (t) x(t)
Simbol Digit
n(t)
Estimasi of mi
ˆ im10101… 10101…
Sistem Komunikasi Digital (Baseband):
Simbol Waveform
Filter (Demod) Decision
sampling st kT=x(t) ˆ im
DetectionFiltering
⊕
( ) is White Gaussian Noise (WGN).n t
Goals:1. Menentukan bentuk filtering yang
optimal.
2. Menentukan bentuk detection yang optimal.
3.2. Representasi Geometris dari Sinyal
Representasi Geometris dari sinyal si(t) :
1
0( ) ( )1,2,...,
Ni ij j
j
t Ts t s ti M
φ=
≤ ≤= ⋅=∑
0
1,2,...,( ) ( )1,2,...,
Tij i j
i Ms s t t dtj N
φ
== ⋅=∫
( ) ; 1, 2,...,j t j Nφ =
Ekspansi
Koefisien Ekspansi
Fungsi Basis Orthonormal
0
1 ;( ) (: )
0 ;
T
i ji j
t tOrthonormal dti j
φ φ=
⋅ = ≠∫
Sintesis
Analisis
N M≤
3.2. Representasi Geometris dari Sinyal – cont.
( )is t
0
T
d t∫
0
T
d t∫
0
T
d t∫
1 ( )tφ
2 ( )tφ
( )N tφ
1is
2is
i Ns
Analisis: Sintesis:
∑
1 ( )tφ
2 ( )tφ
( )N tφ
1is
2is
i Ns
( )is t
01, 2,...,( ) ( ) ;ij i j
Tj Ns s t t dtφ == ⋅∫ 1
0( ) ( ) ;i ij j
N
jt Ts t s tφ
=≤ ≤= ⋅∑
3.2. Representasi Geometris dari Sinyal – cont.
Contoh: Binary PAM (NRZ)
0
1 ;( ) ( )
0 ;
T
i ji j
t t dti j
φ φ=
⋅ = ≠∫
Secara intuitif … fungsi basis:
T
A
s1(t)
-A
s2(t)
T
Tapi, K = ?
T
K
( )tφ
2
0
( ) 1T
i t dtφ =∫
2 2
0
( )T
t dt K Tφ =∫1KT
=
Ingat …
m1 = 1 m2 = 0
3.2. Representasi Geometris dari Sinyal – cont.
Contoh: Binary PAM – cont.
1 1
0
( ) ( )T
t t dt A Ts s φ= ⋅ =∫
T
A
S1(t)
-A
S2(t)
T
T
( )tφ1T
Analisis (koefisien ekspansi):
2 2
0
( ) ( )T
t t dt A Ts s φ= ⋅ = −∫
Sintesis:
1 1( ) ( ) ( )t t A T ts s φ φ= ⋅ = ⋅
2 2( ) ( ) ( )t t A T ts s φ φ= ⋅ = − ⋅
Fungsi Basis:
m1 = 1 m2 = 0
3.2. Representasi Geometris dari Sinyal – cont.
Contoh: Binary PAM – cont.
1 A Ts =
T
A
s1(t)
-A
s2(t)
T
T
( )tφ1T
Representasi Geometris:
2 A Ts = −
( )tφ
Fungsi Basis:
1s2s
A TA T− 0
Signal Space (Konstelasi Sinyal)1-Dimension (1D)
1 fungsi basis
si(t) si ~ sample
m1 = 1 m2 = 0
3.2. Representasi Geometris dari Sinyal – cont.
2
0
( ) 1T
i t dtφ =∫
1KT
=
Fungsi Basis( )tφ
K
T
m1 = 00, m2 = 01 m3 = 10, m4 = 11
Contoh: M-ary PAM (M=4)
s4(t)
-A
Tt
T
s2(t)
A/3
t
s3(t)
T-A/3
tT
s1(t)
A
t
t
3.2. Representasi Geometris dari Sinyal – cont.
4
3
2
1
1
0
2
0
3
0
4
0
( ) ( )
( ) ( )3
( ) ( )3
( ) ( )
T
T
T
T
t t dt A T
A Tt t dt
A Tt t dt
t t
s
dt A T
s
s
s
s
ss
s
φ
φ
φ
φ
= ⋅ = −
= ⋅ = −
= ⋅ =
= ⋅ =
∫
∫
∫
∫
Analisis (koefisien ekspansi):
Contoh: M-ary PAM (M=4) – cont.
Signal Space (Konstelasi Sinyal)1-Dimensi (1D)
1 Fungsi Basis
si(t) si ~ sample
Representasi Geometris:
0
3s4s 2s 1s
A T−3A T
−3A T A T
( )tφ
3.2. Representasi Geometris dari Sinyal – cont.
Point-point penting:
Sinyal Waveform‘dipetakan’ menjadi Sinyal Vektor
is( )i ts ; 1, 2,...,i M=
1, 2, ...;( )j j Ntφ =Fungsi basis berperan sebagai fungsi pemetaaan tersebut.
Fungsi basis bersifat Orthonormal:
2
0 0
1 ;( ) ( ) ( ) 1
0 ;
T T
i j i
i jt t dt t dt
i jφ φ φ
=⋅ = = ⇒
≠∫ ∫
3.3. Optimal Detection: “Maximum Likelihood Detection”
si (t) x(t)
n(t)
+
+⊕
Channel
DetectionFiltering
ˆ imFilter (Demod) Decision
sampling st kT=
z(t) z(kTs )
Baseband (PAM )Demodulation & Detection
x(t) = si (t) + n(t) 0
sT
d t∫
1
sT
( )tφ
Ts
ˆ imDecision
sampling st kT=
z(t) z(kTs )
DetectionMapping
3.3. Optimal Detection: “Maximum Likelihood Detection” – cont.
x(t) = s(t) + n(t) 0
sT
d t∫m̂
Decision
sampling st kT=
z(t)
sample(test statistics)
Detection
MAPPING : Waveform Sample
z(kTs )
Mapping
1
sT
( )tφ
Ts
Konstelasi Sinyal (untuk Binary PAM NRZ):
Binary PAM NRZ
Ts
A
s1(t)
1 1m =-A
s2(t)
Ts
2 0m =
( )tφ1s2s
sA TsA T− 0
3.3. Optimal Detection: “Maximum Likelihood Detection” – cont.
x(t) = s(t) + n(t) Decision
DetectionMapping
Konstelasi Sinyal (untuk Binary PAM NRZ):
( )tφ1s2s
sA TsA T− 0
Binary PAM NRZ
DECISION : Bandingkan test statistic VS. sebuah nilai threshold.
Ts
A
s1(t)
1 1m =
sampling st kT=
z(t) z(kTs ) ( )sz kTm̂
<
<1H
2Hλ
-A
s2(t)
Ts
2 0m =
0
sT
d t∫
1
sT
( )tφ
Ts
3.3. Optimal Detection: “Maximum Likelihood Detection” – cont.
( ) ( )
( ) ( )
( ) ( ) ( ) ( )
( ) ( )
0
0 0
0
s
s
s s
t = kT
t = kT
t = kT t = kT
s
s s
s
T
T T
i
i
z kT z t
x t t dt
s t t dt n t t dt
s k n k
φ
φ φ
=
=
= +
= +
∫
∫ ∫
2igaussian random variable ~ N(s , )nσ
2gaussian random variable ~ N(0, )nσ
mean variance0iz s n= +
( ) ( ) ( )ix t s t n t= +
3.3. Optimal Detection: “Maximum Likelihood Detection” – cont.
0
0
0
0 2
2121( )
2e
n
nn
p nσ
πσ
=
0 0n
1 sTs A=2 sTs A= − 0 z
0
2
02
22
12
2
1( | ) en
nz s
p z sσ
πσ
−
=0
2
01
12
12
2
1( | ) en
nz s
p z sσ
πσ
−
=
P D F o f W G N
2Conditional PDF of ( dikirim) z s 1Conditional PDF of ( dikirim)z s
3.3. Optimal Detection: “Maximum Likelihood Detection” – cont.
1 sTs A=2 sTs A= −
0 z
2( | )p z s 1( | )p z s
Likelihood R atio T est:
0z
0 1( | )p z s
0 2( | )p z s
Likelihood s1Likelihood s2
( )( )
0 10
0 2
|( )
|p z s
zp z s
Λ = <
<1H
2H
( )( )
1
2
p sp s
1 1
2 2
H s d ik irim .
H s d ik irim .
=
=1
3.3. Optimal Detection: “Maximum Likelihood Detection” – cont.
Likelihood R atio T est:
( )( )
10
2
|( )
|p z s
zp z s
Λ = <
<1H
2H
( )( )
1
2
1p sp s
=
0
0
0
0
2
2
2
1
2
2
1
2
1
2
2
2
1 exp
1 exp
n
n
n
n
z s
z s
πσ
πσ
σ
σ
− =
−
<<1H
2H
1
0 0 0
0 0 0
2 20 0 11
2 2 2
2 20 0 22
2 2 2
2exp exp exp
2 2 2
2exp exp exp
2 2 2
n n n
n n n
z z ss
z z ss
σ σ σ
σ σ σ
− ⋅ − ⋅ −
= − ⋅ − ⋅ −
<
<1H
2H1
( ) ( )1 2
Untuk 'equi-probable' binary simbol digit:
1/ 2p s p s= =
3.3. Optimal Detection: “Maximum Likelihood Detection” – cont.
Likelihood R atio T est:
<
<1H
2H
1
<<1H
2H[ ]ln 1 0=
1 2
2s s+
0z <
<1H
2H
Maximum Likelihood (ML) Detection Ruleuntuk Transmisi Binary PAM
( )0 0
2 20 1 2 1 2
0 2 2
( ) ( )exp2n n
z s s s szσ σ
− −Λ = −
( )0 0
2 20 1 2 1 2
0 2 2
( ) ( )ln2n n
z s s s szσ σ− −
Λ = −
0
0 1 22
( )
n
z s sσ−
<
<1H
2H 0
2 21 2
2
( )2 n
s sσ−
3.3. Optimal Detection: “Maximum Likelihood Detection” – cont.
Likelihood R atio T est:
Contoh: Binary PAM
1 2 ( ) 0
2 2s sA T A Ts s + −+
= =0z <
<1H
2H
Konstelasi Sinyal:
Ts
A
s1(t)
1 1m =-A
s2(t)
Ts
2 0m =
( )tφ1s2s
sA TsA T− 0
3.3. Optimal Detection: “Maximum Likelihood Detection” – cont.
x(t) = s(t) + n(t) 0
sT
d t∫
1
sT
( )tφ
Ts
Decision
MLDetectionMappingBinary PAM NRZ
Ts
A
s1(t)
1 1m =-A
s2(t)
Ts
2 0m =
sampling st kT=
z(t) z(kTs ) zm̂
<
<1H
2H0
( )tφ1s2s
sA TsA T− 0
Decision Region IDecision Region II
(Optimal) Maximum Likelihood Detection untuk Binary PAM NRZ:
3.3. Optimal Detection: “Maximum Likelihood Detection” – cont.
Likelihood R atio T est:
Contoh: 4-Ary PAM
4 3 ( 3) 2 2 2 3
s ss
A T A Ts sA T
− + −+= = −0z <
<3H
4H
0
3s4s 2s 1s
sA T−
3sA T
−3
sA T sA T( )tφ
3 2 3 3 0
2 2s sA T A Ts s − ++
= =0z <
<2H
3H
2 1 3 2 2 2 3
s ss
A T A Ts s A T++
= =0z <
<1H
2H
3 Nilai Threshold
3.3. Optimal Detection: “Maximum Likelihood Detection” – cont.
x(t) = s(t) + n(t) 0
sT
d t∫
1
sT
( )tφ
Ts
Decision
MLDetection
Mapping
sampling st kT=
z(t) z(kTs )
z
m̂
<
<3H
4H
23 sA T−
(Optimal) Maximum Likelihood Detection untuk M-ary PAM:
z <
<2H
3H0
z <
<1H
2H
23 sA T
0
3s4s 2s 1s
sA T−3
sA T−
3sA T
sA T( )tφ
IIIII IIV
3.3. Optimal Detection: “Maximum Likelihood Detection” – cont.
x(t) = s(t) + n(t) 0
sT
d t∫
1
sT
( )tφ
Ts
MLDetectionFungsi : Mapping
Hardware: Correlator
zDecisionsampling st kT=
z(t) z(kTs )m̂
<<1H
2H0
Correlator Receiver dengan ML Detection untuk Binary PAM NRZ
z
3.4. Energy/Symbol, Energy/Bit, dan Minimum Distance.
1
min
.;M = Jumlah simbol di dalam alfabetEnergy/Symbol,
Energy/Bit,
Minimum Distance,
1
jarak antara 2 simbol yang terdekat .
k
M
s sk
bb s
s
E EM
TE E
T
D
=
=
=
=
∑
2min , 2
2
b s s s
b
E E A T D A T
E
= = =
=
( )tφ1s2s
sA TsA T− 0
Contoh: Binary PAM NRZ
Ts
A
s1(t)
1 1m =-A
s2(t)
Ts
2 0m =
Konstelasi Sinyal:
3.4. Energy/Symbol, Energy/Bit, dan Minimum Distance – cont.
Contoh: 4-ary PAM
2 2min
5 1 5 2 , = , 9 2 18 3
2 = 25
s s b s s s
b
E A T E E A T D A T
E
= = =
s4(t)
-A
Tst
Ts
s2(t)
A/3t
s3(t)
Ts-A/3t
Ts
s1(t)
A
t
0
3s4s 2s 1s
sA T−3
sA T−
3sA T sA T
( )tφKonstelasi Sinyal:
3.5. Probabilitas Error untuk Transmisi Binary PAM dengan (Optimal) Maximum Likelihood Detection
1 2
1 2
2
1 1 2 2
2
( | ) ( | ) ( | ) ( | )
s sz
s sz
P e s p z s dz P e s p z s dz
+=
∞
+−∞ =
= =∫ ∫
2( | )p z s 1( | )p z s
Probabilitas Error:
1 2
2s s+
z
1 1 2 2( | ) ( ) ( | ) ( )eP P e s P s P e s P s= ⋅ + ⋅ - Probabilitas Total Rata2
[ ]1 21 ( | ) ( | )2P e s P e s= + - equi-probable simbol digit
2 1( | ) ( | )P e s P e s= = - conditional PDF simetrik
1 2
2s s+z <
<1H
2H
ML Detection
2s 1s
3.5. Probabilitas Error untuk Transmisi Binary PAM dengan (Optimal) Maximum Likelihood Detection – cont.
1 2 1 2 0 0
2
22
2 2
1 1( | ) exp22e
s s s s n nz z
z sP p z s dz dzσσ π
∞ ∞
+ += =
− = =
∫ ∫
0
0 0
2n
1 n n
z s duu du dzdz
σσ σ
−= ⇒ = → =
- pergantian variabel
1 2
0
2
2
1 1 exp22
n
s su
u du
σ
π
∞
−=
= ∫
- Complementary Error Function (Q-Function) - ditabulasikan -
1 2 0
0
2 1 2e
2
1 1P exp Q2 22
n
s s n
s su du
σ
σπ
∞
−
− = = ∫
Probabilitas Error:
3.5. Probabilitas Error untuk Transmisi Binary PAM dengan (Optimal) Maximum Likelihood Detection – cont.
Probabilitas Error:
Contoh: Binary PAM
0
0
0
1 2e
min
0 0 0
P Q2
( ) Q
2
2 2 Q Q Q Q
2
n
s s
n
s s b
n
s s
A T A T
A T E E DN N N
σ
σ
σ
−=
− −=
= = = =
( )tφ1s2s
sA TsA T− 0
Symbol-Error Rate (SER)Bit-Error Rate (BER)
3.6. Optimal Filter: “Matched Filter” or “Correlator”
( )Respon Impuls: ( )sh t s T t= −
,dimana ( ) adalah sinyal input, adalah durasi dari s(t).ss t T
Kriteria optimal untuk filtering:
Bentuk demodulator filter yang optimal adalah filter yang me-maksimalkan Signal-to-Noise Power Ratio (SNR) pada output-nya.
Filter yang memenuhi kriteria di atas: Matched Filter
Untuk Binary PAM NRZ:
Ts
Ah(t)
ˆ imMatched Filter Decision
sampling st kT=
z(t) z(kTs )⊕Si (t)
n(t)
DetectionFiltering
3.6. Optimal Filter: “Matched Filter” or “Correlator” – cont.
Matched Filter sebagai Correlator – cont.
( )x t ix0
sT
d t∫
( )tφ
ix( )x t
( ) ( )sh t T tφ= −
h(t)
st kT=
Matched FilterCorrelator
Ekuivalensi antara Correlator dan Matched Filter :
0
( ) ( )sT
ix x t t d tφ= ⋅∫
( )y t
( ) ( ) ( )y x t h t d tτ τ= ⋅ −∫
( ) ( )sx t T t dtφ τ= ⋅ − +∫( ) ( ) ( )s iy T x t t d t xφ= ⋅ =∫
3.6. Optimal Filter: “Matched Filter” or “Correlator” – cont.
x(t) = s(t) + n(t) 0
sT
d t∫
1
sT
( )tφ
Ts MLDetectionCorrelation
Optimal Receiver dengan ML Detection untuk Binary PAM NRZ:
z
m̂Matched Filter z(t) z(kTs )
Filtering
x(t) = s(t) + n(t)
Decisionsampling st kT=
z(t) z(kTs ) m̂<
<1H
2H0z
Decision
<
<1H
2H0z
MLDetection
Correlator Receiverwith ML Detection
Matched Filter Receiverwith ML Detection
( ) ( )sh t T tϕ= −
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