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 Block Diagram dari Sistem Komunikasi Digital: Transmitter 100101…
10101…
Encoder
Modulator
RF Modulator Kanal noise
100101…
1011…
Decoder
Demodulator & Detector Filtering < Mapping < Detection <
Receiver
RF Demodulator
3.1. Pendahuluan – cont. Sistem Komunikasi Digital (Baseband): Simbol Digit
mi
Estimasi of mi
Simbol Waveform
Transmiter
Si (t)
x(t) (Channel)
Receiver
10101…
mˆ i 10101…
n(t) mi adalah simbol digit yang me-representasikan informasi digital (message).
mi ∈ [m1, m2,..., mM ]
← Alfabet simbol
Contoh: 1. Binary PAM: 2. 4-ary PAM:
m1 = 0, m2 = 1 m1 = 00, m2 = 01 , m3 = 10 , m4 = 11
3.1. Pendahuluan – cont. Sistem Komunikasi Digital (Baseband): Simbol Digit
Estimasi of mi
Simbol Waveform Kanal AWGN
mi
Transmiter
Si (t)
10101…
⊕
x(t) Receiver
mˆ i 10101…
n(t)
n ( t ) is White Gaussian Noise (WGN). Goals: 1. Menentukan bentuk filtering yang optimal. 2. Menentukan bentuk detection yang optimal.
tsampling = kTs x(t)
Filter (Demod) Filtering
Decision Detection
mˆ i
3.2. Representasi Geometris dari Sinyal Representasi Geometris dari sinyal si(t) : N
Ekspansi
si(t ) = ∑ sij ⋅φ j(t ) j =1
Sintesis 0 ≤ t ≤ T i = 1,2,..., M
T
Koefisien Ekspansi
sij = ∫ si(t ) ⋅φ j(t ) dt 0
N≤M
i = 1,2,..., M j = 1,2,..., N
Analisis Fungsi Basis Orthonormal
φ j (t ) ; j = 1, 2,..., N
1 ∫0 φ i(t ) ⋅ φ j (t ) dt = 0
T
Orthonormal :
;i = j ;i ≠ j
3.2. Representasi Geometris dari Sinyal – cont. Analisis:
Sintesis:
T
∫
dt
s i1
s i1
0
φ1 ( t )
φ1 ( t ) T
∫
si ( t )
dt
si 2
si 2
0
φ 2 (t )
φ 2 (t )
T
∫
dt
si N
∑
si ( t )
si N
0
φ N (t ) T
sij = ∫ si (t ) ⋅φ j (t ) dt 0
φ N (t )
; j = 1, 2,..., N
N
si (t ) = ∑ sij ⋅φ j (t ) j =1
;0 ≤ t ≤ T
3.2. Representasi Geometris dari Sinyal – cont.
Contoh: Binary PAM (NRZ)
Ingat …
1 ;i = j ∫0 φ i(t ) ⋅ φ j (t ) dt = 0 ; i ≠ j
T
s1(t)
s2(t)
A T T
T
2 φ ∫ i (t ) dt = 1
-A
m1 = 1
m2 = 0
0
Secara intuitif … fungsi basis:
φ (t ) Tapi, K = ?
K T
T
2 2 φ ( t ) dt = K T ∫ 0
K=
1 T
3.2. Representasi Geometris dari Sinyal – cont.
Contoh: Binary PAM – cont.
Analisis (koefisien ekspansi): T
S1(t)
S2(t)
A T T
Fungsi Basis:
φ (t )
0
T
s 2 = ∫ s 2(t ) ⋅ φ (t ) dt = − A
m2 = 0 Sintesis:
s1(t ) = s1 ⋅ φ (t ) = A
1
T
0
-A
m1 = 1
s1 = ∫ s1(t ) ⋅ φ (t ) dt = A T
T ⋅ φ (t )
T
T
s 2(t ) = s 2 ⋅ φ (t ) = − A T ⋅ φ (t )
3.2. Representasi Geometris dari Sinyal – cont.
Contoh: Binary PAM – cont. s1(t)
s2(t)
Representasi Geometris:
s1
s2
A
−A T
T T
0
φ (t )
A T
-A
m1 = 1
m2 = 0
Signal Space (Konstelasi Sinyal) 1-Dimension (1D)
Fungsi Basis:
φ (t )
¾ 1 fungsi basis
s1 = A
1
T
T
T
s2 = − A T
¾ si(t) Æ si ~ sample
3.2. Representasi Geometris dari Sinyal – cont. Contoh: M-ary PAM (M=4) s4(t)
s3(t)
s2(t)
s1(t) A
A/3 t
T
T
t
T
t
T
-A/3 -A
m1 = 00,
φ (t )
m2 = 01
Fungsi Basis
m3 = 10, T
2 φ i ∫ (t ) dt = 1 0
K T
t
1 K= T
m4 = 11
t
3.2. Representasi Geometris dari Sinyal – cont. Contoh: M-ary PAM (M=4) – cont. Representasi Geometris:
Analisis (koefisien ekspansi): T
s4 = ∫ s1(t ) ⋅ φ (t ) dt = − A
T
0
T
s3 = ∫ s 2(t ) ⋅ φ (t ) dt = − 0
T
s2 = ∫ s 3(t ) ⋅ φ (t ) dt = 0
A T 3
A T 3
T
s1 = ∫ s 4(t ) ⋅ φ (t ) dt = A 0
T
s4
s3
s2
s1
−A T
A T − 3
0 A T
A T
3
Signal Space (Konstelasi Sinyal) 1-Dimensi (1D) ¾ 1 Fungsi Basis ¾ si(t) Æ si ~ sample
φ (t )
3.2. Representasi Geometris dari Sinyal – cont.
Point-point penting: ¾ Sinyal Waveform‘dipetakan’ menjadi Sinyal Vektor
JK
si
si (t ) ¾ Fungsi basis
φ j(t ) ; j = 1, 2, ...N
; i = 1, 2,..., M
berperan sebagai fungsi
pemetaaan tersebut. ¾ Fungsi basis bersifat Orthonormal:
1 ;i = j ∫0 φ i(t ) ⋅ φ j (t ) dt = 0 ; i ≠ j
T
T
⇒
∫φ
i
0
2
(t ) dt = 1
3.3. Optimal Detection: “Maximum Likelihood Detection” Channel
⊕+
tsampling = kTs
+ si (t)
Filter (Demod) z(t)
x(t)
z(kTs )
Filtering
n(t)
∫
dt
0
1 Ts
Detection
tsampling = kTs
Ts
x(t) = si (t) + n(t)
Decision
z(t)
mˆ i
z(kTs )
Decision
mˆ i
φ (t ) Ts Mapping
Baseband (PAM ) Demodulation & Detection Detection
3.3. Optimal Detection: “Maximum Likelihood Detection” – cont. MAPPING : Waveform Æ Sample
tsampling = kTs Ts
∫
x(t) = s(t) + n(t)
dt
0
1 Ts
z(t)
z(kTs )
Decision
φ (t )
mˆ
sample (test statistics)
Ts Mapping
Binary PAM NRZ s1(t)
s2(t)
Detection
s1
s2
A Ts Ts
m1 = 1
-A
m2 = 0
− A Ts
0
φ (t )
A Ts
Konstelasi Sinyal (untuk Binary PAM NRZ):
3.3. Optimal Detection: “Maximum Likelihood Detection” – cont. DECISION : Bandingkan test statistic VS. sebuah nilai threshold.
Ts
∫
x(t) = s(t) + n(t)
dt
0
1 Ts
z(t)
z(kTs )
φ (t )
Decision H1
z( kTs ) <
<
tsampling = kTs
λ
mˆ
H2
Ts Mapping
Binary PAM NRZ s1(t)
s2(t)
Detection
s1
s2
A Ts Ts
m1 = 1
-A
m2 = 0
− A Ts
0
φ (t )
A Ts
Konstelasi Sinyal (untuk Binary PAM NRZ):
3.3. Optimal Detection: “Maximum Likelihood Detection” – cont.
z ( kTs ) = z ( t ) t = kT
x ( t ) = si ( t ) + n ( t )
s
Ts
= ∫ x ( t ) φ ( t ) dt 0
t = kTs
Ts
= ∫ si ( t ) φ ( t ) dt 0
+ t = kTs
Ts
∫ n ( t ) φ ( t ) dt 0
t = kTs
= si ( k ) + n0 ( k )
z = si + n0
mean
variance
gaussian random variable ~ N(0, σ n2 ) gaussian random variable ~ N (s i , σ n2 )
3.3. Optimal Detection: “Maximum Likelihood Detection” – cont.
PD F of W G N
p ( n0 ) =
1 2 πσ
e
2 n0
1 n0 2 σ n0
p ( z | s2 ) =
z (s2
1 2 πσ n2
e
dikirim)
1 2
z − s2 n0
Conditional PDF of
2
σ
p ( z | s1 ) =
0
s2 = − A Ts
0
2
n0
0 Conditional PDF of
s1 = A Ts
z (s1
1 2πσ
2 n0
e
dikirim)
1 2
z − s1
σ n0
z
2
3.3. Optimal Detection: “Maximum Likelihood Detection” – cont.
L ik e lih o o d R a tio T est:
p(z | s2 )
p(z | s1 )
p(z0 | s1 ) p(z0 | s2 ) Likelihood s1
Likelihood s2
0
s2 = − A Ts
p ( z0 | s2 ) p ( s2 )
H1
<
Λ(z0 ) =
p ( z0 | s1 ) p ( s1 )
z0
s1 = A Ts
z
<
H2
1
H 1 = s 1 d ik irim . H 2 = s 2 d ik irim .
3.3. Optimal Detection: “Maximum Likelihood Detection” – cont.
L ik e lih o o d R a tio T est:
=
2πσ n2
0
1 2πσ n2
0
H2
p ( s2 )
=1
1 z − s 1 exp 2 σ n0 1 z − s 2 exp 2 σ n0
z 02 exp − 2σ n2 0 = z 02 exp − 2σ n2 0
Untuk 'equi-probable' binary simbol digit:
p ( s1 ) = p ( s2 ) = 1/ 2
2
2
s12 ⋅ exp − 2 2 σ n 0 s22 ⋅ exp − 2 2 σ n0
H1
<
1
H2
2 z 0 s1 ⋅ exp − 2 2 σ n 0 2 z 0 s2 ⋅ exp − 2 2 σ n0
H1
<
1
<
p ( s1 )
<
p ( z | s2 )
H1
<
Λ(z0 ) =
p ( z | s1 )
<
H2
1
3.3. Optimal Detection: “Maximum Likelihood Detection” – cont.
L ik e lih o o d R a tio T est:
z 0 ( s1 − s2 )
σ n2
0
H1
<
z0
<
H2
s1 + s2 2
1
H2 H1
<
ln [1] = 0
H2 H1
<
σ
2 n0
( s12 − s22 ) − 2σ n20
<
<
ln Λ ( z 0 ) =
z 0 ( s1 − s2 )
H1
<
z 0 ( s1 − s2 ) ( s12 − s22 ) Λ ( z 0 ) = exp − 2 2 σ n0 2σ n0
<
H2
( s12 − s22 ) 2σ n20
Maximum Likelihood (ML) Detection Rule untuk Transmisi Binary PAM
3.3. Optimal Detection: “Maximum Likelihood Detection” – cont.
L ik e lih o o d R a tio T est: Contoh: Binary PAM Konstelasi Sinyal: s1(t)
s2(t)
s1
s2
A Ts Ts
m1 = 1 H1
<
z0
<
H2
− A Ts
0
-A
m2 = 0
A Ts + ( − A Ts ) s1 + s2 = = 0 2 2
A Ts
φ (t )
3.3. Optimal Detection: “Maximum Likelihood Detection” – cont. (Optimal) Maximum Likelihood Detection untuk Binary PAM NRZ:
tsampling = kTs
Decision
∫
x(t) = s(t) + n(t)
dt
z(t)
0
1 Ts
z(kTs )
z
φ (t )
s1(t)
<
0
mˆ
H2
Ts
Binary PAM NRZ
H1
<
Ts
ML Detection
Mapping
s2(t) Decision Region II
A Ts Ts
m1 = 1
s1
s2
-A
m2 = 0
Decision Region I
− A Ts
0
A Ts
φ (t )
3.3. Optimal Detection: “Maximum Likelihood Detection” – cont.
L ik e lih o o d R a tio T est: Contoh: 4-Ary PAM
s4
s3
− A Ts
A Ts
−
3
s2 0 A Ts
s1
φ (t )
A Ts
3
3 Nilai Threshold H1
<
z0
<
H2
A Ts 3 + A Ts s2 + s1 = 2 2
2 = A Ts 3
− A Ts 3 + A Ts 3 s3 + s 2 = = 0 < 2 2 H
H2
<
z0
3
− A Ts + ( − A Ts 3) s 4 + s3 2 = = − A Ts < 2 2 3 H
H3
<
z0
4
3.3. Optimal Detection: “Maximum Likelihood Detection” – cont. (Optimal) Maximum Likelihood Detection untuk M-ary PAM:
tsampling = kTs
Decision
dt
z(t)
0
z(kTs )
φ (t )
H2
z
Ts
IV
III
II
I
s4
s3
s2
s1
−
A T s 0 A Ts 3 3
A Ts
mˆ
H2
<
0
H3
Mapping
− A Ts
2 < 3 A Ts
z φ (t )
H3
2 − < 3 A Ts
<
1 Ts
z
H1
<
∫
x(t) = s(t) + n(t)
<
Ts
H4
ML Detection
3.3. Optimal Detection: “Maximum Likelihood Detection” – cont.
Correlator Receiver dengan ML Detection untuk Binary PAM NRZ
Ts
∫
x(t) = s(t) + n(t)
dt
0
1 Ts
z(t)
φ (t )
z(kTs )
z
Decision H1
z
<
tsampling = kTs
<
0
H2
Ts
Fungsi : Mapping Hardware: Correlator
ML Detection
mˆ
3.4. Energy/Symbol, Energy/Bit, dan Minimum Distance.
1 Energy/Symbol, E s = M
M
∑E k =1
sk
;M = Jumlah simbol di dalam alfabet.
Tb Es Energy/Bit, Eb = Ts Minimum Distance, Dmin = jarak antara 2 simbol yang terdekat. Contoh: Binary PAM NRZ s1(t)
Konstelasi Sinyal:
A Ts Ts
m1 = 1
s1
s2
s2(t)
-A
m2 = 0
− A Ts
Eb = E s = A 2 Ts
0
,
φ (t )
A Ts
Dmin = 2 A Ts = 2 Eb
3.4. Energy/Symbol, Energy/Bit, dan Minimum Distance – cont.
Contoh: 4-ary PAM s4(t)
s2(t)
s3(t)
s1(t) A
A/3 Ts
t -A/3
Ts
t
t
Ts
Ts
t
-A
Konstelasi Sinyal:
5 2 E s = A Ts 9
,
s4
s3
− A Ts
A Ts
−
s2 0 A Ts
φ (t )
A Ts
3
3
1 5 2 Eb = E s = A Ts 2 18
s1
,
Dmin
2 = A Ts 3 2 = 2 Eb 5
3.5. Probabilitas Error untuk Transmisi Binary PAM dengan (Optimal) Maximum Likelihood Detection Probabilitas Error:
p ( z | s1 )
p ( z | s2 ) ML Detection
s1 + s2 z < 2 H H1
s2
<
P ( e | s1 ) =
s1 + s2 2
∫
∞
p ( z | s1 ) dz
−∞
Pe = P ( e | s1 ) ⋅ P ( s1 ) + P ( e | s2 ) ⋅ P ( s2 ) =
z
s1 + s 2 2
2
z=
s1
1 [ P (e | s1 ) + P (e | s2 ) ] 2
= P ( e | s2 ) = P ( e | s1 )
∫
P (e | s2 ) = z=
p ( z | s2 ) dz
s1 + s2 2
- Probabilitas Total Rata2 - equi-probable simbol digit - conditional PDF simetrik
3.5. Probabilitas Error untuk Transmisi Binary PAM dengan (Optimal) Maximum Likelihood Detection – cont.
Probabilitas Error: ∞
∫
Pe =
s +s z= 1 2 2
z − s2 u= σn 0
∞
∫
= u=
∞
Pe =
∫
s1 − s2 2 σ n0
1 1 z − s2 exp p ( z | s2 ) dz = ∫ 2 σ 2π s1 + s2 σ n0 n0 z= 2 ∞
s1 − s2 2 σ n0
⇒
du 1 = dz σ n0
→
σ n du = dz 0
dz
- pergantian variabel
1 exp u 2 du 2π 2 1
s1 − s2 1 2 exp u du = Q 2σ n 2π 2 0 1
2
- Complementary Error Function (Q-Function) - ditabulasikan -
3.5. Probabilitas Error untuk Transmisi Binary PAM dengan (Optimal) Maximum Likelihood Detection – cont.
Probabilitas Error:
s1
s2
φ (t )
Contoh: Binary PAM
− A Ts s1 − s2 Pe = Q 2σ n 0
0
A Ts
A Ts − ( − A Ts ) = Q 2σ n0 A Ts = Q σn 0
2 Es = Q N0
2 Eb = Q N0
Dmin = Q 2N0
Bit-Error Rate (BER) Symbol-Error Rate (SER)
3.6. Optimal Filter: “Matched Filter” or “Correlator” Kriteria optimal untuk filtering: Bentuk demodulator filter yang optimal adalah filter yang memaksimalkan Signal-to-Noise Power Ratio (SNR) pada output-nya. Filter yang memenuhi kriteria di atas: Matched Filter
Respon Impuls:
h ( t ) = s (Ts − t )
,dimana s (t ) adalah sinyal input, Ts adalah durasi dari s(t). h(t)
Untuk Binary PAM NRZ: A
n(t) Si (t)
⊕
tsampling = kTs
Ts
Matched Filter
z(t)
Filtering
z(kTs )
Decision Detection
mˆ i
3.6. Optimal Filter: “Matched Filter” or “Correlator” – cont. Matched Filter sebagai Correlator – cont. Ekuivalensi antara Correlator dan Matched Filter : Matched Filter
Correlator
t = kTs Ts
∫
x(t )
0
dt
xi
φ (t )
xi =
Ts
∫ x (t ) ⋅ φ (t ) d t 0
x(t )
h(t)
y (t )
xi
h ( t ) = φ (T s − t ) y (τ ) =
∫ x ( t ) ⋅h (τ
− t) dt
= ∫ x(t ) ⋅φ (τ − Ts + t ) dt y (T s ) =
∫ x ( t ) ⋅φ ( t ) d t
= xi
3.6. Optimal Filter: “Matched Filter” or “Correlator” – cont. Optimal Receiver dengan ML Detection untuk Binary PAM NRZ:
Ts
dt
0
Correlator Receiver with ML Detection
1 Ts
z(t)
z(kTs )
φ (t ) Ts
Matched Filter Receiver
Matched Filter
h ( t ) = ϕ (Ts − t )
with ML Detection Filtering
<
0
mˆ
H2
ML Detection
Correlation
x(t) = s(t) + n(t)
zz
H1
Decision z(t)
z(kTs )
z
H1
<
∫
x(t) = s(t) + n(t)
Decision
<
tsampling = kTs
<
0
H2
ML Detection
mˆ
