continuous time (sürekli zaman) (zamanda Öteleme veya...
TRANSCRIPT
(Zamanda Öteleme veya Kaydırma)
t -1 0 1
2
1
-
-
10
101
0122
10
)(
t
t
tt
t
tx
t 0 1 2
2
1
-
-
---
--
-
2110
211101
1001122)1(2
0110
)1(
tt
tt
tttt
tt
tx
t -2 -1 0
2
1
-
---
--
0110
011101
12011422)1(2
2110
)1(
tt
tt
tttt
tt
tx
Continuous Time (Sürekli Zaman)
(Zamanda Ölçekleme)
t -1 0 1
2
1
-
-
10
101
0122
10
)(
t
t
tt
t
tx
t -1/2 0 1/2
2
1
--
--
2/1120
2/101201
02/1021242)2(2
2/1120
)2(
tt
tt
tttt
tt
tx
t -2 -1 0 2
2
1
--
--
212
10
2112
101
0202
1122)
2
1(2
212
10
)2
1(
tt
tt
tttt
tt
tx
(Zamanda Tersine Çevirme)
t -1 0 1
2
1
-
-
10
101
0122
10
)(
t
t
tt
t
tx
t -1 0 1
2
1
-
-
----
--
-
10
011
1001222)(2
110
)(
t
t
tttt
tt
tx
Örnek:
))2/1(2(2)12(2 --- txtx
t -1 0 1
2
1
t -1 0 1
2
1
t - ½ 0 ½
2
1
t 0 ½ 1
2
4
x(t)
x1(t)=x(-t)
x2(t)=x1(2t)= x(-2t)
x3(t)=2x2(t- ½)= 2x(-2t+1)
Discrete Time (Ayrık Zaman)
n -3 -2 -1 0 1 3 4
1 1 1
. . .
2
-1
2 . . .
-
-
-
-
40
31
21
10
01
12
21
30
][
n
n
n
n
n
n
n
n
nx
-
-
--
-
-
--
---
---
-
5410
4311
3211
2110
1011
0112
1211
2310
]1[
nn
nn
nn
nn
nn
nn
nn
nn
nx n -3 -2 -1 0 1 2 4 5
1 1 1
. . .
2
-1
. . . 3
n -3 -2 -1 0 1 3 4
1 1 1
. . .
2
-1
2 . . .
-
-
-
-
40
31
21
10
01
12
21
30
][
n
n
n
n
n
n
n
n
nx
n -4 -3 -1 0 1 2 3
1 1 1
. . .
2
-1
2 . . .
--
--
---
--
-
--
--
--
-
440
331
221
110
001
112
221
330
][
nn
nn
nn
nn
nn
nn
nn
nn
nx
n -3 -2 -1 0 1 3 4
1 1 1
. . .
2
-1
2 . . .
-
-
-
-
40
31
21
10
01
12
21
30
][
n
n
n
n
n
n
n
n
nx
-
--
--
---
2420
2/3321
1221
2/1120
0021
2/1122
1221
22/3320
]2[
nn
nn
nn
nn
nn
nn
nn
nnn
nx
(n: tamsayı!)
(n: tamsayı!)
(n: tamsayı!)
(n: tamsayı!)
n -2 -1 0 2
1 1
. . .
-1
1 . . .
n -3 -2 -1 0 1 3 4
1 1 1
. . .
2
-1
2 . . .
-
-
-
-
40
31
21
10
01
12
21
30
][
n
n
n
n
n
n
n
n
nx
-
--
--
--
84210
63211
42211
21210
00211
21212
42211
63210
]2
1[
nn
nn
nn
nn
nn
nn
nn
nn
nx
n-5
n-3
n-1
n1
n3
n5
n7
Genlik değeri
tanımlanmamış
aradaki n’ler için
x[½n]0
n -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
1 1 1
. . .
2
-1
4 . . .
Örnek:
n -3 -2 -1 0 1 3 4
1 1 1
. . .
2
-1
2 . . .
-
-
-
-
40
31
21
10
01
12
21
30
][
n
n
n
n
n
n
n
n
nx
-
-
---
-
-
--
---
---
-
2413002
3/4313212
12132)1(2
3/2113002
3/1013 212
0113 422
3/1213 212
1313 002
]13[2
nn
nn
nn
nn
nn
nn
nn
nn
nx
n -1 0 2
4
. . .
-2
1 . . .
RoC
z=x+jy
x=Re{z}
y=Im{z}
|z|=(x2+y2)1/2 Re{z}
Complex z-plane
jIm{z}
z
x
jy
q
q =arctan(y/x)
x=|z|cos(q)
y=|z|sin(q)
z=|z|ejq=|z|{cos(q)+jsin(q)}
Electronic Communications: Telephone, wireless phone, TV, radar, etc.
Concept and Model of Communications
General Communication Model
Source Transmitter Transmission
System Receiver Destination
Microphone Telephone Computer Scanner
Transformer Encoder Compress Modulator
Line/Cable Fiber/Air Satellite Network
Transformer Decoder Uncompress Demodulator
Speaker Earphone Computer Printer
Basic Communication Criteria: Performance, Reliability, Security
Important Reasons for Modulation:
• Ease of Radiation: c=l×f
• Simultaneous Transmission of Several Signals
FDM/TDM
• Effecting the Exchange of SNR with B
Channel Capacity Channel Bandwidth
Signal-to-Noise Ratio: S_
N
Modulation
Carrier: Acos(2πfct+φ) where fc is called carrier frequency
Modulation: change or modify values of A, fc, φ according to input signal m(t) - modify A A[m(t)]: Amplitude Modulation (AM)
- modify fc fc[m(t)]: Frequency Modulation (FM) - modify φ φ[m(t)]: Phase Modulation (PM)
Modulator m(t)
Acos(2πfct+φ)
modulated signal: s(t)
• Sayısal sinyaller analog sinyallere göre gürültü ve parazit sinyallerinden
daha az etkilenirler.
• Sayısal sinyallerdeki bozulmalar tekrar ediciler (regenerative repeaters)
tarafından giderilebilir.
• Hata sezme (error detection) ve düzeltme (correction) teknikleri sayesinde az hata
oranlı sinyal iletimi yapılabilir.
• Sayısal sinyallere parazit ve karıştırıcı sinyal etkilerinden korunabilmek için
güvenlik ve kriptolama gibi sinyal işleme teknikleri uygulanabilir.
• Sayısal devreler analog devrelere göre daha esnek, daha dayanıklı, ve daha az
maliyetli olarak tasarlanabilir.
Neden
Sayısal
Haberleşme?
Modulator, Demodulator & Modem
Modulator accepts bit sequence and modulates a carrier.
Demodulator accepts a modulated signal and regenerates bit sequence.
Modem is a single device which includes both modulator and demodulator.
Multiplexing, Multiplexer & Demultiplexer
Multiplexing is a technique that allows simultaneous transmissions of multiple
signals across a single data link.
CompA1
CompB1
CompC1
Da
Db
Dc D ≥ Da+Db+Dc
D E M U X
CompA2
CompB2
CompC2
1 shared link: rate D
Multiplexer Demultiplexer
FDM – Frequency Division Multiplexing
- A set of signals are put in different frequency positions of a link/medium. - Bandwidth of the link must be larger than the sum of signal bandwidths. - Each signal is modulated using its own carrier frequency. - Examples: radio, TV, satellite, etc.
A1
B1
C1
Mod
Mod
Mod
1
2
3
+
f
Dem
Dem
Dem
1
2
3
A2
B2
C2
1
2
3
1
2
3
f1
f2
f3
TDM – Time Division Multiplexing
- Multiple data streams are sent in different time in single data link/medium. - Data rate of the link must be larger than a sum of the multiple streams. - Data streams take turn to transmit in a short interval. - Widely used in digital communication networks.
CompA1
CompB1
CompC1
CompA2
CompB2
CompC2
D E M U X
… C1 B1 A1 C1 B1 A1 …
For no aliasing:
Bit rate:
Bandwidth of PCM waveform:
R=n.fs=(bit sayısı/örnek veri)x(örnek veri/sn )=bit/s=bps
A-tipi sıkıştırma eğrisinin parçalı gösterimi
Amaç; giriş genliğinin herhangi bir değeri için belirli sınırlar içinde kalan bir kuantalama
hatası elde etmektir.
Lokal kuantalama
seviye (adım) sayısı: M
Encoder Transmission
System/Channel Bandwidth=B
Decoder
t
0 1 0 0 1 0
Maximum Signal Rate
Channel Capacity
Shannon Theorem (1948):
For a system/channel bandwidth B and signal-to-noise ratio S/N, its channel capacity is,
C = Blog2(1+S/N) bits/sec (bps, bit rate)
C is the maximum number of bits that can be transmitted per second with a Pe=0.
To transmit data in bit rate D, the channel capacity of a system/channel must be
C ≥ D
+
Noise n(t)
s(t)
t
Relationship between Transmission Speed and Noise
Information / Hz: 1 Hertz can transmit a maximum of 2 pieces of information per second.
2 bits / sec / Hz
Shannon theorem C = Blog2(1+S/N) shows that the maximum rate or channel Capacity of a system/channel depends on bandwidth, signal energy and noise intensity. Thus, to increase the capacity, three possible ways are 1) increase bandwidth; 2) raise signal energy; 3) reduce noise. Shannon theorem tell us that we cannot send data faster than the channel capacity, but we can send data through a channel at the rate near its capacity.
Examples
1. For an extremely noise channel S/N 0, C 0, cannot send any data regardless of bandwidth
2. If S/N=1 (signal and noise in a same level), C=B
3. The theoretical highest bit rate of a regular telephone line where B=3000Hz and S/N=35dB. 10log10(S/N)=35 log2(S/N)= 3.5x log210
C= Blog2(1+S/N) =~ Blog2(S/N) =3000x3.5x log210=34.86 Kbps If B is fixed, we have to increase signal-to-noise ratio for increasing transmission rate.
Channel Capacity