Transcript Multi-Channel Communication in Systems with AWGN (cont.)

Advanced Wireless Networks
Lecture 4: Multi-Channel and Multi-Carrier Systems
Multi-Channel Communication in Systems with AWGN
We consider multi-channel signaling over fixed channels that differ only in
attenuation and phase shift. In such a model, the signal waveforms are
generally can be expressed as:


( n)
( n)
sm
(t )  Re slm
(t )e i 2fct ,
0t T
.
The equivalent baseband signals
received from the L channels can be expressed
in the following form:
( n)
rl( n) (t )   n slm
(t )e int  z n (t )
The decision variables of the coherent detection can be found via the correlation
metrics:
  T ( n)

( n )
C m   Re  gˆ n  rl (t ) slm
(t )dt ,
n 1
 0

L
Lectures 1 & 2:
Overview
m  1, 2,...., M
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Multi-Channel Communication in Systems with AWGN (cont.)
In non-coherent detection, no attempt is made to estimate the channel parameters.
The demodulator may base its decision either on the sum of the envelopes (envelope
decision) or the sum of the squared envelopes (square-law detection). We continue our
attention to square-law detection of the received signals of the L channels, which
produces the decision variables
Cm 
2
L T
( n )
(t )dt
  rl(n) (t )slm
m  1, 2,...., M
,
n 1 0
An error is occurred if C2  C1 , or equivalently, if the difference D  C1  C2  0
For non-coherent detection, this difference can be expressed as
D
with

n 1
T
X n   rl (t ) s
(n)
0
Lectures 1 & 2:
Overview
L
X
2
n
 Yn
2

T
( n )
l1
(t )dt
Yn   rl( n ) (t )sl(2n ) (t )dt
0
Adv. Wireless Comm. Sys.
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Multi-Channel Communication in Systems with AWGN (cont.)
For coherent detection, the difference D can be expressed as

1 L
D   X nY n*  X *n Yn
2 n1

where
Yn  gˆ n
n  1, 2,..., L

T

X n   rl( n) (t ) sl(1n ) (t )  sl(2n ) (t ) dt
0
Now we will find the probability that the general quadratic form in complex-valued
Gaussian variables is less than zero, i.e.,
D
 A X
L
n1
2
n
 B Yn
2

 CX nY n*  C * X *n Yn  0
This probability is the probability of error of binary multi-channel signaling in
AWGN. In coherent PSK, the probability of error takes the simple form:
P  Q
b
Lectures 1 & 2:
Overview
2 b
,

b 
N0
L
 gn
n 1
Adv. Wireless Comm. Sys.
2


N0
L
 n2
n 1
3
Multi-Channel Communication in Systems with AWGN (cont.)
If channels are all identical, i.e.,  n   for all n, we get:
L 2
b 

N0
EL is the total transmitted signal energy for the L signals, that is, above formula shows
that the receiver combines the energy from the L channels in an optimum manner without
any loss during division of the total transmitted signal energy among the L channels.
The same performance is obtained as in the case in which a single waveform having
energy EL.
Other extreme case, is binary DPSK, the probability of error of which can be presented
in the following form:
e  b L 1
1 L1n  2 L  1
2
cn 
Pb  2 L 1 cn b ,
  k 
n
!
2

k 0 
n 1

Figure shows the probability from square-law noncoherent combining of the L signals as a function of L
for various  b. As follows from illustrations, it is easily
to obtain a form of the curve as a function of  b , i.e.,
Lectures 1 & 2:
Overview
Adv. Wireless Comm. Sys.
1  b
Pb  e
2
4
Characterization of Multipath Channels with Fading
Usually researchers dealt only
with classical AWGN channels
which are not time-varied. Other
situation will occur in ionospheric
channels at 3-30 MHz due to
scattering from ionosphere, as a
time-varying media:
Base
Another example is cellular land
communication, stationary and
mobile at 300 MHz-3 GHz due to
multipath propagation signals
station
Ionosphere
Subscriber
(cellphone)
Fading Dispersive
Subchannels
R
T
So, the response of the channel becomes time-dependent and the total signal is now
corrupted not only by Gaussian white noise n(t), but also by multiplicative noise due
to fading, fast and slow:

y (t ) 
Lectures 1 & 2:
Overview
 h(t , ) x( )d  n(t )

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Characterization of Multipath Channels with Fading (cont.)
Time-invariant (stationary) channel:
The corresponding parameters of fading
obtained empirically are shown in Table:
  [sec]
Channel
HF
10 3
Ultrasound
Underwater
10 2
SHF
10
fd
[Hz]
0.5
5
10
6
Time-varied (time-dispersive) channel:
1 Bd
Lectures 1 & 2:
Overview
Adv. Wireless Comm. Sys.

6
Performance of the AWGN Channel without Fading
The received signal in time t, s(t), is then given by
s(t )  Ag(t )  n(t )
where n(t) is the noise waveform, g(t) is the modulated signal and
A is overall path loss, assumed not to vary in time. The power of
Gaussian noise is:
Pnoise 
1
n(t )n * (t )   n2  BN 0  N 0 / Ts
2
The signal-to-noise ratio (SNR) at the input of the receiver is then:
  SNR 
A 2 g 2 (t )
2 Pn

A 2 g 2 (t )
2 n2
s
A 2Ts


N0
2N 0
or in terms of the corresponding SNR per bit:
b 
Lectures 1 & 2:
Overview

m

b
N0
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Performance of the AWGN Channel without Fading (cont.)
Let us now calculate the bit error rate (BER) performance for binary phase shift
keying (BPSK) signals in AWGN channel. We, first of all, will consider the 2-D
(two-dimensional) case of BPSK signals, where two signals correspond to a
binary 1 and 0. Their complex baseband presentation is
g1 
2 s
2 s
, g0  
Ts
Ts
The error rate performance of digital modulation scheme in AWGN channel with N 0
depends on the Euclidean distance d between the transmitted waveforms,
and is determined by the probability of error:

Pe   Pe (  ) p(  )d
0
or accounting for Q-function
 A2 d 2 

  Q
Pe  Q


2
N


0 
and for d  2  s will get
 d2 
  Q 4b
Pe  Q
 2N
 2N0 
0



Lectures 1 & 2:
Overview
d2 

2 N 0 
 

  Q 2


Adv. Wireless Comm. Sys.
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Performance of the Channel with Flat Rayleigh Fading
Since the fading varies with time, the SNR at the input of the receiver also
varies with time. It is necessary, in contrast with AWGN case, to distinguish
between the instantaneous SNR,  , and the mean SNR, denoted as
.
Then, the signal r(t) can be presented as:
r (t )  A (t ) g (t )  n(t )
where  (t ) is the complex fading coefficient at time t.
If the fading is assumed constant over the transmitted symbol duration,
then is also constant over a symbol and SNR is given by:
 (t ) 
A 2 |  (t ) | 2 | g (t ) | 2
2 Pnoise
A 2 |  (t ) | 2

2 Pnoise
Then the average SNR for flat fading, having unit variance, equals:
A2
   (t ) 
2 Pnoise
Lectures 1 & 2:
Overview
Adv. Wireless Comm. Sys.
9
Performance of the Channel with Flat Rayleigh Fading
For Rayleigh distribution the PDF of error equals:
p(  ) 
1
 
exp  
 

Here   ( s / N 0 )  2 , where in slow flat fading channel, where    2s / N0 .
If so, for the cumulative density function CDF we have:
  
CDF (  )  Pr(    s )   p(  )d  1  exp  s 
 
0
s
The average bit error probability for BPSK signal can be defined as:




1
1
 
 
PeBPSK   Pe ( ) p( )d   Q 2
exp   d  1 



2
1






0
0
Lectures 1 & 2:
Overview
Adv. Wireless Comm. Sys.
10
Performance of the Channel with Flat Rayleigh Fading (cont.)
The average bit error probability for coherent binary frequency shift keying
(FPSK) signal
1
 
PeFPSK  1 

2
2
The average bit error probability for differential phase shift keying:
PeDPSK
1

2(1   )
BER for various modulations vs. Ws / N 0
in a flat slow fading channel with respect to
that for AWGN.
and for incoherent binary FPSK
PeICFPSK 
1
(2   )
Lectures 1 & 2:
Overview
Adv. Wireless Comm. Sys.
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