Transcript MS PowerPoint
Digital Transmission through the AWGN Channel
ECE460 Spring, 2012
Geometric Representation
Orthogonal Basis 1. Orthogonalization (Gram-Schmidt) 2. Pulse Amplitude Modulation a. Baseband b. Bandpass c.
Geometric Representation 3. 2-D Signals a. Baseband b. Bandpass 1) Carrier Phase Modulation (All have same energy) 1) Phase-Shift Keying 2) Two Quadrature Carriers 2) Quadrature Amplitude Modulation 4. Multidimensional a. Orthogonal 1) Baseband 2) Bandpass b. Biorthogonal 1) Baseband 2) Bandpass 2
Geometric Representation
Gram-Schmidt Orthogonalization 1. Begin with first waveform,
s
1 (
t
) with energy ξ 1: 1 1 1 2. Second waveform a. Determine projection,
c
21,
c
21
s
2 onto ψ 1 b. Subtract projection from
s
2 (t)
d
2
s
2
c
21 1 c.
Normalize 2
d
2 2 where 2
d
2 2 3. Repeat
c ki
s k d k
k
s k
d k
k
k i
1 1
c ki
i
where
k
d k
2 3
Example 7.1
4
Pulse Amplitude Modulation
Baseband Signals
Binary PAM • • Bit 1 – Amplitude +
A
Bit 0 – Amplitude
- A M-ary
PAM
M
2
k s m
m
A g m T
T
2
s m A
2
m
T A
2
m
g g T
2 M-ary PAM Binary PAM Fixed
R b
1
T b
k kT
5
u m
Pulse Amplitude Modulation
Bandpass Signals
Baseband Signal
s m
X Bandpass Signal
s m
cos 2
A g m T
cos 2
f t c
m
1, 2, ... ,
M U m
A m
2
G T
f
f c
G T
f
f c
What type of Amplitude Modulation signal does this appear to be?
m
u m
2
A
2
m A
2
m
2
t g T
2
g T
2 cos 2 2
c
A
2
m
2
g T
2 cos 4
c
6
PAM Signals
Geometric Representation
M-ary PAM waveforms are one-dimensional
s m
s m
m
1, 2,...,
M
where
s m
1
g g T
g A m
0
T m
1, 2,...,
M d d d
0
d d
d = Euclidean distance between two points 7
Optimum Receivers
Start with the transmission of any one of the
M-ary
waveforms: g
M
s m
,
m
signal 1, 2,...,
M
g g Transmitted within timeslot 0 Corrupted with AWGN:
s m T
s m
Demodulator Sampler 1 , ,..., 2
r N
Detector
r
s m
Output Decision 1. Demodulators a. Correlation-Type b. Matched-Filter-Type 2. Optimum Detector 3. Special Cases (Demodulation and Detection) a. Carrier-Amplitude Modulated Signals b. Carrier-Phase Modulation Signals c.
Quadrature Amplitude Modulated Signals d. Frequency-Modulated Signals 8
Demodulators
Correlation-Type r k
0
T
0
T s m
0
T
s mk s m
n k
k
0
T k
1, 2,...,
N
r
s
m
n
Next, obtain the joint conditional PDF
f
m
N
0 1
N
/ 2 exp
k N
1
r k
s mk
2 /
N
0
N
0 1
N
/ 2 exp
r s
m
2 /
N
0
m
1, 2,...,
M
9
Demodulators Matched-Filter Type
Instead of using a bank of correlators to generate {
r
k }, use a bank of
N
linear filters.
The Matched Filter
Key Property: if a signal
s
(
t
) is corrupted by AGWN, the filter with impulse response matched to
s
(
t
) maximizes the output SNR Demodulator 10
Optimum Detector
Decision based on transmitted signal in each signal interval based on the
Maximum a Posterior Probabilities (MAP)
P
signal
s
m
was transmitted |
r
m
1, 2,...,
M
m
|
r
f m N
1
f
r
|
s m
m
m
s
m
m
the denominator is a constant for all
M
, this reduces to
M
s m
and
D
D
m
k N
1
r k
s mk
2
m
2
r s
m
s
m
2 minimum distance detection minimize
C
m
2
r s s
m
2 maximize (correlation metric) 11
Probability of Error
Binary PAM Baseband Signals
Consider binary PAM baseband signals 1
s
2
g T
interval and zero elsewhere. This can be pictured geometrically as
b
b s
2 0
s
1 Assumption: signals are equally likely and that
s
1 transmitted. Then the received signal is was
r
1
b
n
Decision Rule:
r s
1
s
2 0 The two conditional PDFs for
r
are | | 2 1 1
N
0 e 1
N
0 e
b
2 /
N
0
b
2 /
N
0 12
Example 7.5.3
Consider the case of binary PAM signals in which two possible the metrics for the optimum MAP detector when the transmitted signal is corrupted with AWGN.
13
Probability of Error
M-ary PAM Baseband Signals
Recall baseband
M-ary
PAM are geometrically represented in 1 D with signal point values of
s m
g A m m
1, 2,...,
M
And, for symmetric signals about the origin,
A m
2
m M
m
1, 2,...,
M
g
Each signal has a different energies. The average is
av P av
1
M
g M
g m M
1
m m M
1 2
m
cos 1 2 1
M M
av T
2 3 1 3
g
M
2 3 1
M
g T
2 14
Demodulation and Detection
Carrier-Amplitude Modulated Signals
Demodulation of bandpass digital PAM signal Received Signal
r
(
t
) Oscillator Transmitted Signal:
u m
A g m T
Received Signal: wh er e cos 2 cos 2
T
f t c
0 2
T
0 Crosscorrelation 0
T
A m
A m
Optimum Detector o r 2
g
0
T g T
2
g
2
n
cos 2 2
c
m m
r
s m
2
r s m
2
s m
2 0
T T
15
Two-Dimensional Signal Waveforms
Baseband Signals •
Are these orthogonal?
•
Calculate ξ.
•
Find basis functions of (b).
16
Problem 7.22
In an additive white Gaussian noise channel with noise power-
N
spectral density of , two equiprobable messages are transmitted by 1 2
At T
0, , 0
T
otherwise
s
2
A
0, 1
t T
' 0
T
otherwise 1. Determine the structure of the optimal receiver 2. Determine the probability of error.
17
Two-Dimensional Bandpass Signals
Carrier-Phase Modulation 1. Given M-two-dimensional signal waveforms
s m
,
u m
s m
cos 2 0
T m
1, 2,...,
M
2. Constrain bandpass waveforms to have same energy
m
T
0
u m
2
T
0
s m
2 1 2
s T
0
s
2
m
m
cos 2 2 1 2
c
T
0
s
2
m
cos 4
c
18
Demodulation and Detection
Carrier-Phase Modulated Signals
The received signal: [
u m
T
t n t
[
T
wher e
m
0,1,...
M
1 Giving basis vectors as 1 2
g g T
cos 2
f t c
2 2
g g T
sin 2
f t c
Outputs of correlators:
r
s m
n
s
cos 2
n c
,
s
sin 2
n s
19
Two-Dimensional Bandpass Signals
Quadrature Amplitude Modulation
u m
T
T
sin 2
m
1, 2,...,
M
20
Multidimensional Signal Waveforms
Orthogonal
Multidimensional means multiple basis vectors • •
Baseband Signals
Overlapping (Hadamard Sequence) Non-Overlapping o Pulse Position Mod.
(PPM)
s m
A g T
t
m
1 where
m
m
1, 2,..., 1
M
21
Multidimensional Signal Waveforms
Orthogonal
Bandpass Signals
As before, we can create bandpass signals by simply multiplying a baseband signal by a sinusoid:
u m
s m
f t c
0
T
Carrier-frequency modulation: Frequency-Shift Keying (FSK)
u m
2
b T
2
m
f t
m
0,1,...,
M
, 0
T
mn
1
s T
u m
0 sin 2 2
m
m
n
n
f T
f T
22
Multidimensional Signal Waveforms
Biorthogonal
Baseband
Begin with
M
/2
s
1
s
2 orthogonal vectors in
N
=
M
/2 0,
s
, 0, 0,..., 0
s
, 0,..., 0 dimensions.
s
M
/ 2 0, 0, 0,...,
s
Then append their negatives
s
M
1 2
s
, 0, 0,..., 0
s
M
0, 0, 0,...,
s
Bandpass
As before, multiply the baseband signals by a sinusoid.
23
Multidimensional Signal Waveforms
Simplex
Subtract the average of M orthogonal waveforms
s
m
s m
s
T
0
s
m
1
M k M
1
s k
2
dt
1 1
M
s
In geometric form (e.g., vector)
s
m
s
m
1
M k M
1
s
k
Where the mean-signal vector is
s
1
M k M
1
s
k
Has the effect of moving the origin to reducing the energy per symbol
s
s
m
2
s
s
m
s
1 1
M
2
s
24
Demodulation and Detection
Carrier-Amplitude Modulated Signals
Demodulation of bandpass digital PAM signal Received Signal
r
(
t
) Oscillator Transmitted Signal:
u m
A g m T
Received Signal: wh er e cos 2 cos 2
T
f t c
0 2
T
0 Crosscorrelation 0
T
A m
A m
Optimum Detector o r 2
g
0
T g T
2
g
2
n
cos 2 2
c
m m
r
s m
2
r s m
2
s m
2 0
T T
25