principles of communications lecture 12: noise in...
TRANSCRIPT
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Principles of CommunicationsLecture 12: Noise in Modulation Systems
Chih-Wei Liu National Chiao Tung [email protected]
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Outlines
Signal-to-Noise Ratio
Noise and Phase Errors in Coherent Systems
Noise in Angle Modulation
Threshold Phenomenon
Noise in Pulse Code Modulation
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Signal-to-Noise Ratio
SNR is one of the most important parameters in communications. (The other is bandwidth).
Channel Model: Additive, White, Gaussian Noise (AWGN). Independent of signal.
Compare different receivers (modulations):Same SNR at A, compare SNR at B.
at a certain point
Signal powerSNRNoise power
=
N0 / 2
Receivermessage+
+ A B
n(t) noise
signal(containingmessage)
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Noise in Baseband System
The power spectral density is . (Some books use N0)
Physical meaning: N0 is the average noise power per unit bandwidth (single-sided psd) at the front end of the receiver.
20N
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Baseband model: a basis for comparison (benchmark)
This is a basic operation in many comm systems We filter out the out-of-band noise.
This filtering does not change signal (power).
0 00
0 00
Signal power = watts. (transmitted power, modulated signal)1Filter input noise power = , and ( ) . 2
1Filter output noise power= , ( ) .2
(SNR enhancement
T
BT
iB
WT
oW
PPN df N B SNR
N BPN df N W SNR
N W
= =
= =
) . (The filter reduces part of noise.)
( )o
i
SNR BSNR W
=
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A. DSB-SC System
Assume coherent detection.
2
( ) ( )cos( ) ( )
( ) ( )cos( ) ( )cos( ) ( )sin( )
r c c
c c c c s c
x t A m t t n t
e t A m t t n t t n t t
= + +
= + + + +
Bandpass noiseBandpass filtering
2 cos(ct +)
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2 2 20 0
22
2 2
0
3 2
1 1Noise power: ( ) ( ) ( ) 2 .2 2
Transmitted signal power: ( ).2
Pre-detection SNR: ( ) . (For DSB-SC)4
( ) ( ) 2cos( )( ){1 cos(2( ))}
( ){1 cos(
c s
c
cT
c
c c
c
n t n t n t N W
A m t
A mSNRWN
e t e t tA m t t
n t
= + =
=
= += + ++ +
0
2( ))} ( )sin(2( )).Through LPF ( ) ( ) ( ).
c s c
c c
t n t ty t A m t n t + +
= +
Passband
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We assume coherent detection, i.e., the LO signal has the same freq and phase as the carrier.
Does this detection (demodulation) provide a 3dB gainNot quite. Compare the equivalent baseband system.
20
2 2
2 2
0
Noise power: ( ) 2 . (Noise power remains the same.)
Signal power: . (Signal power "doubled")
Post-detection SNR: ( ) . (For DSB-SC)2
( )Detection gain 2 ( 3 ).( )
c
c
cD
D
T
n t N W
A m
A mSNRN W
SNR dBSNR
=
=
= = =
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What is the equivalent baseband system?
Why? The baseband noise is N0W (BW=W).
The passband noise is 2N0W (BW=2W).
The detection gain cancels the noise BW increase.
2 2
0
2 2
0
1transmitted power2
( )2
Tc
cD
P A mN W
A m SNRN W
= =
= =
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B. SSB System
Assume coherent detection The received signal ( for the upper; for the lower sideband):
( ) [ ( )cos( ) ( )sin( )] ( ) where : the Hilbert transform of ( ).
r c c cx t A m t t m t n tm m t
+
= + + +
fw0-w
2 cos(ct +)
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2 2 20 0
2
Noise component: (Decompose into I and Q components)( ) ( )cos( ) ( )sin( )
Let N
BP noiseThus, ( ) [ ( ) ( )]cos( ) [ ( ) ( )]sin( )
c c s c
T c s
c c c c s c
n t n t t n t t
n n n N W
e t A m t n t t A m t n t t
= + +
= = =
= + + +m
3
don't care 2cos( )
LPF ( ) ( ) ( ) noise term.
c
D c c
t
ey t A m t n t
+
= +
Key: SSB BW = W not 2W. Hence, the received noise power is reduced.
coherent
ffcfc-w
N0 / 2
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2 2
20
2 2
0
Assume that ( ) is independent of (t).
Post-detection signal power: ( ).
Post-detection noise power: ( ) .
Post-detection SNR: . (For SSB-SC)
What is the pre-dete
c
D c
D c
cD
m t n
S A m
N n N W
A mSNRN W
=
= =
=
2
2 2 2
ction SNR? The transmitted signal power
{ [ ( )cos( ) ( )sin( )]}
{[ ( )cos( )] [ ( )sin( )]
2 ( ) ( )cos( )sin( ) ( ( ) and ( ) are ortho
T c c c
c c c
c c
S A m t t m t t
A m t t m t t
m t m t t tm t m t
= + +
= + + +
+ +gonal.)
( cos( ) and sin( ) are orthogonal.)c ct t
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2 2 2 2 2
2 2 2
{ ( ) cos ( ) ( ) sin ( )
1 1 1 1 cos(2 2 ) cos(2 )2 2 2 21 1 { ( ) ( )2 2
T c c c
c c
c
S A m t t m t t
t t
A m t m t
= + + +
+ + +
= + 2 2
2 2
0
} (Notice that ( ) ( ).)
( ( )) ! The detection gain is
( ) 1. That is, .( )
( )No gain! No loss ( )! (In DSB, 2 )
c D
D
D DD T
TTT
T D T D
m t m t
A m t S
SSNR N SNR SNRSSNR
N N WN N N N
=
= =
= = =
== =Q
ffc-fc+w
N0 / 2
-fc fc-w0
2)2
( 0 WN
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C. AM System -- CoherentAssume coherent detection
cc c0
carrier
Mn(f)
W0-W
e2 (x)
e3 (x)
BPF
xr(t) = xc(t) + n(t)
LPF ( ) ( ) ( ) . ( is removed.)D c n c c cy t A am t n t A A= + +
X 2 cos( ct + )
Received signal ( ) [1 ( )]cos( ). : modulation index
: normalized messagec c n c
n
x t A am t t am
= + +
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2 2 2
is removed.
1) assume ( ) 0, we can thus remove dc term; 2) In reality, we cannot recover dc term of ( ) anyway.
Thus we simply remove it.
Post-detection signal power: ( ( ))
c
D c n c
A
m tm t
S A am t A a m
=
= =
Q
2
20
2 2 2
0
.
Post-detection noise power: 2 .
Post-detection SNR: ( ) . (For AM)2
n
D c
c nD
N n N W
A a mSNRN W
= =
=
WWN 00 2N2)22( =
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2
2 2 2 2 2 2
2 2 2
2
Now, let's calculate the pre-detection SNR. The Tx signal power:
{ [1 ( )]cos( )}
cos ( ) ( ) cos ( ).1 ( ( ) , cos ( ) cos(2 ).)2
12
T c n c
c c c n c
n n c c
T T c
S A am t t
A t A a m t t
m t m t t
P S A
= + +
= + + +
= + = + +
= = + 2 2 2
2 2 2 2
00
2 2 2 2 2
2 22 2 2 2
2 2
2 2
1 .2
1Bandpass noise power 2 . ( ) .2 2
( ) 2 The detection gain 1.1( ) 1( )2
( )Recall, (power) efficiency .(1
c n
c c nT T
D c n n
T nc c n
n Dff
n
A a m
A A a mN N W SNRN W
SNR A a m a mSNR a mA A a m
a m SNRESa m
+= =
= = =