Transcript PLL and Noise

PLL and Noise in Analog Systems
Analog and Digital Communications
Autumn 2005-2006
Oct 13, 2005
CS477: Analog and Digital Communications
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FM Detection: Phase Lock Loops
Basic Task of PLL:
Lock the instantaneous angle of a VCO with the
instantaneous angle of an external signal. i.e.,
lock (or track) the phase and frequency.
x c(t) = A c cosòc(t)
v(t) = A v cosòv(t)
e(t)
LPF
Ka
y(t) = 12A cA vK a sin ï (t)
VCO
y( t ) = 12A cA vK a cos[òc( t ) à òv( t )]
= 12A cA vK a cos[ï ( t ) à 90î ] = 12A cA vK a sin ï ( t )
Oct 13, 2005
CS477: Analog and Digital Communications
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Phase Lock Loops

Used in





Modulators and demodulators
Synchronizers
Frequency synthesizers
Multiplexers
Locks or synchronizes the external angle with
the output angle of VCO

Oct 13, 2005
A negative feedback system based on phase
comparison
CS477: Analog and Digital Communications
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PLL: Phase Comparator
x c(t) = A c cosòc(t)
e(t)
LPF
y(t) = 12A cA v sin ï (t)
v(t) = A v cosòv(t)
x c( t ) = A c cosòc( t ) = A c cos(2ùf ct + þ )
v(t ) = A v cosòv(t ) ù A v cos(2ùf ct + þê + 90î )
= à A v sin(2ùf ct + þê)
e(t ) = à A cA v cos(2ùf ct + þ) sin(2ùf ct + þê)
= A cA v[sin( þ à þê) à sin(4ùf ct + þ + þê)]
2
y( t ) = 12A cA v sin( þ à þê) = 12A cA v sin ï ( t )
Oct 13, 2005
CS477: Analog and Digital Communications
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Linearized PLL Models
Assumptions:
1. Incoming frequency is approximately the same
as the VCO free running frequency
2. Phase error is small such that:
sin(þ à þê) = sin ï (t ) = ï (t )
Consider phases as the signals for further analysis
þ(t) +
h(t)
à
Ka
y(t)
þê( t ) = 2ùK v
þê(t)
R
t
y( ü) dü
Integrator
Oct 13, 2005
CS477: Analog and Digital Communications
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Linearized PLL Models
Ð(f ) +
H(f )
à
Y(f )
Ka
ê (f )
Ð
K v=j f
Y( f ) =
Oct 13, 2005
K aH (f )
1+ K aH (f )(K v=j f ) Ð( f
=
j f K aH (f )
j f + K aK vH (f ) Ð( f
=
1
Kv
j f K H (f )
)
)
áj f + K H (f ) Ð( f )
(K = K aK v)
CS477: Analog and Digital Communications
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FM Detection using Linear PLL Model
þç(t ) = 2ùf É x (t ) ) Ð(f ) =
Y( f ) =
1
Kv
j f K H (f )
fÉ
j f X (f
áj f + K H (f ) áfj fÉ X ( f ) =
)
fÉ
K vH L ( f
(for FM input)
)X (f )
If H (f ) is a lowpass filter, i.e., H (f ) = 1;
Then H L ( f ) is a first-order lowpass filter
H L (f ) = 1+ j (f1 =K ) ; j f j < W
jf j < W
Y( f ) ù Kf ÉvX ( f );
K > W
y( t ) ù Kf Évx ( t )
First order PLL requires loop gain to be greater than
the frequency deviation K õ f É
Thus second order PLLs are used in practice.
Oct 13, 2005
CS477: Analog and Digital Communications
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Non-linearities in FM
FM transmission is robust against channel variations!
vo( t ) = a1vi ( t ) + a2v2i( t ) + a3v3i( t ) (Non-linear channel)
vi ( t ) = x c( t ) = A c cos(2ùf ct + þ ( t ))
vo( t ) = K 0 + K 1(cos2ùf ct + þ ( t )) + K 2(cos4ùf ct + 2þ ( t ))
+ K 3(cos6ùf ct + 3þ ( t ))
We need a BPF; what are the requirements?
f c + f É + W ô 2f c à (2f É + W) (Carson’s rule)
f c õ 3f É + 2W
With this BPF, output is: vêo( t ) = K 1(cos2ùf ct + þ ( t ))
Unlike AM, FM can combat amplitude non-linearities!
Oct 13, 2005
CS477: Analog and Digital Communications
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Noise

White Noise
SN ( f ) =
R N ( ü) =

N0
2
N0
2 î ( ü)
Filtered/Colored Noise
SY( f ) =
N0
2;
R Y( ü) =
N0
2 2Wsinc2Wü
N (t )
LPF
Y( t )
jf j < W
Y2( t ) = R Y(0) = N 0W
Oct 13, 2005
CS477: Analog and Digital Communications
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Noise Equivalent Bandwidth
Average power in filtered noise:
R N ( ü) =
N0
2 î ( ü)
N (t )
h(t )
Y( t )
N ñ Y2( t ) = R Y(0)
1
1
R
N0 R
2
j H ( f )j df = N 0 j H ( f )j 2df
N= 2
BN =
1
g
à1
1
R
0
j H ( f )j 2df
(Noise equivalent bandwidth)
0
g = j H ( f )j 2max
) N = gN 0B N
Oct 13, 2005
CS477: Analog and Digital Communications
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