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# Costas Loop - Notes, Electronics and Communication Engineering Notes | EduRev

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## : Costas Loop - Notes, Electronics and Communication Engineering Notes | EduRev

``` Page 1

1

Costas Loop

Modules: Sequence Generator, Digital Utilities, VCO, Quadrature Utilities (2), Phase Shifter,
Tuneable LPF (2), Multiplier

Phase-shift keying that employs two discrete phases (0 and   radians) is often called binary
phase-shift keying (BPSK).

0.1 Binary Phase-Shift Keying
BPSK has the mathematical form
( )  ( )    (

)
(1)
where  ( )    is the sequence of bipolar voltages representing the data and

is the carrier
frequency.  During a bit period in which the polarity is   , the carrier has its nominal phase.
During a bit period in which the polarity is   , the carrier phase is different from the nominal by

.  At those points in time corresponding to a change in bit from 0 to 1 or vice versa, there is
a phase shift of

.

( ): sequence of bipolar voltages representing the data

BPSK carrier
Page 2

1

Costas Loop

Modules: Sequence Generator, Digital Utilities, VCO, Quadrature Utilities (2), Phase Shifter,
Tuneable LPF (2), Multiplier

Phase-shift keying that employs two discrete phases (0 and   radians) is often called binary
phase-shift keying (BPSK).

0.1 Binary Phase-Shift Keying
BPSK has the mathematical form
( )  ( )    (

)
(1)
where  ( )    is the sequence of bipolar voltages representing the data and

is the carrier
frequency.  During a bit period in which the polarity is   , the carrier has its nominal phase.
During a bit period in which the polarity is   , the carrier phase is different from the nominal by

.  At those points in time corresponding to a change in bit from 0 to 1 or vice versa, there is
a phase shift of

.

( ): sequence of bipolar voltages representing the data

BPSK carrier
2

A BPSK modulator can be implemented (for a relatively small

) with a multiplier.

At the receiver the data can be recovered with synchronous demodulation.  If a stolen carrier is
available, the received signal is multiplied by this stolen carrier.  It is assumed here that the
stolen carrier is      (

).  If this is the case, then the signal processing in the receiver is
{ ( )    (

)      (

)}  ( )
(2)
where  { } represents the lowpass filtering of the multiplier output.  In practice, the bandwidth of
( ) is much smaller than the carrier frequency.  The filter passes  ( ) while blocking the
double-frequency term.
In the field, where the receiver is usually remote from the transmitter, no stolen carrier is
available.  For synchronous demodulation to work, the receiver must somehow reconstruct a
copy of the (unmodulated) carrier from the received signal.  It is important to note that this
reconstructed copy must match the arriving carrier in phase as well as frequency.
As an example of what doesn’t work, consider what happens if synchronous demodulation is
attempted with a copy of the (unmodulated) carrier that is offset in phase from the arriving
carrier by   /  radians.  An example of this is      (

).  From trigonometry,
( )    (

)      (

)  ( )    (

)
(3)
The difference-frequency term is absent in this case; only a double-frequency term is present.
Therefore,
{ ( )    (

)     (

)}
(4)
In this case, nothing passes the filter and the demodulation fails.  This demonstrates the
importance of getting the phase, as well as the frequency, right in the receiver.  This is known as
carrier synchronization.  When BPSK is employed, carrier synchronization is done in the
0.2 Phase-Locked Loop
A simple phase-locked loop is designed to track a sinusoid.  The VCO produces a sinusoid.
When the loop is tracking properly, this VCO sinusoid and the input sinusoid have the same
frequency.  The multiplier produces a difference-frequency term and a sum-frequency term, but
only the former passes through the lowpass filter.  The output of the filter is an error signal, and
it is amplified and then placed at the input to the VCO, completing the loop.
X
( )
(

)
(

)
Page 3

1

Costas Loop

Modules: Sequence Generator, Digital Utilities, VCO, Quadrature Utilities (2), Phase Shifter,
Tuneable LPF (2), Multiplier

Phase-shift keying that employs two discrete phases (0 and   radians) is often called binary
phase-shift keying (BPSK).

0.1 Binary Phase-Shift Keying
BPSK has the mathematical form
( )  ( )    (

)
(1)
where  ( )    is the sequence of bipolar voltages representing the data and

is the carrier
frequency.  During a bit period in which the polarity is   , the carrier has its nominal phase.
During a bit period in which the polarity is   , the carrier phase is different from the nominal by

.  At those points in time corresponding to a change in bit from 0 to 1 or vice versa, there is
a phase shift of

.

( ): sequence of bipolar voltages representing the data

BPSK carrier
2

A BPSK modulator can be implemented (for a relatively small

) with a multiplier.

At the receiver the data can be recovered with synchronous demodulation.  If a stolen carrier is
available, the received signal is multiplied by this stolen carrier.  It is assumed here that the
stolen carrier is      (

).  If this is the case, then the signal processing in the receiver is
{ ( )    (

)      (

)}  ( )
(2)
where  { } represents the lowpass filtering of the multiplier output.  In practice, the bandwidth of
( ) is much smaller than the carrier frequency.  The filter passes  ( ) while blocking the
double-frequency term.
In the field, where the receiver is usually remote from the transmitter, no stolen carrier is
available.  For synchronous demodulation to work, the receiver must somehow reconstruct a
copy of the (unmodulated) carrier from the received signal.  It is important to note that this
reconstructed copy must match the arriving carrier in phase as well as frequency.
As an example of what doesn’t work, consider what happens if synchronous demodulation is
attempted with a copy of the (unmodulated) carrier that is offset in phase from the arriving
carrier by   /  radians.  An example of this is      (

).  From trigonometry,
( )    (

)      (

)  ( )    (

)
(3)
The difference-frequency term is absent in this case; only a double-frequency term is present.
Therefore,
{ ( )    (

)     (

)}
(4)
In this case, nothing passes the filter and the demodulation fails.  This demonstrates the
importance of getting the phase, as well as the frequency, right in the receiver.  This is known as
carrier synchronization.  When BPSK is employed, carrier synchronization is done in the
0.2 Phase-Locked Loop
A simple phase-locked loop is designed to track a sinusoid.  The VCO produces a sinusoid.
When the loop is tracking properly, this VCO sinusoid and the input sinusoid have the same
frequency.  The multiplier produces a difference-frequency term and a sum-frequency term, but
only the former passes through the lowpass filter.  The output of the filter is an error signal, and
it is amplified and then placed at the input to the VCO, completing the loop.
X
( )
(

)
(

)
3

Phase-locked loop
Loop gain is an important parameter in a phase-locked loop.  The loop gain is defined as the
product of the VCO sensitivity and the amplification in the loop.  The behavior of the loop
depends on whether the loop gain is positive or negative.  In the following discussion, it is
assumed that the loop gain for this simple phase-locked loop is positive.  (In the TIMS
instrument, the VCO sensitivity is negative and the amplifier gain is also negative, so the minus
signs cancel and the loop gain is indeed positive.)
With positive loop gain, a positive error signal (appearing at the output of the lowpass filter)
causes the VCO output frequency to be larger than its nominal value (the frequency with zero
input to the VCO).  A negative error signal causes the VCO output frequency to be smaller than
its nominal value.
The input to the loop is here modeled as     (

).  The VCO output is modeled as
(

).  The multiplier produces a difference-frequency term     (

), and this
is the error signal.  (The sum-frequency term is blocked by the lowpass filter.)  The error signal
is plotted below as a function of the phase difference

.

Phase-locked loop: identifying the lock point
There is a stable lock point at the positive-going zero crossing:

.  The following
reasoning shows this to be a point of phase lock.  If

, the error signal is positive
(assuming

is not larger than  ) and therefore the VCO is forced to produce phase at a
faster rate (that is, to produce a larger frequency).  This means the feedback action of the loop
pushes the loop back to the point

.  If

, the error signal is negative and
X
LPF
VCO
Page 4

1

Costas Loop

Modules: Sequence Generator, Digital Utilities, VCO, Quadrature Utilities (2), Phase Shifter,
Tuneable LPF (2), Multiplier

Phase-shift keying that employs two discrete phases (0 and   radians) is often called binary
phase-shift keying (BPSK).

0.1 Binary Phase-Shift Keying
BPSK has the mathematical form
( )  ( )    (

)
(1)
where  ( )    is the sequence of bipolar voltages representing the data and

is the carrier
frequency.  During a bit period in which the polarity is   , the carrier has its nominal phase.
During a bit period in which the polarity is   , the carrier phase is different from the nominal by

.  At those points in time corresponding to a change in bit from 0 to 1 or vice versa, there is
a phase shift of

.

( ): sequence of bipolar voltages representing the data

BPSK carrier
2

A BPSK modulator can be implemented (for a relatively small

) with a multiplier.

At the receiver the data can be recovered with synchronous demodulation.  If a stolen carrier is
available, the received signal is multiplied by this stolen carrier.  It is assumed here that the
stolen carrier is      (

).  If this is the case, then the signal processing in the receiver is
{ ( )    (

)      (

)}  ( )
(2)
where  { } represents the lowpass filtering of the multiplier output.  In practice, the bandwidth of
( ) is much smaller than the carrier frequency.  The filter passes  ( ) while blocking the
double-frequency term.
In the field, where the receiver is usually remote from the transmitter, no stolen carrier is
available.  For synchronous demodulation to work, the receiver must somehow reconstruct a
copy of the (unmodulated) carrier from the received signal.  It is important to note that this
reconstructed copy must match the arriving carrier in phase as well as frequency.
As an example of what doesn’t work, consider what happens if synchronous demodulation is
attempted with a copy of the (unmodulated) carrier that is offset in phase from the arriving
carrier by   /  radians.  An example of this is      (

).  From trigonometry,
( )    (

)      (

)  ( )    (

)
(3)
The difference-frequency term is absent in this case; only a double-frequency term is present.
Therefore,
{ ( )    (

)     (

)}
(4)
In this case, nothing passes the filter and the demodulation fails.  This demonstrates the
importance of getting the phase, as well as the frequency, right in the receiver.  This is known as
carrier synchronization.  When BPSK is employed, carrier synchronization is done in the
0.2 Phase-Locked Loop
A simple phase-locked loop is designed to track a sinusoid.  The VCO produces a sinusoid.
When the loop is tracking properly, this VCO sinusoid and the input sinusoid have the same
frequency.  The multiplier produces a difference-frequency term and a sum-frequency term, but
only the former passes through the lowpass filter.  The output of the filter is an error signal, and
it is amplified and then placed at the input to the VCO, completing the loop.
X
( )
(

)
(

)
3

Phase-locked loop
Loop gain is an important parameter in a phase-locked loop.  The loop gain is defined as the
product of the VCO sensitivity and the amplification in the loop.  The behavior of the loop
depends on whether the loop gain is positive or negative.  In the following discussion, it is
assumed that the loop gain for this simple phase-locked loop is positive.  (In the TIMS
instrument, the VCO sensitivity is negative and the amplifier gain is also negative, so the minus
signs cancel and the loop gain is indeed positive.)
With positive loop gain, a positive error signal (appearing at the output of the lowpass filter)
causes the VCO output frequency to be larger than its nominal value (the frequency with zero
input to the VCO).  A negative error signal causes the VCO output frequency to be smaller than
its nominal value.
The input to the loop is here modeled as     (

).  The VCO output is modeled as
(

).  The multiplier produces a difference-frequency term     (

), and this
is the error signal.  (The sum-frequency term is blocked by the lowpass filter.)  The error signal
is plotted below as a function of the phase difference

.

Phase-locked loop: identifying the lock point
There is a stable lock point at the positive-going zero crossing:

.  The following
reasoning shows this to be a point of phase lock.  If

, the error signal is positive
(assuming

is not larger than  ) and therefore the VCO is forced to produce phase at a
faster rate (that is, to produce a larger frequency).  This means the feedback action of the loop
pushes the loop back to the point

.  If

, the error signal is negative and
X
LPF
VCO
4

therefore the VCO is forced to produce phase at a slower rate.  In this case also, the feedback
action pushes the loop back to the point

.  Of course, if

the error signal
is zero, and the loop tends to stay where it is.
It should be noted that a negative-going zero crossing is not a stable lock point.  If

moves slightly off the point

, the feedback action pushes the loop away from the
point

.  This is called a point of unstable equilibrium.
As described above there is one stable lock point per cycle of carrier phase.  This point occurs at

.  In other words, phase lock corresponds to

.  Remember that the input
sinusoid is modeled here as a sine and the VCO output is modeled as a cosine.  Therefore, when
in phase lock (and with positive loop gain) the VCO sinusoid leads the input sinusoid by

.
0.3 Costas Loop
A Costas loop is a type of phase-locked loop that is used for carrier synchronization in a receiver
when the modulation is BPSK.

Costas loop
Some mathematics will demonstrate how the Costas loop works.  The input to the Costas loop is
modeled here as
( )    (

)
(5)
where  ( )    is the sequence of bipolar voltages representing the data,

is the carrier
frequency, and

is an implicit function of time representing that part of the total signal phase
that is not included in

.
The VCO output is modeled here as
(

)
(6)
where

is an implicit function of time representing that part of the total signal phase that is not
included in

.  The output of the filter in the upper channel is
/
X
data
Page 5

1

Costas Loop

Modules: Sequence Generator, Digital Utilities, VCO, Quadrature Utilities (2), Phase Shifter,
Tuneable LPF (2), Multiplier

Phase-shift keying that employs two discrete phases (0 and   radians) is often called binary
phase-shift keying (BPSK).

0.1 Binary Phase-Shift Keying
BPSK has the mathematical form
( )  ( )    (

)
(1)
where  ( )    is the sequence of bipolar voltages representing the data and

is the carrier
frequency.  During a bit period in which the polarity is   , the carrier has its nominal phase.
During a bit period in which the polarity is   , the carrier phase is different from the nominal by

.  At those points in time corresponding to a change in bit from 0 to 1 or vice versa, there is
a phase shift of

.

( ): sequence of bipolar voltages representing the data

BPSK carrier
2

A BPSK modulator can be implemented (for a relatively small

) with a multiplier.

At the receiver the data can be recovered with synchronous demodulation.  If a stolen carrier is
available, the received signal is multiplied by this stolen carrier.  It is assumed here that the
stolen carrier is      (

).  If this is the case, then the signal processing in the receiver is
{ ( )    (

)      (

)}  ( )
(2)
where  { } represents the lowpass filtering of the multiplier output.  In practice, the bandwidth of
( ) is much smaller than the carrier frequency.  The filter passes  ( ) while blocking the
double-frequency term.
In the field, where the receiver is usually remote from the transmitter, no stolen carrier is
available.  For synchronous demodulation to work, the receiver must somehow reconstruct a
copy of the (unmodulated) carrier from the received signal.  It is important to note that this
reconstructed copy must match the arriving carrier in phase as well as frequency.
As an example of what doesn’t work, consider what happens if synchronous demodulation is
attempted with a copy of the (unmodulated) carrier that is offset in phase from the arriving
carrier by   /  radians.  An example of this is      (

).  From trigonometry,
( )    (

)      (

)  ( )    (

)
(3)
The difference-frequency term is absent in this case; only a double-frequency term is present.
Therefore,
{ ( )    (

)     (

)}
(4)
In this case, nothing passes the filter and the demodulation fails.  This demonstrates the
importance of getting the phase, as well as the frequency, right in the receiver.  This is known as
carrier synchronization.  When BPSK is employed, carrier synchronization is done in the
0.2 Phase-Locked Loop
A simple phase-locked loop is designed to track a sinusoid.  The VCO produces a sinusoid.
When the loop is tracking properly, this VCO sinusoid and the input sinusoid have the same
frequency.  The multiplier produces a difference-frequency term and a sum-frequency term, but
only the former passes through the lowpass filter.  The output of the filter is an error signal, and
it is amplified and then placed at the input to the VCO, completing the loop.
X
( )
(

)
(

)
3

Phase-locked loop
Loop gain is an important parameter in a phase-locked loop.  The loop gain is defined as the
product of the VCO sensitivity and the amplification in the loop.  The behavior of the loop
depends on whether the loop gain is positive or negative.  In the following discussion, it is
assumed that the loop gain for this simple phase-locked loop is positive.  (In the TIMS
instrument, the VCO sensitivity is negative and the amplifier gain is also negative, so the minus
signs cancel and the loop gain is indeed positive.)
With positive loop gain, a positive error signal (appearing at the output of the lowpass filter)
causes the VCO output frequency to be larger than its nominal value (the frequency with zero
input to the VCO).  A negative error signal causes the VCO output frequency to be smaller than
its nominal value.
The input to the loop is here modeled as     (

).  The VCO output is modeled as
(

).  The multiplier produces a difference-frequency term     (

), and this
is the error signal.  (The sum-frequency term is blocked by the lowpass filter.)  The error signal
is plotted below as a function of the phase difference

.

Phase-locked loop: identifying the lock point
There is a stable lock point at the positive-going zero crossing:

.  The following
reasoning shows this to be a point of phase lock.  If

, the error signal is positive
(assuming

is not larger than  ) and therefore the VCO is forced to produce phase at a
faster rate (that is, to produce a larger frequency).  This means the feedback action of the loop
pushes the loop back to the point

.  If

, the error signal is negative and
X
LPF
VCO
4

therefore the VCO is forced to produce phase at a slower rate.  In this case also, the feedback
action pushes the loop back to the point

.  Of course, if

the error signal
is zero, and the loop tends to stay where it is.
It should be noted that a negative-going zero crossing is not a stable lock point.  If

moves slightly off the point

, the feedback action pushes the loop away from the
point

.  This is called a point of unstable equilibrium.
As described above there is one stable lock point per cycle of carrier phase.  This point occurs at

.  In other words, phase lock corresponds to

.  Remember that the input
sinusoid is modeled here as a sine and the VCO output is modeled as a cosine.  Therefore, when
in phase lock (and with positive loop gain) the VCO sinusoid leads the input sinusoid by

.
0.3 Costas Loop
A Costas loop is a type of phase-locked loop that is used for carrier synchronization in a receiver
when the modulation is BPSK.

Costas loop
Some mathematics will demonstrate how the Costas loop works.  The input to the Costas loop is
modeled here as
( )    (

)
(5)
where  ( )    is the sequence of bipolar voltages representing the data,

is the carrier
frequency, and

is an implicit function of time representing that part of the total signal phase
that is not included in

.
The VCO output is modeled here as
(

)
(6)
where

is an implicit function of time representing that part of the total signal phase that is not
included in

.  The output of the filter in the upper channel is
/
X
data
5

( )    (

)
(7)
where

is the DC gain of the upper-channel filter.  The term     (

) by itself would be
a suitable error signal for a carrier synchronization loop.  However, the presence of  ( ) means
that the signal of Eq. (7) cannot by itself serve as the error signal for the loop.
The local oscillator applied to the lower channel is      (

).  The output of the filter in
the lower channel is

( )    (

)
(8)
where

is the DC gain of the lower-channel filter.
The third multiplier (the one on the right side of the diagram) produces a suitable error signal:

[ (

)]
(9)
Eq. (9) was obtained by noting that

( )   and by using the trigonometric identity

( )   ( )

(  )
(10)
The error signal of Eq. (9) passes through an amplifier with gain   on the way to the VCO input.
The loop gain is the product of all gains in the signal path and the VCO sensitivity

.  The
loop gain is therefore proportional to

.  For this experiment   and

are
negative and

and

are positive.  Therefore, the loop gain is positive.  For a loop with
positive loop gain, a positive error signal (the signal at the output of the third multiplier) causes
the VCO output frequency to increase and a negative error signal causes the VCO output
frequency to decrease.
The error signal for the Costas loop is proportional to    [ (

)].  Below that expression is
plotted as a function of

.

Costas loop: identifying the lock points
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