Noise in Modulation

The primary figure of merit for signals in analog

systems is signal-to-noise ratio. The signal-to-
noise ratio is the ratio of the signal power to the
noise power.
While the amount of external ambient noise is not
always within the control of a communications
engineer, the way in which we can distinguish
between the signal an the noise is.
The demodulation process for any modulation
system seeks to recreate the original modulating
signal with as little attenuation of the signal and as
much attenuation of the noise as possible.
The signal-to-noise ratio changes as the modulated
signal goes through the demodulation process. We
wish the signal-to-noise ratio to increase as a result
of the demodulation process.
The extent to which the signal-to-noise ratio
increases is called the signal-to-noise improvement,
or SNRI. This signal-to-noise improvement is equal
to the ratio of the demodulated signal-to-noise ratio
to the input signal-to-noise ratio.
The calculation of the signal-to-noise improvement
(SNRI) is shown in the following diagram.
Demodulator
s
i
(t)
s
o
(t)
n
i
(t) n
o
(t)
> < = > < =
> < = > < =
2 2
2 2
o no i ni
o so i si
n P n P
s P s P
. .
no
so
o
ni
si
i
P
P
SNR
P
P
SNR = =
.
no
ni
si
so
si
ni
no
so
i
o
P
P
P
P
P
P
P
P
SNR
SNR
SNRI = = =
The notation < > means time average.
In determining the ability of a demodulator to
discriminate signal from noise, we calculate P
si
, P
so
,
P
ni
and P
no
and plug these values (or expressions)
into the expression

.
no
ni
si
so
P
P
P
P
SNRI =
Noise in Amplitude
Modulated Systems
The SNRI will be calculated for DSB-SC
and DSB-LC.
A DSB-SC demodulator is linear : the
signal and the noise components can
be considered separately.
A DSB-LC demodulator is non-linear :
the signal and the noise components
cannot be treated separately.
X
s
o
(t)
cose
c
t
s
i
(t) = x
c
(t)
. cos ) ( ) ( t t m t x
c c
e =
The expression for the modulated carrier for DSB-
SC is
Demodulation of this signal is performed by
multiplying x
c
(t) by cos e
c
t and low-pass filtering the
result.
LPF
d(t)
The result of this demodulation can be seen by
analysis:
]. 2 cos 1 [ ) (
cos ) (
cos ) ( ) (
2
1
2
t t m
t t m
t t x t d
c
c
c c
e
e
e
+ =
=
=
Low-pass filtering the result eliminates the cos 2e
c
t
component.
). ( ) (
2
1
t m t s
o
=
Based upon these relationships, we can find the
ratio of the signal input power and the signal output
power.
2
1
2 2 2
) ( cos ) ( t m t t m P
c si
= = e
( )
4
1
2
2
2
1
2
) ( ) ( t m t m P
so
= =
.
2
1
) (
) (
2
1
2
4
1
2
= =
t m
t m
P
P
si
so
So, it seems, half of our SNRI calculation is done.

The other half of the calculation deals with the
noise.
Let us use the quadrature decomposition to
represent the noise:
. sin ) ( cos ) ( ) ( t t n t t n t n
c s c c
e e =
It is this signal which will represent n
i
(t).
The noise input to the demodulator (in the above
form) is demodulated along with the signal.
Because the demodulator is linear, we can treat the
noise separately from the signal.
X
n
o
(t)
cose
c
t
n
i
(t) = n
c
cose
c
t n
s
sine
c
t
LPF
d
n
(t)
| |
t n t n
t t n t n
t t n t n
t t n t d
c s c c
c c s c c
c c s c c
c n
e e
e e e
e e e
e
2 sin ] 2 cos 1 [
cos sin cos
cos sin cos
cos ) ( ) (
2
1
2
1
2
+ =
=
=
=
After d
n
(t) passes through the low-pass filter, all we
have left is
). ( ) (
2
1
t n t n
c o
=
Now, we calculate the power in the noise.
( ) . ) ( ) ( ) (
2
4
1
4
1
2 2
2
1
2
t n t n t n P
c c no
= = =
The ratio of the noise powers becomes
. 4
) (
) (
2
4
1
2
= =
t n
t n
P
P
no
ni
Finally, our signal-to-noise improvement becomes
. 2 4
2
1
= = =
no
ni
si
so
P
P
P
P
SNRI
Thus, the DSB-SC demodulation process improves
the signal-to-noise ratio by a factor of two.
| | . cos ) ( 1 ) ( t t m t x
c c
e + =
The expression for the modulated carrier for DSB-
LC is
(This is a simplified version of the more accurate
expression which takes into account the modulation
index. In this simplified version, the modulation
index is equal to one.)
The demodulation process extracts m(t) from x
c
(t).
The input signal power is
| |
| |
| |
| |
2
1
2
2
1
2
2
1
2
2
2
) ( 1
) ( ) ( 2 1
) ( 1
cos ) ( 1
t m
t m t m
t m
t t m P
c si
+ =
+ + =
+ =
+ = e
| | . ) ( 1
2
1
2
t m + =
The modulating signal m(t) is assumed here to have
zero mean.
If m(t) = cose
m
t, then
| | . 1
4
3
2
1
2
1
= + =
si
P
The output power is simply
) (
2
t m P
so
=
which is equal to if m(t) = cose
m
t.
.
3
2
4
3
2
1
= =
si
so
P
P
Thus, when m(t) is a sinewave, our signal power
ratio is
Now, we deal with the noise.
As mentioned previously, since the demodulation
process is non-linear, we cannot treat the signal and
the noise separately. To deal with the noise, we
retain the carrier, but set the modulating signal to
zero. Thus, the signal that we demodulate is
| | ). ( cos 1 t n t
c
+ e
The resultant output from demodulating this signal
will be the output noise power.
To demodulate this signal (analytically), we use the
quadrature decomposition for n(t). The input to our
demodulator becomes
. sin ) ( cos ) ( cos t t n t t n t
c s c c c
e e e +
We then work with this expression:
| | . sin ) ( cos ) ( 1
sin ) ( cos ) ( cos
t t n t t n
t t n t t n t
c s c c
c s c c c
e e
e e e
+ =
+
At this point we make an assumption (which turns
out to be quite reasonable in many cases):
| |. ) ( 1 ) ( t n t n
c s
+ <<
In words, the noise (either the in-phase or
quadrature component) is much smaller in
magnitude that that of the signal cose
c
t.
With this assumption the input to the demodulator
becomes
| | . cos ) ( 1 t t n
c c
e +
We recall that when the input
is applied, the output is
| | t t m
c
e cos ) ( 1+
). (t m
Similarly, when the input is
is applied, the output is
| | t t n
c c
e cos ) ( 1+
). (t n
c
Thus, the noise output is
). ( ) ( t n t n
c o
=
The output noise power is
. ) ( ) (
2
2
ni c no
P t n t n P = = =
. 1
3
2
3
2
= = =
no
ni
si
so
P
P
P
P
SNRI
, 1 =
no
ni
P
P
Thus,
and
We see that the signal-to-noise improvement for
DSB-LC is not as good as that of DSB-SC. The
reason for using DSB-LC is that it is relatively easy
to demodulate. (Standard broadcast AM [530-1640
kHz], uses DSB-LC.)
Noise in Frequency
Modulated Systems
The frequency modulation and demodulation process
is non-linear.

As with DSB-LC, we cannot treat the signal and the
noise separately.
]. ) ( cos[ ) (
}
+ = t m k t t x
f c c
e
The expression for the modulated carrier for FM
As with DSB-LC AM, he demodulation process
extracts m(t) from x
c
(t). This extraction process
consists of three steps:
1. Extract argument from cos().
2. Subtract e
c
t.
3. Differentiate what is left to get k
f
m(t).
Based upon these three steps we can quickly get
the input signal power and the output signal power.
). something cos( ) ( ) ( = = t x t s
c i
. ) something ( cos
2
1
2
= =
si
P
). ( ) ( t m k t s
f o
=
. ) ( ) (
2
2
2
2
t m k t m k P
f f so
= =
Without much loss of generality, we can assume that
m(t) = cose
m
t.
.
2
2
1
f so
k P =
.
2
2
1
2
2
1
f
f
si
so
k
k
P
P
= =
We now have the signal power ratio:
As with the DSB-LC AM, the effective noise input
comes along with an unmodulated carrier:
). ( ] ) ( 0 cos[ t n t m t
c
+ +
}
e
If use the quadrature decomposition of the noise, we
have, as the effective noise input
, sin ) ( cos ) ( cos t t n t t n t
c s c c c
e e e +
| | . sin ) ( cos ) ( 1 t t n t t n
c s c c
e e +
or,
This expression can be combined into a single
sinewave
), cos( u e + t R
c
where
.
) ( 1
) (
tan
). ( )] ( 1 [
1
2
2 2
t n
t n
t n t n R
c
s
s c
+
=
+ + =

u
At this point we make a simliar assumption to what
we made with DSB-LC AM:
| | . 1 ) ( 1 ) ( ~ + << t n t n
c s
(The new part is the ~1.)
Using this assumption we have
). ( ) ( tan
1
t n t n
s s
~ ~

u
(This last is true because tan
-1
x ~ x for small values
of x.)
) ) ( cos( t n t R
s c
+ e
With the assumptions given, our effective noise input
becomes
We may now apply our three demodulation steps:
1. Extract argument from cos().
2. Subtract e
c
t.
3. Differentiate what is left to get k
f
m(t).
1.
2.
3.
) (t n t
s c
+ e
) (t n
s
) (t n
s

Thus, the output noise is

). ( ) ( t n t n
s o

=
All that remains to be done is to find the noise power
from the noise signals [n
i
(t) and n
o
(t)].
The noise power will be found from the power
spectral densities of the input noise and the output
noise.
The input noise is additive white Gaussian noise.
The power spectral density of the input noise is
.
2
) (
0
N
f S
ni
=
The output noise is the derivative of the quadrature
component of the (input) noise [n
s
(t)].
We know, from a previous exercise, that
. ) (
0
N f S
ns
=
We need to find the power spectral density of the
derivative of the quadrature component. To find this
we multiply S
ns
(f) by the square of the transfer
function for the differentiation operation.
The effect of the differentiation operation is shown in
three ways:
d
dt
j2tf
n
s
(t) n
s
(t)
.
N
s
(f) N
o
(f)
|2tf|
2
S
ns
(f) S
no
(f)
Thus, the power spectral density of the output noise
is
( ) . 2 ) (
0
2
0
N f f S
n
t =
Now that we have the power spectral densities of
the input and the output spectra, all that remains to
find the power is to integrate the respective power
spectral densities over the appropriate frequencies.
The input noise is a bandpass process. Let B
T
be
the bandwidth.
f
S
ni
(f)
N
0
/2
B
T
Integrating the power spectral density, we have
. ) (
2
2
0
0
T T ni
B N B
N
P = =
The input noise is a lowpass process. Let W be the
bandwidth.
f
S
no
(f)=N
0
4t
2
f
2
W

-W

Integrating the power spectral density, we have
W
W
W
W
no
f
N
df f N P

=
=
}
3
4
4
3
2
0
2 2
0
t
t
.
3
8
3
4 2
3
0
2
3
2
0
W N
W
N
t
t
=
=
.
8
3
3 2
3
8
0
3 2
0
W
B B N
P
P
T
W N
T
no
ni
t
t
= =
Thus, the ratio of the noise powers is
Finally, the signal-to-noise improvement is
.
8
3
3 2
2
W
B
k
P
P
P
P
SNRI
T
f
no
ni
si
so
t
= =
k
f
peak frequency deviation
B
T
modulated carrier bandwidth
W demodulated bandwidth
Exercise: Suppose that the modulated carrier
bandwidth is given by Carsons rule:
). 1 ( 2 + = |
m T
f B
Further suppose that the demodulated bandwidth is
the bandwidth of the modulating signal:
m
f W =
Show that the resultant signal-to-noise improvement
is
). 1 ( 3
2
+ = | | SNRI
Use
.
2
m
f
f
k
t
| =
Exercise: Suppose that the modulated carrier
bandwidth is simply twice the modulating frequency:
. 2
m T
f B =
Further suppose that the demodulated bandwidth is
the bandwidth of the modulating signal:
m
f W =
Show that the resultant signal-to-noise improvement
is
. 3
2
| = SNRI
Improvement of FM SNRI
Using De-Emphasis
The signal-to-noise improvement for FM, while good,
can be improved by minimizing the noise.

The weakest link where it comes to the noise
amplification is the differentiator: the noise increases
as the cube of the bandwidth.

If the high-pass effect of the differentiator can be
offset by a low-pass filter, the noise will be
attenuated, and the SNRI will be improved.
The weakest link:
f
S
no
(f)=N
0
4t
2
f
2
W

-W

.
3
8
4
3
0
2
2 2
0
}

= =
W
W
no
W N
df f N P
t
t
Now, let us insert a low-pass filter whose transfer
function is
.
1
1
) (
1
f
jf
LP
f H
+
=
Our output noise power becomes
( )
.
1
4
2
2 2
0
'
1
}

+
=
W
W
f
f
no
df
f N
P
t
We can integrate this function by letting
. tan
1
u =
f
f
}
}
}

=
=
+
=
1
1
1
1
1
1
1
1
1
1
1
1
tan
tan
2 2
0
3
1
tan
tan
2
2
2
2
0
3
1
tan
tan
2
2
2 2
0
3
1
'
] 1 [sec 4
sec
sec
tan
4
sec
tan 1
tan 4
f
W
f
W
f
W
f
W
f
W
f
W
d N f
d N f
d
N
f P
no
u u t
u u
u
u
t
u u
u
u t
]. tan )[ 2 ( 4
1 1
1 2
0
3
1
f
W
f
W
N f

= t
We now define a quantity called the signal-to-noise
improvement improvement (no mistake). This is the
improvement in the SNRI as a result of de-
emphasis:
.
'
no
no
P
P
= I
The I factor becomes
.
] tan [ 3
] tan )[ 2 ( 4
3
8
1 1
1 1
1
3
1
3
1 2
0
3
1
3
0
2
'
f
W
f
W
f
W
f
W
no
no
f
W
N f
W N
P
P

= = I
t
t
A plot of this I factor is plotted on the following slide.
10
-1
10
0
10
1
0
2
4
6
8
10
12
14
16
Increase in SNRI Using De-Emphasis
W / f
1
I

(
d
B
)