Noise in Modulation

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Noise in Modulation
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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.
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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.
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The calculation of the signal-to-noise
improvement (SNRI) is shown in the following
diagram.
si(t)
si(t)
Demodulator
ni(t)
no(t)
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The notation lt gt means time average.
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In determining the ability of a demodulator to
discriminate signal from noise, we calculate Psi,
Pso, Pni and Pno and plug these values (or
expressions) into the expression
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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.

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The expression for the modulated carrier for
DSB-SC is
Demodulation of this signal is performed by
multiplying xc(t) by cos wct and low-pass
filtering the result.
d(t)
LPF
X
so(t)
si(t) xc(t)
coswct
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The result of this demodulation can be seen by
analysis
Low-pass filtering the result eliminates the cos
2wct component.
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Based upon these relationships, we can find the
ratio of the signal input power and the signal
output power.
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So, it seems, half of our SNRI calculation is
done.
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The other half of the calculation deals with the
noise.
Let us use the quadrature decomposition to
represent the noise
It is this signal which will represent ni(t).
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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.
ni(t) nccoswct nssinwct
LPF
dn(t)
X
no(t)
coswct
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After dn(t) passes through the low-pass filter,
all we have left is
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Now, we calculate the power in the noise.
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The ratio of the noise powers becomes
Finally, our signal-to-noise improvement becomes
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Thus, the DSB-SC demodulation process improves
the signal-to-noise ratio by a factor of two.
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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 xc(t).
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The input signal power is
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The modulating signal m(t) is assumed here to
have zero mean.
If m(t) coswmt, then
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The output power is simply
which is equal to ½ if m(t) coswmt.
Thus, when m(t) is a sinewave, our signal power
ratio is
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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
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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
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We then work with this expression
At this point we make an assumption (which turns
out to be quite reasonable in many cases)
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In words, the noise (either the in-phase or
quadrature component) is much smaller in
magnitude that that of the signal coswct.
With this assumption the input to the demodulator
becomes
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We recall that when the input
is applied, the output is
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Similarly, when the input is
is applied, the output is
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Thus, the noise output is
The output noise power is
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Thus,
and
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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.)
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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.

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The expression for the modulated carrier for FM
As with DSB-LC AM, he demodulation process
extracts m(t) from xc(t). This extraction
process consists of three steps
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1. Extract argument from cos().
2. Subtract wct.
3. Differentiate what is left to get kfm(t).

Based upon these three steps we can quickly get
the input signal power and the output signal
power.
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Without much loss of generality, we can assume
that m(t) coswmt.
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We now have the signal power ratio
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As with the DSB-LC AM, the effective noise input
comes along with an unmodulated carrier
If use the quadrature decomposition of the noise,
we have, as the effective noise input
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or,
This expression can be combined into a single
sinewave
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where
At this point we make a simliar assumption to
what we made with DSB-LC AM
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(The new part is the ?1.)
Using this assumption we have
(This last is true because tan-1 x ? x for small
values of x.)
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With the assumptions given, our effective noise
input becomes
We may now apply our three demodulation steps

1. Extract argument from cos().
2. Subtract wct.
3. Differentiate what is left to get kfm(t).

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1.
2.
3.
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Thus, the output noise is
All that remains to be done is to find the noise
power from the noise signals ni(t) and no(t).
The noise power will be found from the power
spectral densities of the input noise and the
output noise.
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The input noise is additive white Gaussian noise.
The power spectral density of the input noise is
The output noise is the derivative of the
quadrature component of the (input) noise ns(t).
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We know, from a previous exercise, that
We need to find the power spectral density of the
derivative of the quadrature component. To find
this we multiply Sns(f) by the square of the
transfer function for the differentiation
operation.
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The effect of the differentiation operation is
shown in three ways
d dt
.
ns(t)
ns(t)
j2pf
Ns(f)
No(f)
2pf2
Sns(f)
Sno(f)
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Thus, the power spectral density of the output
noise is
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.
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The input noise is a bandpass process. Let BT be
the bandwidth.
Sni(f)
N0/2
f
BT
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Integrating the power spectral density, we have
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The input noise is a lowpass process. Let W be
the bandwidth.
Sno(f)N04p2f2
f
W
-W
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Integrating the power spectral density, we have
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Thus, the ratio of the noise powers is
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Finally, the signal-to-noise improvement is
kf peak frequency deviation
BT modulated carrier bandwidth
W demodulated bandwidth
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Exercise Suppose that the modulated carrier
bandwidth is given by Carsons rule
Further suppose that the demodulated bandwidth is
the bandwidth of the modulating signal
Show that the resultant signal-to-noise
improvement is
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Use
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Exercise Suppose that the modulated carrier
bandwidth is simply twice the modulating
frequency
Further suppose that the demodulated bandwidth is
the bandwidth of the modulating signal
Show that the resultant signal-to-noise
improvement is
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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.

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Sno(f)N04p2f2
The weakest link
f
W
-W
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Now, let us insert a low-pass filter whose
transfer function is
Our output noise power becomes
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We can integrate this function by letting
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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
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The G factor becomes
A plot of this G factor is plotted on the
following slide.
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