Module 4 Noise in Analog Modulation

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4.1 Introduction

Idea of Modeling:
1. To study of all physical systems
2. Understand the capabilities and limitations of communication systems.
Characteristics of receiver model:
1. Provide an adequate description of receiver noise
2. Accounts for inherent filtering and modulation characteristics of the system
3. Simple for statistical analysis

4.2 Receiver Model

To undertake analysis of noise in continuous-wave modulation systems, we need a receiver model.

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The customary practice is to model the receiver noise (channel noise) as additive, white, and
Gaussian. These simplifying assumptions enable us to obtain a basic understanding of the way in
which noise affects the performance of the receiver.
Channel model: additive white Gaussian noise with PSD = N0/2.
Receiver model: ideal BPF and ideal demodulator
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Fig. 1 Noisy receiver model
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From the fig.1, s(t) denotes the incoming modulated signal and w(t) denotes front-end
receiver noise. The power spectral density of the noise w(t) is denoted by N0/2, defined for
both positive and negative frequencies. N0 is the average noise power per unit bandwidth
measured at the front end of the receiver. Assume the BPF is ideal, having a bandwidth, BT
equal to the transmission bandwidth, W of the modulated signal s(t) and a mid-band
frequency equal to the carrier frequency fc , fc >> BT. The noise after the BPF is thus
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narrowband noise.

Fig. 2 Idealized characteristic of band-pass filtered noise

The filtered noise n(t) may be treated as a narrow band noise represented in the canonical
form:
………………….. (4.1)
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where nI(t) is the in-phase noise component and nQ(t) is the quadrature noise component,
both measured with respect to the carrier wave c(t) = Ac cos(2πfct).
The filtered signal x(t) available for demodulation is defined by
x(t) = s(t) + n(t) ………………….. (4.2)
Where, n(t) is noise which is the band-pass filtered version of w(t).
The average noise power may be calculated from the power spectral density SN( f
):

………………….. (4.3)
Signal to Noise Ratio (SNR)
SNR is a measure of the degree to which a signal is contaminated by noise.
Input signal-to-noise ratio (SNR)I is defined as:

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Output signal-to-noise ratio (SNR)O is defined as:

This ratio depends on the type of modulation in the transmitter and the type of demodulation in the
receiver.
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Channel signal-to-noise ratio (SNR)C is defined as:

As a reference, this ratio may be viewed as SNR from direct transmission of the message
signal m(t) without modulation, as shown in the fig.3.
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Fig. 3 Base band transmission Model

To compare across different modulators, we assume that (see fig. 3):

• The modulated signal s(t) of each system has the same average power
• Channel noise w(t) has the same average power in the message bandwidth W
Figure of merit: For the purpose of comparing different continuous-wave modulation systems
(AM, FM or PM), we normalize the receiver performance by dividing the (SNR)O ratio by the
(SNR)C ratio. The higher the value of the figure of merit, the better will be the noise
performance of the receiver. The figure of merit may equal to one, be less than one, or be
greater than one, depending on the type of modulation used.

………………….. (4.4)
4.3 Noise in DSB SC Receivers

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The noise model of a DSB-SC receiver using a coherent detector is as shown in the fig. 4.
 The amplitude of the locally generated sinusoidal wave is assumed to be unity.
 For the demodulation scheme to operate satisfactorily, it is necessary that carrier wave in
the receiver side be synchronized both in phase and in frequency with the carrier wave in
the transmitter.
 We assume that this synchronization has been achieved.

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Fig.4 Noisy model of a DSB-SC receiver using a coherent detector

The DSB-SC component of the modulated signal s(t) is expressed as

………………….. (4.5)
where, C is the system dependent scaling factor The purpose of C is to ensure that the signal
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component s(t) is measured in the same units as the additive noise component n(t). m(t) is
the sample function of a stationary process of zero mean, whose power spectral density SM( f
) is limited to a maximum frequency W (the message signal bandwidth).
The average power P of the message signal is the total area under the curve of power
spectral density, defined as
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………………….. (4.6)
The average power of the DSBSC modulated signal s(t) is
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………………….. (4.7)
The (two sided) noise power spectral density is N0/2, then the average noise power in the
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message bandwidth 2 W is

………………….. (4.8)
The (SNR)C ratio of the DSB-SC modulation system is

………………….. (4.9)
To determine the (SNR)O ratio:
Using the narrowband representation of the filtered noise n(t), the total signal at the coherent
detector input may be expressed as: x(t) = s(t) + n(t) i.e, using Eq (4.1) and
Eq(4.5),
………………….. (4.10)
The output of the product modulator component of the coherent detector is:

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….(4.11)
Equation (4.10) indicates the following:
1. The message signal m(t) and in-phase noise component nI(t) of the filtered noise n(t)
appear
additively at the receiver output.
2. The quadrature component nQ(t) of the noise n(t) is completely rejected by the coherent
detector.

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We note that these two results are independent of the input SNR ratio. Thus, coherent
detection distinguishes itself from other demodulation techniques in the important property:
the output message component is un-mutilated and the noise component always appears
additively with the message, irrespective of the input SNR ratio.
The receiver output signal: as in Eq (4.10)
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The average power of message component:

The average power of the noise at the receiver:

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Output SNR for DSBSC:

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Figure of merit:

4. 4 Noise in AM Receivers

A standard AM signal is given by

………………….. (4.12)
Where, Accos(2πfct) is the carrier wave, m(t) is the message signal and bandwidth is W, ka is
a constant that determines the percentage modulation.
The noise model of standard AM receiver using a envelope detector is as shown in the fig. 5.

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Fig. 5 Noisy model of a AM receiver using a envelope detector

The noise analysis of the AM receiver is performed by determining

1. (SNR)C -- channel signal-to-noise ratio and
2. (SNR)O -- output signal-to-noise ratio
Average power of the AM signal:

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…………….. (4.12)
The average power of noise in the message band-width (same as the DSB-SC system)

The channel signal-to-noise ratio for AM

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Fig. 6 Phasor diagram for AM plus narrowband noise for high CNR

The filtered signal x(t) applied to the envelope detector in the receiver is by (see phasor
diagram),
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……….. (4.13)
The DC term Ac may be removed simply by means of a blocking capacitor. If we ignore the
term Ac, we find that the remainder has a form similar to the output of a DSB-SC receiver
using coherent detection. The output signal-to-noise ratio of an AM using an envelope
detector is approximately

………………….. (4.14)
Eq. (4.14) is valid only if the following two conditions are satisfied.
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1. The SNR is high
2. The amplitude sensitive ka is adjusted for 100% modulation or less, so there is no
distortion of the signal envelope

Figure of merit: Figure of merit of an AM receiver using envelope detection is always less
than unity. Because,

………………….. (4.15)
4.4.1 Threshold Effect
If the carrier SNR ratio is high enough then the signal dominates and the noise effect will be under
control. However, if the carrier SNR ratio is small then the noise dominates (no term proportional to
the message signal see fig.) in turn, performance of envelope detector completely changes which

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results in a complete loss of information. That means, loss of message signal in an envelope detector
due to small carrier SNR ratio, is called threshold effect.
There is some carrier SNR ratio, above which message corruption is negligible. But below some
carrier power level, the performance of the envelope detector deteriorates very rapidly. Every
nonlinear detector (ex: envelope detector) exhibits a threshold effect.
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Fig.7 phasor representation for CNR ratio is small and
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noise dominates

Note: Noise is multiplicative here. No term proportional to the message.

4.5 Noise in FM Receivers

The receiver model used in the noise analysis of angle modulated signals is shown in Fig.8.
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Fig. 8 Noisy model of FM receiver

The noise w(t) is modeled as white Gaussian noise of zero mean and power spectral density
N0/2.
The received FM signal s(t) has a carrier frequency fc and transmission bandwidth BT, such
that only a negligible amount of power lies outside the frequency band fc ± BT /2 for positive
frequencies.
The BPF has a mid-band frequency fc and bandwidth BT and therefore passes the FM signal
essentially without distortion. Ordinary, BT is small compared with the mid-band frequency fc

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so that we may use the narrowband representation for n(t), where n(t) is the filtered version of
receiver noise w(t) and expressed in terms of its in-phase and quadrature components.
In an FM system, the message information is transmitted by variations of the instantaneous
frequency of a sinusoidal carrier wave, and its amplitude is maintained constant. If any
variations of the carrier amplitude occurred at the receiver input the only cause is from noise.
The limiter is used to remove amplitude variations by clipping the modulated wave at the
filter output.
Discriminator is a device with output proportional to the deviation in the instantaneous
frequency it recovers the message signal. The discriminator consists of two components:
1. A slope network or differentiator with a purely imaginary transfer function that varies linearly with
frequency. It produces a hybrid-modulated wave in which both amplitude and frequency vary in
accordance with the message signal.
2. An envelope detector that recovers the amplitude variation and thus reproduces the

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message signal.
The slope network and envelope detector are usually implemented as integral parts of a
single physical unit.
Base-band LPF: has a bandwidth of W. It passes the message signal and removes out-of-
band noise at the discriminator output and thereby keeps the effect of output noise to a
minimum.

Analysis: The filtered noise at the band-pass filter output is defined as

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………………….. (4.16)
The incoming FM signal s(t) is given by
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………………….. (4.17)
The noisy signal at the band-pass filter output is
………………….. (4.18)

Fig. 9 Phasor diagram for FM plus narrowband noise for high CNR

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The envelope of x(t) is of no interest to us, because any envelope variations at the band pass
output are removed by the limiter. Therefore, our motivation is to determine the error in the
instantaneous frequency of the carrier wave caused by the presence of the filtered noise n(t).
Assuming the discriminator is an ideal, and its output is proportional to θ’(t)/2π, where θ’(t) is
the derivative of θ(t) with respect to time.
We need to make certain simplifying approximations so that our analysis may yield useful
results

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……….. (4.19)
This means that the additive noise nd(t) appearing at the discriminator output is determined
effectively by the carrier amplitude Ac and the quadrature component nQ(t) of the narrowband
noise n(t).
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To determine the average output noise power, we note
that the
noise nd(t) at the discriminator output is proportional to the
time
derivative of the quadrature noise component nQ(t).
The differentiation of a function respect to time
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corresponds to
multiplication of its Fourier transform by j2πf. We may
obtain the noise nd (t) by passing nQ (t) through a linear
transfer function

Fig.10 Noise analysis of FM receiver.

(a) PSD of quadrature component of

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narrowband noise (b) PSD at
discriminator output (c) PSD of noise at
receiver output.

For wide-band FM, we usually find that W is smaller than

BT /2 where BT is transmission bandwidth of the FM signal.
This means that out-of-band components of noise n (t) will

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be
rejected.
In FM system, increasing the carrier power has a noise-quieting effect.

………………….. (4.20)
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The average power in the modulated signal s(t) is A2c /2, and the average noise power in the
message bandwidth is WNo. The channel signal to noise ratio (SNR)C,FM is
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Figure of merit for frequency modulation

Capture Effect
FM has ability to minimize the effect of unwanted signals (noise or interference). Interference is one
that signal has closer frequency to the carrier. Suppose the following cases occurred:
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i) If interference strength is weaker than the desired FM, then FM receiver works well.
ii) If interference strength is stronger than the desired FM, then FM receiver locks into stronger signal
(capture the unwanted FM signal). This causes to suppress the desired FM.
iii) If both have nearly equal strength, FM receiver fluctuates back and forth between them.
The above phenomenon is known as capture effect.
4.6 Threshold Effect
Output SNR of FM receiver is defined as

………………….. (4.21)

Eqn. (4.21) is valid only if the Carrier to Noise Ratio (CNR), measured at the discriminator input, is
high compared to unity. If the CNR is lowered, the FM receiver breaks down (pause). At first,
individual “clicks” are heard in the (audio) receiver output and as the CNR decreases further, the clicks

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merge to a “crackling” or sputtering sound. Near the breakdown point, Eq. (4.21) fails to accurately
predict the (SNR)O,FM . This phenomenon is known as the threshold effect.
Qualitative Discussion: Below the threshold the FM receiver breaks (i.e., significantly deteriorated).
This can be analyzed by examining the phasor diagram as shown in the fig. 11

Fig.11 Phasor diagram interpretation of Eqn 4.22

Assume message signal m(t) = 0 in the FM, so that carrier is un-modulated. i.e, only carrier and noise

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is available at the FM receiver.
x(t) = c(t) + n(t)

………………….. (4.22)
Eqn (4.22) consists of real component [Ac+nI(t)] along x-axis and imaginary component [nQ(t)] along
y-axis, as shown in fig. r(t) is envelope of noise with its phase angle .
As the noise changes randomly, the point P1 wanders around P2.
– High SNR: change of angle θ(t) is small and P1 spends most of its time near P2.
– Low SNR: P1 occasionally sweeps around origin, (i.e, closer to P2) resulting in θ(t) increases or
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decreases by 2π radians.
Clicks are produced only when θ(t) changes by 2π radians. Two conditions to occur clicks are:
1. Positive going click occurs, if envelope of noise r(t) and phase angle of noise satisfies:
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2. Negative going click occurs, if envelope of noise r(t) and phase angle of noise satisfies:
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To characterize threshold performance CNR is defined as

………………….. (4.23)
When ρ is decreased, average number of clicks/unit time are increases. Otherwise, ρ is large, threshold
is said to occur. This is illustrated in the fig.12.

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Fig. 12 Illustrating impulse like components in  (t)  d(t)/dt produced by changes of 2 in (t); (a) and
(b) are graphs of (t) and  (t) respectively , respectively.

4.6.1 FM threshold reduction

In applications where power is very limited and bandwidth is abundant, such as space
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communications, threshold extension circuit can be employed to make it possible to use the available
bandwidth more efficiently. This can be achieved by using
1. FM demodulator with negative Feed Back (FMFB) or
2. PLL demodulator.
The FMFB circuit is also called a threshold extension circuit and a signal enhancer. Block diagram of
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an FMFB demodulator is shown in the fig.13.
Idea is to achieve extended threshold using an FMFB demodulator. To understand the operation of this
receiver, suppose that VCO is removed from the circuit and the feedback path is left open.
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Fig.13 Block diagram of an FMFB demodulator

Without feedback: Assume that the wide band FM is applied to the receiver input terminal 1, and a
second FM from the same source but with a modulation index (β <1) is applied to terminal 2 of the
mixer (see the fig. ). The output of the mixer consists of sum and difference frequency components.
The BPF is designed to pass only difference frequency component. The frequency deviation of the
output of BPF would be small, although the frequency deviation of both input FM waves is large. The
FM wave with reduced β passed by the BPF is then frequency - demodulated by the combination of
limiter/discriminator and finally processed by the base band PLF.
It is now apparent that the second FM wave required at terminal 2, may be obtained by feeding the
output of LPF via VCO as in the fig.13.
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Fig.14 FM Threshold
extension

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In the combined presence of an un-modulated carrier Accos2πfc and a narrow band noise

As long as the CNR is sufficiently large, the FMFB receiver does not respond to the in-phase noise
component nI(t), but it would demodulate the quadrature noise nQ(t) like a signal. Then at the
discriminator input composite signal x(t) = Ac/ nQ(t) (see phasor diagram). The FMFB uses a very
important priori information that even though the carrier frequency of the incoming FM wave will
usually have large frequency deviations, its rate of change will be at the base band rate.
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With feedback: VCO generates an FM wave that reduces nQ(t) component of noise in the output of
base band LPF. Thus even if CNR is sufficiently large, FMFB demodulator does not respond to nI(t),
but it demodulates nQ(t) component in the same fashion as it would demodulate a signal modulation.
An FMFB demodulator is essentially a tracking filter that can
i) Track only the slowly varying the frequency of wideband FM waves.
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ii) Respond only to a narrow band of noise centered about the instantaneous carrier frequency.
The bandwidth of noise that responded by FMFB demodulator is precisely same as the band of noise
that tracks by VCO. As a result, FMFB receivers allow a threshold extension approximately by 5-7 dB.
PLL is also a tracking filter is relatively simple and hence it provides threshold extension, but
improvement on order of 2 - 3dB is lower than FMFB demodulator.

4.7 Pre-Emphasis and De-Emphasis in FM

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1
H pe ( f )  1  j ( f / f 0 ) and H de ( f )  , where f 0  1 /( 2Cr ).
1  j( f / f0 )

Fig. 15 Pre-Emphasis and De-Emphasis in FM

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Illustration of Pre-Emphasis and De-Emphasis in FM is shown in the fig.15. For many signals of
common interest, such as speech, music etc., most of the energy concentration is in the low
frequencies and the frequency components lies in the band-width W have very little energy.
When these low energy, high frequency components frequency modulate a carrier, they will
not give rise to full- frequency deviation and hence the message will not be utilizing fully the
allocated bandwidth. The net result is a low SNR at the high frequency end of the message
spectrum. To offset this undesirable occurrence, pre-emphasis and de-emphasis technique is
used. These circuits can improve the output SNR by around 13 dB.
Pre-emphasis filter boosts the amplitude of high modulating signal prior to modulation at the
transmitter side before noise is introduced. This is accomplished by passing the message
signal m(t), through the pre-emphasis filter (HPF), denoted Hpe (f ). This filter increases the
energy content of message signal so that signal strength will be stronger than the noise and
in turn, high SNR.

AM
The inverse operation, de-emphasis, is referred to attenuating the amplitude of high
frequency components of message signal used to bring back the original amplitude value.
This is accomplished by passing the discriminator output through the de-emphasis filter,
denoted Hde (f ).
In order to produce an undistorted version of the original message at the receiver output, we must
have:
H pe ( f ) H de ( f )  1 for  W  f  W .

This relation guarantees the intactness of the message power. Next, we need to find Hde(f) such that the
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noise power is optimally suppressed.
Under the assumption of high carrier-to-noise ratio, the noise PSD at the de-emphasis filter output is
given by:
 N0 f 2
 | H de ( f ) |2 , | f | W
| H de ( f ) |2 S N ( f )   Ac2
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o

 0, otherwise

W N0 f 2
 Average noise power   | H de ( f ) |2 df
W Ac2

Since the message power remains the same, the improvement factor of the output signal-to-noise ratio
after and before pre/de-emphasis is:
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W N0 f 2
W Ac2
df

W
f 2 df 2W 3
I 2
 W
W
 W
N0 f
 3
W
f 2 | H de ( f ) |2 df f 2 | H de ( f ) |2 df ………….. (4.24)

W Ac2
| H de ( f ) |2 df W W

This improvement factor assumes the use of high CNR at the discriminator input in the receiver.

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