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Chapter 5 Effect of Noise on Analog Communication Systems

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Chapter 5Effect of Noise on Analog Communication Systems

- Chapter 5 Effect of Noise on Analog Communication Systems
- 5.1 Effect of Noise on linear-Modulation Systems
- 5.1.1 Effect of noise on a baseband system
- 5.1.2 Effect of noise on DSB-SC AM
- 5.1.3 Effect of noise on SSB AM
- 5.1.4 Effect of noise on conventional

- 5.2 Carrier-Phase Estimation with a phase-Locked Loop (PLL)
- 5.2.1 The phase-locked loop (PLL)
- 5.2.2 Effect of additive noise on phase estimation

- 5.3 Effect of noise on angle modulation
- 5..3.1 Threshold effect in angle modulation
- 5.3.2 Pre-emphasis and De-emphasis filtering

- 5.4 Comparison of analog-modulation systems

- 5.1 Effect of Noise on linear-Modulation Systems

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- 5.5 Effects of transmission losses and noise in analog communication systems
- 5.5.1 Characterization of thermal noise sources
- 5.5.2 Effective noise temperature and noise figure
- 5.5.3 Transmission losses
- 5.5.4 Repeaters for signal transmission

- 5.6 Further reading

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- The effect of noise on various analog communication systems will be analysis in this chapter.
- Angle-modulation systems and particularly FM, can provide a high degree of noise immunity, and therefore are desirable in cases of severe noise and/or low signal power.
- The noise immunity is obtained at the price of sacrificing channel bandwidth because he bandwidth requirements of angle-modulation systems is considerably higher than amplitude-modulation systems.

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- (1) Baseband system

baseband

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- (2) DSB-SC AM

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- Coherent demodulation

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- If PLL is employed
- Assume
- Message power :

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- Noise power :
Note that :

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- (3) SSB-AM
with ideal-phase coherent demodulator

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- (4) Conventional AM
for synchronous demodulation (similar to DSB, except using

instead of m(t) )

by a dc block

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but

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In practical application :

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Envelope detector

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- Case 1
After removing the dc component,

The same as y(t) for the synchronous demodulation, without the Â½ coefficient

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- Case 2

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Notes: i) Signal and noise components are no longer additive.

ii) Signal component is multiple by the noise and is no

longer distinguishable.

iii) no meaningful SNR can be defined.

- It is said that this system is operating below the threshold
==> threshold effect

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- Consider DSB-SC AM
assume , zero-mean (i.e. no dc component)

the average power at the output of a narrow band filter

tuned to the carrier frequency fc is zero.

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Let

There is signal power at the frequency , which can

be used to drive a PLL.

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- The mean value of the output of Bandpass filter is a sinusoid with frequency , phase , and amplitude
.

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- If input to the PLL is and output of the
VCO is , represents the estimate of , then

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- Loop filter is a lowpass filter, dc is reserved as is removed.
- It has transfer function
- n(t) provide the control voltage for VCD (see section 5.3.3).
- The VCO is basically a sinusoidal signal generation with an instantaneous phase given by
where is a gain constant in radians/ volt-sec.

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- The carrier-phase estimate at the output of VCO is :
and its transfer function is .

- Double-frequency terms resulting from the multiplication of the input signal to the loop with the output of the VCD is removed by the loop filter
PLL can be represented by the closed-loop system model as follows.

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- In steady-state operation, when the loop is tracking the phase of the received carrier, is small.
- With this approximation, PLL is represented the linear model shown below.

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closed-loop transfer function

where the factor of Â½ has been absorbed into the gain parameter k

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- The closed-loop system function of the linearized PLL is second order when the loop filter has a signal pole and signal zero.
- The parameter determines the position of zero in H(s), while K, and control the position of the closed-loop system poles.

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- The denominator of H(s) may be expressed in the standard form
where : loop-damping factor

: natural frequency of the loop

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- The magnitude response as a function of the normalized frequency is illustrated, with the damping factor as a parameter and .

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- The one-side noise equivalent bandwidth
- Trade-off between speed of response and noise in the phase estimate

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- PLL is tracking a signal as
which is corrupted by additive narrowband noise

, are assumed to be statistically independent stationary Gaussian noise.

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- Problem 4.29 ïƒ a phase shift does not change the first two moments of nc(t) and ns(t).
i.e. xc(t) and xs(t) have exactly the same statistical characteristics as nc(t) and ns(t).

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- The equivalent model is shown as below

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- When the power of the incoming signal is much larger than the noise power, .
- Then we may linearize the PLL shown as bellow.

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is additive at the input to the loop, the variance of the phase error , which is also the variance of the VCO output phase is

Bneq : one-sided noise equivalent bandwidth of the loop, given by

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- Note that is the power of the input sinusoid, and is simply proportional to the total noise power with the bandwidth of the PLL divided by the input signal power, hence
where is defined as the SNR

Thus, the variance of is inversely proportional to the SNR.

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- Note : the variance of linear model is close to the exact variance for

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- By computing the autocorrelation and power spectral density of these two noise component, one can show that both components have spectral power in frequency band centered at 2 fc.
- Letâ€™s select Bneq << Bbp, then total noise spectrum at the input to PLL may be approximated by a constant with the loop bandwidth.

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- This approximation allows us to obtain a simple expression for the variance of the phase error as
where SL is called the squaring loss and is given as

since SL<1 , we have an increase in the variance of the phase error caused by the added noise power that results from the squaring operation.

- E.g. if the loss is 3dB or equivalently, the variance in the estimate increase by a factor of 2.

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- Costas loop

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- These terms (signal noise and noise noise) are similar to the two noise terms at the input of the PLL for the squaring method.
- In fact, if the loop filter in the Costas loop is identical to that used in the squaring loop, the two loops are equivalent.
- Under this condition the pdf of the phase error, and the performance of the two loops are identical.
- In conclusion, the squaring PLL and the Costas PLL are two practical methods for deriving a carrier-phase estimation for synchronous demodulation of a DSB-SC AM signal.

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- RX for a general angle-modulated signal

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- Assume: signal power is much higher than noise power
i.e.

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Note that

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- The output of the demodulator is
where

(noise component)

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- Noise component
- The autocorrelation function

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- Assume m(t) is a sample function of a zero-mean stationary
Gaussian process M(t) with autocorrelation

then

- is a zero-mean, stationary Gaussian process

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- At any fixed time t, the random variable
is the difference between two jointly Gaussian random variables.

- is a Gaussian R.V. with zero mean and variance

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- Autocorrelation function

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- Power spectral density
where and G(f) is its Fourier transform.

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- The bandwidth of is half the bandwidth Bc of the angle-modulated signal, which for high modulation indices is much larger than W, the message bandwidth.
- Since the bandwidth of the angle-modulated signal is defined as the frequencies that contain 98%-99% of the signal power, G(f) is very small in the neighborhood of and

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- A typical example of G(f), and the result of their convolution is shown below.

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- Because G(f) is very small in the neighborhood of ,
the resulting SY(f) has almost a flat spectrum for , the bandwidth of the message.

i.e. for | f |<W , the spectrum of the noise components in the PM and FM.

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- The power spectrum of the noise component at the output of the demodulator in the frequency interval | f | < W for PM and FM is shown below.

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- PM has a flat noise spectrum and FM has a parabolic noise spectrum.
- The effect of noise in FM for higher-frequency components is much higher than the effect of noise on low-frequency components.
- The noise power at the output of the lowpass filter is the noise power in the frequency range [ -W, +W ].

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- Output signal power
- The SNR

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- Note that is the received signal power, denote by PR.
- The output SNR

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- Let
- The output SNR of a baseband system with the same received power

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- Note that in the above expression is the average-to-peak power ratio of the message signal (or, equivalently, the power content of normalized message, ).
- Therefore,

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- Carsonâ€™s rule
- The ration of the channel bandwidth to the message
bandwidth

- We can express the output SNR in terms of the bandwidth expansion factor

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- In both PM and FM, the output SNR is proportional to the square of the modulation index .
- Therefore, increasing increases the output SNR even with low received power.
- This is in contrast to amplitude modulation where such an increase in the received SNR is not possible.

- The increase in the received SNR is obtained by increasing the bandwidth.
- Therefore, angle modulation provides a way to trade-off bandwidth for transmitted power.

- The relation between the output SNR and the bandwidth expansion factor, , is a quadratic relation.
- This is far from optimal

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- In fact, if we increase such that the approximation
( ) does not hold, a phenomenon known as the threshold effect will occur, and the signal will be lost in the noise.

- A comparison of the above result with the SNR in amplitude modulation shows that, in both case, increasing the transmitter power will increase the output SNR, but the mechanisms are totally different.
- In AM, any increase in the received power directly increases the signal power at the output of the receiver.
- In FM, the effect of noise is higher at higher frequencies.

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- The noise analysis of angle demodulation schemes is based on the assumption that the SNR at the demodulator input is high.
- This assumption of high SNR is a simplifying assumption that is usually made in analysis of nonlinear-modulation systems.
- Threshold effect
- There exists a specific signal to noise ratio at the input of the demodulator known as the threshold SNR, beyond which signal mutilation occurs.
- The existence of the threshold effect places an upper limit on the trade-off between bandwidth and power in an FM system.
- This limit is a practical limit in the value of the demodulation index .
- Threshold in an FM system

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- In general, there are two factors that limit the value of the demodulation index .
- The limitation on channel bandwidth which effects through Carsonâ€™s rule.
- The limitation on the received power that limits the value of to less than what is derived from Equation .

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- In order to reduce the threshold, in other words, in order to delay the threshold effect to appear at lower-received signal power, it is sufficient to decrease the input-noise power at the receiver.
- This can be done by increasing the effective system bandwidth at the receiver.

- Two approaches to FM threshold extension are to employ FMFB or PLL-FM at the receiver.
- In application where power is very limited and bandwidth is abundant, these systems can be employed to make it possible to use the available bandwidth more efficiently.
- Using FMFB, the threshold can be extended approximately by 5 - 7 dB.

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- The objective in pre-emphasis and de-emphasis filtering is to design a system which behaved like an ordinary frequency modulator â€“ demodulator pair in the low frequency band of the message signal and like a phase modulator â€“ demodulator pair in the high-frequency band of the message signal.
- At the demodulator , we need a filter that at low frequencies has a constant gain and at high frequencies behaves as an integrator.
- The demodulator filter which emphasizes high frequencies is called the pre-emphasis filter and the demodulator filter which is the inverse of the modulator filter is called the de-emphasis filter.

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- The characteristics of the pre-emphasis and de-emphasis filters depend largely on the power-spectral density of the message process.
- In commercial FM broadcasting of music and voice, the frequency response of the receiver (de-emphasis) filter is
where

is the 3-dB frequency of the filter

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- The noise component after the de-emphasis filter has a power spectral density
- The noise power at the output of the demodulator

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- The ratio of the output SNRs

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- Linear modulation system
- DSB-SC
- Conventional AM
- SSB-SC
- VSB

- Nonlinear modulation system
- FM
- PM

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- The bandwidth efficiency of the system
- SSB-SC > SSB > VSB > DSB-SC > DSB
- PM = FM
- The most bandwidth-efficient analog communication system is the SSB-SC system with a transmission bandwidth equal to the signal bandwidth.
- PM and particularly FM are least-favorable systems when bandwidth is the major concern, and their use is only justified by their high level of noise immunity.

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- The power efficiency of the system
- Angle-modulation scheme and particularly FM provide a high level of noise immunity and, therefore, power efficiency.
- Conventional AM and VSB+C are least power-efficient system.

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- The case of implementation of system
- The simplest receiver structure is the receiver for conventional AM, and the structure of receiver for VSB+C system is only slightly more complicated.
- FM receivers are also easy to implement.
- DSB-SC and SSB-SC require synchronous demodulation and their receiver structure is much more complicated.

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- Based on quantum mechanics, the power-spectral density of thermal noise is
where h is Planckâ€™s constant, k is Boltzmannâ€™s constant, and T is the temperature of the resistor in degree Kelvin, i.e., T=273+C, where C is in degrees Centigrade.

- by

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- The power spectral density of noise voltage across the load resistor is

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- Thermal noise converted to amplifier and load

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- The noise power at the output of the network is
- Noise-equivalent bandwidth of the filter is defined as
- The output noise power from an idea amplifier that introduces no additional noise

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- The noise figure of the amplifier is

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- Transmission Loss
L = PT / PR

Ldb=10log L = 10log PT â€“ 10log PR

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L

- PR= PT / L
- P0=gPR= gPT/ L
- By g = L andP0= PT
- The SNR at the output of the repeater is

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