In this chapter we study the noise performance of analog (continuous wave) modulation system. We begin the study by describing signal to noise ratios that provide basis for evaluating the noise performance of an analog communication receiver.
The output signal to noise ratio is a measure of describing the fidelity with which the demodulation process in the receiver recovers the original message from the received modulated signal in the presence of noise.
It is defined as the ratio of the average power of the message signal to the average power of the noise, both measured at the receiver output. It is expressed as
The output signal to noise ratio is unambiguous as long as the recovered message and noise at the demodulator output is additive. This requirement is satisfied exactly by linear receivers such as coherent detectors and approximately by using non-linear receivers such as envelope detector and frequency discrimination used in FM, provided that input noise power is small compared with average carrier power.
The calculation of the output signal to noise ratio involves the use of an idealized receiver model and its details depends on the channel noise and the type of demodulation used in the receiver. Therefore, (SNR)o does not provide sufficient information when it is necessary to compare the output signal to noise ratios of different analog modulation – demodulation schemes. To make such comparison, baseband transmission model is used. The baseband transmission model is shown in fig 8.1 below,
The baseband transmission model assumes:
1. The transmitted or modulated message signal power is fixed
2. The baseband low pass filter passes the message signal, and rejects out of band noise.
According to the baseband transmission model, we may define the channel signal to noise ratio, referred to the receiver input as
Itis independent of the type of modulation or demodulation used.
This ratio may be viewed as a frame of reference for comparing different modulation systems and we may normalize the noise performance of a specific modulation – demodulation system by dividing the output signal to noise ratio of the system by channel signal to noise ratio. This ratio is known as figure of merit for the system. It is expressed as
The above equation shows that, the higher the value that the FOM has, the better the noise performance of the receiver.
It is customary to model channel noise as a sample function of white noise process whose mean is zero and power spectral density is constant.
The channel noise is denoted by w(t), and its power spectral density is No/2 defined for both negative and positive frequencies.
Where No is the average noise power per unit bandwidth measured at the front end of the receiver.
The received signal consists of an amplitude modulated signal s(t) corrupted by the channel noise w(t). To limit the effect of noise on signal, we may pass the received signal through a band pass filter of bandwidth wide enough to accommodate s(t).
AM receiver is of superheterodyne type, this filtering is performed in two sections: RF section and IF section. Fig 8.2 below shows the idealized receiver model for amplitude modulation.
IF filter accounts for the combinations of two effects:
1. The filtering effect of the actual IF section in the super heterodyne AM receiver
2. The filtering effect of actual RF section in the receiver translated down to the IF band.
i.e ., the IF section provides most of the amplification and selectivity in the receiver.
The IF filter bandwidth is just wide enough to accommodate the bandwidth of modulated signal s(t) and is tuned so that its mid band frequency is equal to the carrier frequency except SSB modulation. The ideal band pass characteristics of IF filter is shown in fig below,
The band limited noise n(t) may be considered as narrow band noise.
The modulated signal s(t) is a band pass signal , its exact description is depends on the type of modulation used.
The time domain representation of narrow band noise can be expressed in two different ways,
1. Represented in terms of in-phase and quadrature components.
This method is well suited for the noise analysis of AM receivers using coherent detector and also for envelope detection.
2. Represented in terms of envelope and phase.
This method is well suited for the noise analysis of FM receivers.
We begin the noise analysis by evaluating (SNR)o, (SNR)c for an AM receiver using coherent detection.In put signal is either DSBSC or SSB modulated wave and pass it through IF filter. The use of coherent detection requires multiplication of IF filter output x(t) by Locally generated sinusoidal wave cos2rrfct and then, low pass filtering the product. For convenience, we assume that amplitude of locally generated signal is unity. For satisfactory operation local oscillator must be synchronized to carrier both in frequency and phase, we assume that perfect synchronization.
Coherent detection has the unique feature that for any signal-to-noise ratio, an output strictly proportional to the original message signal always present. That is., the output message component is unmutilated and the noise component always appears additively with the message irrespective of the signal-to-noise ratio.
Consider a DSBSC wave defined by
1. The message m(t) and in-phase noise component nI(t)of the narrow band 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.
The message signal component at the receiver output equals 1/2 Ac m(t). Hence, the average power of message signal at the receiver output is equal to Ac2P/4, where P is the average power of the original message signal m(t).
The noise component at the receiver output is 1/2 nI(t). Hence, the power spectral density of the output noise equals ¼ times nI(t). To calculate the average power of the noise at the receiver output, we first determine the power spectral density of the in-phase noise component nI(t).
The power spectral density SN(f) of the narrow band noise n(t) is of the form shown in fig(8.5) below.
Evaluating the area under the curve of power spectral density of fig 8.6 above and multiplying the result by ¼, we find that the average noise power at the receiver output is equal to WN0/2.
Then, the output signal to noise ratio for DSBSC modulation is given by
EXERCISE 8.1
Consider Eq8.8 that defines the signal x(t) at the detector input of a coherent DSBSC receiver. Show that:
a. The average power of the DSBSC modulated signal component s(t) is Ac2P/2.
b. The average power of the filtered noise component n(t) is 2WN0.
c. The signal to noise ratio at the detector input is
Solution:
Given,
The signal at the input of DSBSC coherent receiver has Signal component and narrow band noise n(t) component. The narrow band noise is expressed in terms of its in-phase nI(t) and quadrature component nQ(t).
a. The average power of the signal component is Ac2P/2 Where P is the average power of the message
b. The average power of the filtered noise component n(t) consists of average noise powers of its in-phase component nI(t) and quadrature component nQ(t).
From fig 8.6, The power spectral density of in-phase component nI(t) and quadrature component nQ(t) of narrow band noise n(t) is
SSB Receivers:
Let us consider the incoming signal is an SSB wave and assume that only the lower side band is transmitted, then we may express the modulated wave as
We may make following observations with respect to in-phase and quadrature component of eqn 8.13:
1. The two components are uncorrelated with each other. Therefore, their power spectral densities are additive.
2. The Hilbert transform m (t) is obtained by passing m(t) through a linear filter with transfer function –jsgn(f). The squared magnitude of this transfer function is equal to one for all frequencies. Therefore, m(t) and m (t) have the same average power.
Next, procedure is similar to DSBSC receiver.
The average power of s(t) is given by Ac2P/4, contributed by in-phase and quadrature component of SSB wave s(t).
i.e., Average power of in-phase and quadrature component is Ac2P/8. The average noise power in the message bandwidth W is WN0
Then, channel signal to noise ratio is
Next is to find (SNR)o,
The transmission bandwidth B = W. The mid band frequency of the power spectral density SN(f) of the narrow-band noise n(t) is differ from the carrier frequency fc by W/2. Therefore, we may express n(t) as
Equation 8.16 consists of both in-phase and quadrature component of narrow band noise n(t) and the quadrature component m� (t) of the modulated signal is completely eliminated.
The message component in the receiver output is 1/4 Ac m(t) so that the average power of the recovered message is Ac2P/16.
Exercise 2
Consider the two elements of the noise component in the SSB receiver output from eqn (8.16)
a. Sketch the power spectral density of the in-phase noise component nI(t) and quadrature component nQ(t).
b. Show that the average power of the modulated noise nI(t)cosΠWt or
nQ(t)sinrrWt is WN0/2
c. Hence, show that the average power of the output noise is WN0/4
Solution:
a. The power spectral density of the narrow band noise n(t) is shown in fig below:
Exercise 3
The signal x(t) at the detector input of a coherent SSB receiver is defined by
x(t) = s(t) + n(t)
Where the signal component s(t) and noise component n(t) are themselves defined by eqn 8.13 and eqn 8.15 respectively, show that
a. The average power of the signal component s(t) is Ac2P/4
b. The average power of the noise component n(t) is WN0
c. The signal to noise ratio at the detector input is
Noise in AM receivers using Envelope detection:
In standard AM wave both sidebands and the carrier are transmitted. The AM wave may be written as
Where Ac cos2πfct is the carrier wave, m(t) is the message signal, and kais the sensitivity of the modulator determines the percentage of modulation.
The average power of the modulated signal s(t) is A2 [ 1+ k2a P ] / 2 Where P is the average power of the message m(t).
The average noise power in the message bandwidth W is WN0 Then, the channel signal to noise ratio is
The received signal x(t) at the envelope detector input consists of the modulated message signal s(t) and narrow band noise n(t). then,
Fig 8.9Phasor diagram for AM wave plus narrow-band noise for the case of high carrier –to- noise for the case of high carrier-to-noise ratio.
From this phasor diagram the receiver output can be obtained as y(t) = envelope of x(t)
When the average carrier power is large compared with the average noise
i.e., the signal term Ac +Ackam(t) is large compared with the nI(t), nQ(t). Then, we may approximate the output y(t) as
The term Ac is Dc component and is neglected.
The output signal power is A2ck2a P.
The output noise power is 2WN0.
The output signal to noise ratio is
This expression valid only if,
1. The noise at the receiver is small compared to the signal
2. The amplitude sensitivity ka is adjusted for a percentage modulation less than or equal to 100%
Then, FOM is
The FOM of AM receiver using envelope detection is always less than unity.
When compared to DSBSC and SSB the noise performance of AM is always inferior.
Consider the message m(t) is a single- tone or single frequency signal as m(t) = Amcos2πfmt
where Am – Amplitude of the message and
fm – frequency of the message.
Then, the AM wave is given by
s(t) = Ac[ 1+ µ cos2πfmt] cos2πfct
where µ = Amka is the modulation factor.
The average power of the modulating wave m(t) is P = ½ A2m
Therefore, Eqn 8.25 becomes
This means that, AM system must transmit three times as much average power as a suppressed carrier system in order to achieve the same quality of noise performance.
The carrier-to-noise ratio of a communication receiver is defined by
Threshold effect:
When the carrier-to-noise ratio at the receiver input of a standard AM is small compared to unity, the noise term dominates and the performance of the envelope detector changes completely.
In this case it is convenient to represent the narrow band noise n(t) in terms of its envelope r(t) and phase Ψ(t), as given by
In the fig, we have used the noise as a reference, because it is a dominant term.
To the noise phasor, we added a phasor representing the signal term Ac[ 1+ kam(t) ], with the angle between them is Ψ(t).
From the fig it is observed that carrier amplitude is small compared to the noise envelope r(t). Then, we may approximate the envelope detector output as
The above relation shows that, the detector output is not proportional to the message signal m(t). The last term contains the message signal m(t) multiplied by noise in the form of cos [ Ψ(t) ]. The phase Ψ(t) of a noise uniformly distributed over 2π radians, and it can have values between 0 to 2π with equal probability.
Therefore, we have a complete loss of information. The loss of a message in an envelope detector that operates at a low carrier to noise ratio is referred to as threshold effect.
It means a value of the carrier to noise ratio below which the noise performance of a detector deteriorates much more rapidly than high carrier to noise rario as in eqn8.24((SNR)o).
