To study the noise performance of a FM receiver idealized FM receiver model is required. Fig below shows the idealized FM receiver model,
The noise w(t) is modelled as white noise of zero mean and power spectral density N0/2. The received FM signal s(t) has a carrier frequency fc and transmission bandwidth B, so that only a negligible amount of power lies outside the frequency
band fc- B/2 ≤ |f| ≤ fc+ B/2. The FM transmission bandwidth B is in excess of twice the message bandwidth W by an amount that depends on the deviation ratio of the incoming FM wave.
The IF filter in fig 8.11 above represents the combined filtering effect of RF sectionand IF sections of an FM receiver of the superheterodyne type. The filter has a midband frequency fc and bandwidth B, passes the FM signal without any distortion. The bandwidth B is small compared with the mid band frequency fc. The transfer function of the IF filter is shown in fig 8.12 above. The filtered narrow band noise n(t) is represented in terms of its in-phase and quadrature components.
The limiter in fig 8.11 is used to remove any amplitude variations at the IF output. The discriminator is assumed to be ideal and its output proportional to the deviation in the instantaneous frequency of the carrier away from fc. The base band low pass filter is ideal and its bandwidth is equal to message bandwidth.
For the noise analysis of FM receivers, the narrow-band noise n(t) at the output of the IF filter is expressed as
The envelope of the x(t) is not considered, because any envelope variation is limited by the limiter.
Now, our aim is to determine error in the instantaneous frequency of the carrier wave caused by the presence of narrow band noise. The discriminator output is proportional to Ɵ' (t), where Ɵ' (t) is the derivative of 8(t) with respect to time.
We assume that the carrier-to-noise ratio measured at the discriminator input is large compared with unity. Then, the expression for relative phase is given by
The first term in eqn 8.39 is proportional to message m(t) as in eqn 8.35. Hence, using eqn 8.35 and 8.39, the discriminator output is given by
where the noise term nd(t) is defined by
provided that the carrier to noise ratio is high, the discriminator output v(t) consists of a scaled version of message m(t) and noise component nd(t).
Accordingly, we may use the output signal to noise ratio to assess the quality of performance of the FM receiver.
From eqn 8.40 the post detection filter output is kf m(t). Hence, the average output signal power is k2fP. Where P is the average power of the message signal m(t).
The calculation of average output noise power is complicated due to the presence of the term sin [Ψ(t) - Φ(t)] in eqn 8.41. The phase Ψ(t) is uniformly distributed over 2π radians, the mean square value of the noise nd(t) will be biased by the message dependent component Φ(t). The Φ(t) produces components outside the spectrum of nd(t), but they are rejected by post detector filter. Therefore, Φ(t) is neglected.
Then, eqn 8.41 becomes
Thus, we may state that under the condition of high carrier to noise ratio, the calculation of the average output noise power in an FM receiver depends only on the carrier amplitude Ac and the quadrature noise component nQ(t).
Recall that, differentiation of a function with respect to time corresponds to multiplication of its Fourier Transform by j2πf.
It means, we may obtain the noise process nd(t) by passing nQ(t) through a linear filter of transfer function equal to
Then, the power spectral density of SNd(f) is related to the power spectral density SNQ(f) of nQ(t) as,
The corresponding power spectral density of the noise nd(t) is shown in fig 8.15 above ,
The discriminator output is applied to a low pass filter of bandwidth equal to message bandwidth W.
For wide band FM, W is very small compare to B/2 where B is the transmission bandwidth of the FM signal. It means out of band components of noise nd(t) will be rejected.
Therefore, power spectral density SNo(f) of output noise no(t) appear at the output of low pass filter is defined by
Note that the average output noise power is inversely proportional to the average carrier power A2c/2.
The output signal to noise ratio is given by
The average power in the modulated signal s(t) is A2c/2, and the average noise power in the message bandwidth is WN0. Thus the channel signal to noise ratio is
FOM is
Therefore, from eqn 8.52 the FOM of a wide band FM system is a quadratic function of the deviation ratio.
In wide band FM, the transmission bandwidth is approximately proportional to the deviation ratio D. Accordingly, we may state that when the carrier to noise ratio is high, an increase in the transmission bandwidth B provides a corresponding quadratic increase in the output signal to noise ratio or FOM of the system.
Consider the case of a sinusoidal wave of frequency fm as the modulating wave, and assume a frequency deviation ∆f. The modulated wave is defined by
Note:
1. The modulation index β =∆f/W is determined by the bandwidth W of the post detector filter and is not related to the message bandwidth fm. But the filter is chosen to pass the spectrum of the message.
2. If fm may lie anywhere b/n 0 and W, it will gives same output SNR. Let us compare the performance of AM and FM systems,
For single tone modulation in AM,
Therefore, β = 0.5 roughly defines the transition from narrow band FM to wide band FM.
In the frequency modulation, the signal can be affected by another frequency modulated signal whose frequency content is close to the carrier frequency of the desired FM wave. The receiver may lock such an interference signal and suppress the desired FM wave when interference signal is stronger than the desired signal. When the strength of the desired signal and interference signal are nearly equal, the receiver fluctuates back and forth between them, i.e., receiver locks interference signal for some times and desired signal for some time and this goes on randomly. This phenomenon is known as the capture effect.
FM Threshold Effect:
The output signal to noise ratio of FM receiver is valid only if the carrier to noise ratio is measured at the discriminator input is high compared to unity. It is observed that as the input noise is increased so that the carrier to noise ratio decreased, the FM receiver breaks. At first individual clicks are heard in the receiver outputand as the carrier to noise ratio decreases still further, the clicks rapidly merge in to a crackling or sputtering sound.
Near the break point eqn8.50 begins to fail predicting values of output SNR larger than the actual ones. This phenomenonis known as the threshold effect.
The threshold effect is defined as the minimum carrier to noise ratio that gives the output SNR not less than the value predicted by the usual signal to noise formula assuming a small noise power.
For a qualitative discussion of the FM threshold effect,
Consider, when there is no signal present, so that the carrier is unmodulated. Then the composite signal at the frequency discriminator input is
Where nI(t) and nQ(t) are inphase and quadrature component of the narrow band noise n(t) with respect tocarrier wave Accos 2πfct. The phasor diagram of fig8.17 below shows the phase relations b/n the various components of x(t) in eqn(8.55)..
As the amplitudes and phases of nI(t) and nQ(t) change randomly with time the point P wanders around the point Q.
When the carrier to noise ratio is large nI(t) and nQ(t) are small compared to Ac, so that point P always around Q. Thus the angle 8(t) small and within a multiple of 2π radians.
The point P occasionally sweeps around the origin and 8(t) increases or decreases by 2π radians, When the carrier to noise ratio is small. The clicks are produced only when 8(t) changes by ±2π radians.
From the phasor diagram of fig above, we may deduce the condition required for clicks to occur.
A positive going click occurs when the envelope r(t) and phase Ψ(t) of the narrow band noise n(t) satisfy the following conditions:
These conditions ensure that 8(t) changes by 2π radians in the time incrementdt, during which the phase of the narrow band noise increases by the incremental amount dΨ(t).
Similarly, the condition for negative going click to occur are
These conditions ensure that 8(t) changes by - 2π radians in the time incrementdt As the carrier to noise ratio decreased, the average number of clicks per unit
time increases. Whenthis number becomes large, the threshold is said to occur. Consequently, the output SNR deviates from a linearfunction of the carrier to noise ratio when the latter falls below the threshold.
This effect is shown in fig below, this calculation is based on the following two assumptions:
1. The output signal is taken as the receiver output measured in the absence of noise. The average output signal poweris calculated for a sinusoidal modulation that produces a frequency deviation ∆fequal to 1/2 of the IF filter bandwidth B, The carrier is thus enabled to swing back and forth across the entire IF band.
2. The average output noise power is calculated when there is no signal present, i.e., the carrier is unmodulated, with no restriction placed on the value of the carrier to noise ratio.
The curve plotted in fig above for the ratio (B/2W) = 5. The linear portion of the curve corresponds to the limiting value 3ρ(B/2W)3. From the fig we may observe that the output SNR deviates appreciably from a linear function of the carrier to noise ratio ρ, when ρ becomes less than a threshold of 10dB.
The threshold carrier to noise ratio ρthdepends on the ratio of IF filter bandwidth to message bandwidth B/W and ρthis influenced by the presence of modulation.
We may state that the loss of message at an FM receiver outputis negligible ifthe carriertonoise ratio satisfies the condition
The IF filter bandwidth B is designed to equal the FM transmission bandwidth. Hence, we may use carson's rule to relate B to the message bandwidth W as follows
B = 2W(1+D)
where D is the deviation ratio, for sinusoidal modulation, the modulation index β is used in place of D. therefore, For no significant loss of message at an FM receiver output as,
In Specific applications such as space communications, It is required toreduce the noise threshold in an FM receiverso as to satisfactorily operate the receiver with the minimum signal power possible.
This can be achieved by using an FM demodulator with negative feedback(FMFB) or by using a phase locked loopdemodulator.
Fig above is a block diagram of an FMFB demodulator. The conventional local oscillator is replaced by VCO. To understand the operation of this receiver, suppose that VCO is removed from the circuit and the feedback path is leftopen.
Assume that the wide band FM is applied to the receiver input, and a second FM from the same source but with amodulation index a fraction smaller is applied to the VCO terminal of the product modulator. The output of theproduct modulator consists of sum and difference frequency components. The IF filter is designed to pass only difference frequency component. The frequency deviation of the IF filter output would be small, although the frequency deviation of both input FM wave is large, since the difference between their instantaneous deviations is small. Hence, the modulation indices would subtract, and the resulting FM wave at the IF filter output have a smaller modulation index than the input FM waves. This means that the IF filter bandwidth in fig above need only be a fraction of that required for either wideband FM wave. The FM wave with reduced modulation index passed by the IF filter is then frequency - demodulated by the combination of limiter/discriminator and finally
processed by the base band filter. It is now apparent thatSecond wide band FM waves replaced by VCO feed by o/p of low pass filter as in fig above.
Now, It will be shown that the SNR of an FMFB receiver is same as that of conventional FM receiver with the same input signal and noise power if the carrier to noise ratio is sufficiently large. Assume for the moment there is no feedback around the demodulator.In the combined presence of an unmodulated carrier Ac cos2πfct and a narrow band noise
n(t) = nI(t)cos2rrfct - nQ(t)sin2rrfct,
the phase of the compositesignal x(t) at the limiter - discriminator input is approximately equal to nQ(t)/Ac. This assumes that the carrier to noise ratio is high. The envelope of x(t) is of no interest to us, because the limiter removes all variations in the envelope. Thus the composite signal at the frequency discriminator input consists of a small index phase - modulated wave with the modulation derived from the component nQ(t) of noise that is in phase quadrature with the carrier. When feedback is applied, the VCO generates a wave that reduces the phase - modulation index of the wave at the IF filter output, i.e., the quadrature component nQ(t) of noise.
Thus we see that as long as the carrier - to noise ratio is sufficiently large, the FMFB receiver does not respond to the in-phase noise component nI(t), but it would demodulate the quadrature noise like a signal. Signal and quadrature noise are reduced in the same proportion by the applied feedback, with the result that the base band SNR is independentof feedback.For large carrier to noise ratio the baseband SNR of an FMFB receiver is same as that of a conventional receiver.
The FMFB users a very important piece of a 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.
An FMFB demodulator is essentially a tracking filter that can track only the slowly varying the frequency of widebandFM waves. Consequently it responds only to a narrow band of noise centered about the instantaneous carrier frequency. The bandwidth of noise to which the FMFB receiver responds is precisely the band of noise that the VCO tracks. As a result, FMFB receivers allow a threshold extention.
Like the FMFB demodulator, the PLL is also a tracking filter and hence it also provides threshold extension.
WKT the power spectral density of the noise at the receiver output has a square law dependence on the operating frequency as shown in fig below.
Fig 8.20 shows the PSD of message source have spectra of this form. We see that the PSD of the message falls off appreciably at higher frequencies. On the other hand PSD of noise increases rapidly with frequency. Thus at ±W, relative spectral density of message is low, where as that of the output noise is very high. It means, the message is not using the allocated bandwidth efficiently. So, to improve the noise performance by slightly reducing the bandwidth of the low pass filter to reject large amount of noise power while losing only a small amount of message power. But this approach is not satisfactory because the distortion of the message.
A more satisfactory approach is the use of pre-emphasis in the transmitter and de-emphasis in the receiver as shown in fig below.
In this method, we artificially emphasize the high frequency components of the message signal prior modulation in the transmitter before the noise is introduced in the receiver. The low frequency and high frequency portions of the PSD of the message are equalized in such a way that the message fully occupies the frequency band allotted to it. Then, at the discriminator output in the receiver perform the inverse operation by de-emphasizing the high frequency components to restore the message. In this process, the high frequency noise is reduced thereby effectively increasing the output SNR of the system.
To produce an undistorted version of the message at the receiver the filter transfer functions are the inverse of each other.
The average message power is independent of filter transfer functions.
The pre-emphasis filter is selected so that the average power of the emphasized message m1(t) (o/p of pre-emphasis ) has the same average power as of the message m(t). Then,
It is a constraint on the transfer function Hpe(f) of pre-emphasis filter.i.e., the bandwidth of the transmitted signal remains same, with or without pre-emphasis.
WKT, the PSD of the nd(t) at the discriminator output is
The bandwidth of the post detection filter is W and is less than B/2, then, the average power of the modified noise at the receiver output is
Average o/p noise power with de-emphasis 
Because the average message power at the receiver output is unaffected by the pre- emphasis and de-emphasis. The improvement in output SNR due to pre-emphasis and de-emphasis is defined by
Note: Improvement factor assumes high carrier to noise ratio at the discriminator input.
A simple and commonly used pre- emphasis filter that emphasizes high frequencies, whose transfer function is defined by
This is closely realized by an RC – Amplifier network shown in fig below.
R << r and 2πfRC<< 1 inside the frequency band, Amplifier is used to make up attenuation produced by the RC n/w at low frequencies. The frequency parameter
The constant k is chosen to satisfy the constraint, which requires that the average power of the pre-emphasized message signal be the same as the average power of the original message signal.
Assume that the PSD of the original message m(t) is
The output signal to noise ratio of an FM receiver without pre-emphasis and de- emphasis is typically 40 – 50 dB. By using the simple pre-emphasis and de- emphasis filter noise performance will improved.
