1. DEMODULATION OF FM SIGNALS
Frequency demodulation involves a frequency discriminator,
whose instantaneous output amplitude is directly proportional to
the instantaneous frequency of the input FM signal.
The frequency discriminator consists of a slope circuit followed
by an envelope detector.
A slope circuit is characterized by an imaginary frequency
response, varying linearly with frequency in a frequency interval.
Consider the frequency response depicted in Figure 2.29a,
where a is a constant.

2. signal s(t) of carrier frequency fc and transmission bandwidth BT.
The response s1(t) of the slope circuit is produced by FM
signal s(t) of carrier frequency fc and transmission bandwidth BT.
The spectrum of s(t) is essentially zero outside the frequency
interval fc - BT/2 < | f | < fc＋BT/2.
We may replace the BPF with frequency response H1(f) with
equivalent LPF with frequency response H1(f) by doing two things:
1). Shifting H1(f) to the right by fc, where fc is the mid-band
frequency of the BPF;
this operation aligns the translated frequency response
of the equivalent LPF with that of the BPF.
2). Seting H1(f)(f - fc) equal to 2H1(f) for f > 0.
Thus for the problem at hand we get

3. Hence, using Eqs. (2.60) and (2.61), we get
which is plotted in Figure 1.19b.
The incoming FM signal s(t) is defined by Equ. (2.26) :
Given that the carrier frequency fc is high compared to
the transmission bandwidth of the FM signal s(t), the complex
envelope of s(t) is

4. the slope circuit defined by Figure 2.29a due to s(t).
Let s1(t) denote the complex envelope of the response of
the slope circuit defined by Figure 2.29a due to s(t).
The Fourier transform of s1(t) can be expressed as:
where S(f) is the Fourier transform of s(t).

5. is equivalent to differentiating the signal in the time domain.
Multiplication of the Fourier transform of a signal by j2pf
is equivalent to differentiating the signal in the time domain.
From Equ. (2.64) we deduce
Substituting Equ. (2.63) into (2.65), we get
The desired response of the slope circuit is therefore

6. Figure 2. 29 (a) Frequency response of ideal slope circuit
Figure 2.29 (a) Frequency response of ideal slope circuit. (b) Frequency response of the slope circuit’s complex low-pass equivalent. (c) Frequency response of the ideal slope circuit complementary to that of part (a).

7. Provided that we choose
we may use an envelope detector to recover the amplitude variations and thus obtain the original message signal.
The resulting envelope-detector output is therefore
The bias term pBTaAc in Equ. (2.68) may be subtracted from the envelope-detector output |~s1(t)|, the output of a 2nd envelope detector preceded by the complementary slope circuit with a frequency response H2(f) as described in Figure 2.29c. That is, the two slope circuits are related by

8. produced by the incoming FM signal s(t). Following similar
Denote s2(t) the response of complementary slope circuit
produced by the incoming FM signal s(t). Following similar
procedure, we may write
where ~s2(t) is the complex envelope of s2(t).
Difference between envelopes in Eqs. (2.68) and (2.70) is
a scaled version of message signal m(t) and free from bias.
Model the frequency discriminator as pair of slope circuits
with complex transfer functions related by Equ. (2.69),
followed by envelope detectors and a summer, as in Figure 2.30.
This scheme is called a balanced frequency discriminator.

9. Figure 2.30 Block diagram of frequency discriminator.

10. 1). Transmission has to operate within the allocated FM channels.
FM STEREO MULTIPLEXING
Specifications for FM stereo transmission is influenced by :
1). Transmission has to operate within the allocated FM channels.
2). It has to be compatible with monophonic radio receivers.
Figure 2.31a shows the block diagram of the multiplexing system
used in an FM stereo transmitter.
Let mL(t) and mR(t) denote the signals picked up by left-hand and
right-hand microphones at the transmitting end of the system.
They are applied to a simple matrixer that generates the sum signal,
mL(t) + mR(t), and the difference signal, mL(t) - mR(t).
The sum signal is left in its baseband form; it is available for
monophonic reception. The difference signal and a 38-kHz subcarrier
(derived from a 19-kHz crystal oscillator by frequency doubling) are
applied to a product modulator, thereby producing a DSB-SC wave.

11. where fc = 19 kHz, and K is the amplitude of the pilot tone.
The multiplexed signal m(t) also includes a 19-kHz pilot to provide reference for coherent detection of the difference signal at the stereo receiver.
The multiplexed signal is described by
where fc = 19 kHz, and K is the amplitude of the pilot tone.
This multiplexed signal m(t) then frequency-modulates the main carrier to produce the transmitted signal.

12. Figure 2. 31 (a) Multiplexer in FM stereo
Figure 2.31 (a) Multiplexer in FM stereo. (b) Demultiplexer in FM stereo.

13. At a stereo receiver, the multiplexed signal m(t) is recovered by frequency demodulating the incoming FM wave.
Then m(t) is applied to the demultiplexing system shown in Figure 2.31b.
The individual components of the multiplexed signal m(t) are separated by three appropriate filters.

14. The recovered pilot (using a narrowband filter tuned to 19-kHz) is frequency doubled to produce the desired 38-kHz subcarrier.
This subcarrier enables the coherent detection of the DSB-SC modulated wave, thereby recovering the difference signal, mL(t) - mR(t).
The baseband LPF in the top path of Figure 2.31b is designed to pass the sum signal, mL(t) + mR(t).
Finally, the simple matrixer reconstructs the left-hand signal mL(t) and right-hand signal mR(t), and applies them to their respective speakers.

15. Amplification, to compensate for the loss of signal power.
2.8 Superheterodywe Receiver
Besides demodulating, the receiver also performing :
Carrier tuning, to select the desired signal (desired radio or TV station).
Filtering, to separate the desired signal from other modulated signals.
Amplification, to compensate for the loss of signal power.
The receiver consists of an RF section, a mixer and LO,
an IF section, demodulator, and power amplifier.
Typical parameters of commercial AM and FM radio receivers
are listed in Table 2.3.
Figure 2.32 shows a super-heterodyne receiver for AM using
envelope detector for demodulation.
The combination of mixer and LO provides heterodyning,
whereby the incoming signal is converted to a predetermined IF,
usually lower than the incoming carrier frequency.

16. where fLO is the LO frequency and fRF is the RF carrier frequency.
The heterodyning is to produce an IF carrier defined by
fIF = fLO – fRF (2.78)
where fLO is the LO frequency and fRF is the RF carrier frequency.
The IF section consists of one or more stages of tuned amplification, with a bandwidth corresponding to that required
for the particular type of modulation that the receiver is intended
to handle.
The IF section provides most of the amplification and
selectivity in the receiver. The output of the IF section is applied
to a demodulator to recover the baseband signal.
If coherent detection is used, then a coherent signal source
must be provided in the receiver.

17. will result in fIF at the mixer output.
In superheterodyne receiver, input frequencies | fLO ± fIF |
will result in fIF at the mixer output.
This introduces possible simultaneous reception of two signals
differing in frequency by twice the fIF.
For example, a receiver tuned to 650 kHz and having
fIF = 455 kHz is subject to an image interference at 1.56 MHz;
any receiver with this fIF, is subject to image interference
at a frequency of 910 kHz higher than the desired station.
The mixer is incapable of distinguishing between the
desired signal and its image in that it produces an IF output
from either one of them.
The practical cure for image interference is to employ highly
selective stages in the RF section to favor the desired signal and
discriminate against the image signal.

18. receivers lies in the use of an FM demodulator such as limiter-
The basic difference between AM and FM superheterodyne
receivers lies in the use of an FM demodulator such as limiter-
frequency discriminator.
In an FM system, the message information is transmitted by
variations of the instantaneous frequency of a sinusoidal carrier,
and its amplitude is maintained constant.
An amplitude limiter, following the IF section, is used to
remove amplitude variations by clipping the modulated wave
at the IF section output.
The resulting rectangular wave is rounded off by a BPF that
suppresses harmonics of the carrier frequency.
Thus the filter output is again sinusoidal, with an amplitude
that is independent of the carrier amplitude at the receiver input.

19. Figure 2.32 Basic elements of an AM radio receiver of the superheterodyne type.

20. 2.9 Noise in CW Modulation Systems
1). Channel model, which assumes a communication channel that
is distortionless but perturbed by additive white Gaussian noise
(AWGN).
2). Receiver model, which assumes a receiver consisting of an
ideal BPF followed by an ideal demodulator; the BPF is used
to minimize the effect of channel noise.
Figure 2.33 shows the noisy receiver model that combines
the above two assumptions.
In this figure, s(t) denotes the incoming modulated signal
and w(t) denotes the channel noise.
The BPF in Figure 2.33 represents the combined filtering
of the tuned amplifiers used in the actual receiver for the signal
amplification prior to demodulation.

21. Figure 2.33 Receiver model.

22. N0 being the average noise power per unit bandwidth of the receiver.
SIGNAL-TO-NOISE RATIOS: BASIC DEFINITIONS
Let the power spectral density (psd) of the noise w(t) be N0/2,
N0 being the average noise power per unit bandwidth of the receiver.
the receiver BPF in Figure 2.33.
For DSB-SC, AM, and FM, with midband frequency of
the receiver BPF in Figure 2.33 to be the carrier frequency fc,
we may model psd SN(f) of noise n(t), resulting from white noise w(t)
through the filter, as shown in Figure 2.34.
Typically, fc >> BT , the filtered noise n(t) can be represented
as a narrowband noise in the canonical form
n(t) = nI(t)cos(2pfct) – nQ(t)sin(2pfct) (2.79)
nI(t) : the in-phase noise component,
nQ(t) : the quadrature noise component,
measured with respect to the carrier wave Accos(2pfct).

23. The details of s(t) depend on the type of modulation used.
The filtered signal x(t) available for demodulation is
x(t) = s(t) + n(t) (2.80)
The details of s(t) depend on the type of modulation used.
The average noise power at the demodulator input is equal
to the total area under the curve of the psd SN(f).
From Figure 2.34, this average noise power is equal to N0BT.
With the demodulated signal s(t) and the filtered noise n(t)
appearing additively at the demodulator input (Equ. (2.80)),
we may define (SNR)I as the ratio of the average power of the
modulated signal s(t) to the average power of the filtered noise n(t).
A useful measure of noise performance is the (SNR)O ,
the ratio of the average power of the demodulated message signal to
the average power of the noise, both measured at the receiver output.

24. Figure 2.34 Idealized characteristic of band-pass filtered noise.

25. frequency discrimination as in FM, the average power of the
When the receiver uses envelope detection as in AM or
frequency discrimination as in FM, the average power of the
filtered noise n(t) is relatively low to justify the output SNR
as a measure of receiver performance.
For output SNRC comparison of different modulation-
demodulation systems, it must be made on an equal basis:
The modulated signal s(t) transmitted by each system has
the same average power.
The channel noise w(t) has the same average power measured
in the message bandwidth W.

26. as the ratio of the average power of the modulated signal to
As a frame of reference we define the channel (SNR)C
as the ratio of the average power of the modulated signal to
the average power of channel noise in the message bandwidth,
both measured at the receiver input. This definition is
illustrated in Figure 2.35.
We define a figure of merit for the receiver as follows:
Figure of merit = (SNR)O/(SNR)C (2.81)
The higher the figure of merit, the better the noise
performance of the receiver.
The figure of merit may = 1, < 1, or > 1, depending on
the type of modulation used.

27. Figure 2.35 The baseband transmission model, assuming a message bandwidth W for calculating the channel SNR.

28. 2.10 Noise in Linear Receivers Using Coherent Detection
AM demodulation depends on whether the carrier is
suppressed or not. When the carrier is suppressed we require
the use of coherent detection, in which case the receiver is linear.
When AM includes transmission of the carrier, demodulation
is accomplished by using an envelope detector, in which case
the receiver is nonlinear.
Figure 2.36 shows the model of a DSB-SC receiver by using
a coherent detector. Coherent detection requires multiplication of
the filtered signal x(t) by a locally generated cos(2pfct) and then
low-pass filtering the product.
The DSB-SC component of the filtered signal x(t) is
s(t) = CAccos(2pfct) m(t) (2.82)
where Accos(2pfct) is the carrier wave and m(t) is the message signal.

29. of a zero mean stationary process, whose psd SM(f) is limited to
In Equ. (2.82), we assume that m(t) is the sample function
of a zero mean stationary process, whose psd SM(f) is limited to
the message bandwidth W.
The average power P of the message signal is the total area
under the curve of the psd, as shown by
The carrier wave is statistically independent of the message signal.
The average power of the DSB-SC modulated signal s(t)
may be expressed as C2A2P/2.
With a noise spectral density of No/2, the average noise power
in the message bandwidth W is equal to WNo.
The channel SNR of the DSB-SC system is therefore
(SNR)C,DSB = C2Ac2P/2WNo (2.84)

30. the total signal at the coherent detector input may be expressed as
Using the narrowband representation of the filtered noise n(t),
the total signal at the coherent detector input may be expressed as
where nI(t) and nQ(t) are the in-phase and quadrature components
of n(t) with respect to the carrier.
The output of the product-modulator component of the coherent detector is
The LPF in the coherent detector in Figure 2.36 removes the
high-frequency components of v(t), yielding the receiver output

31. 1). The message signal m(t) and in-phase noise component nI(t) of
Equ. (2.86) 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.
In coherent detection, the message signal component
at the receiver output is CAcm(t)/2. The average power of
this component may be expressed as C2Ac2P/4,
where P is the average power of the original message signal m(t)
and C is the system-dependent scaling factor.
In DSB-SC modulation, the BPF in Figure 2.36 has bandwidth
BT = 2W to accommodate the sidebands of the modulated signal s(t).
The average power of the filtered noise n(t) is therefore 2WNo.
The average power of the in-phase noise component nI(t) is
the same as that of the filtered noise n(t).

32. Note that the factor C2 is common to both the output and channel SNRC,
From Equ. (2.86) the noise component at the receiver output is nI(t)/2, it follows that the average power of the noise at the receiver output is
(1/2)2 2WNo = (1/2) WNo
The output SNR for a DSB-SC receiver using coherent detection
is therefore
Using Eqs. (2.84) and (2.87), we obtain the figure of merit
[(SNR)O/(SNR)C]DSB-SC = (2.88)
Note that the factor C2 is common to both the output and channel SNRC,
and therefore cancels out in evaluating the figure of merit.

33. we find that the figure of merit is exactly the same with DSC-SC.
Following the noise analysis of a coherent detector for SSB,
we find that the figure of merit is exactly the same with DSC-SC.
The important conclusions are two-fold:
1). For the same average signal power and average noise power in
the message bandwidth, coherent SSB receiver will have the same
output SNR as coherent DSB-SC receiver.
2). In both cases, the noise performance of the receiver is exactly
the same as that obtained by simply transmitting the message signal
in the presence of the same channel noise.
The only effect of the modulation process is to translate the
message to a different frequency band to facilitate its transmission
over a band-pass channel.
Neither DSB-SC nor SSB modulation offers the means for
trade-off between improved noise performance and increased
channel bandwidth.

34. Figure 2.36 Model of DSB-SC receiver using coherent detection.

35. 2.11 Noise in AM Receivers Using Envelope Detection
An AM system using envelope detector is shown in Figure 2.37.
In an AM signal, both sidebands and the carrier are transmitted,
s(t) = Ac[1 + kam(t)] cos(2pfct) (2.89)
where Accos(2pfct) is the carrier wave, m(t) is the message signal,
and ka is a constant that determines the percentage modulation.
The average power of the carrier component in the AM
signal s(t) is Ac2/2.
The average power of the information-bearing component
Ackam(t)cos(2pfct) is Ac2ka2P/2, where P is the average power of
the message signal m(t).
The average power of the full AM signal s(t) is therefore
equal to Ac2(l + ka2P)/2.
The average noise power in the message bandwidth is WN0.

36. Figure 2.37 Model of AM receiver.

37. (SNR)C,AM = Ac2(1 + ka2P)/2WNo (2.90)
The channel SNR for AM is therefore
(SNR)C,AM = Ac2(1 + ka2P)/2WNo (2.90)
To evaluate the output SNR, we represent the filtered noise n(t)
in terms of its in-phase and quadrature components.
Define the filtered signal x(t) applied to the envelope detector
in the receiver model of Figure 2.37 as follows:
x(t) = s(t) + n(t)
= [Ac + Ackam(t) + nI(t)] cos(2pfct) – nQ(t) sin(2pfct) (2.91)
From the phasor diagram in Figure 2.38a, the receiver output
is obtained as
y(t) = envelope of x(t)
= {[ Ac + Ackam(t) + nI(t)]2 + nQ2(t)}1/ (2.92)
The signal y(t) defines the output of an ideal envelope detector.

38. Figure 2.38 (a) Phasor diagram for AM wave plus narrowband noise for the case of high CNR. (b) Phasor diagram for AM wave plus narrowband noise for the case of low CNR.

39. average noise power, the signal term Ac[1+kam(t)] will be large
When the average carrier power is large compared with the
average noise power, the signal term Ac[1+kam(t)] will be large
compared with the noise terms nI(t) and nQ(t).
Then we may approximate the output y(t) as :
y(t)〜Ac + Ackam(t) + nI(t) (2.93)
The presence of the constant term Ac in the envelope detector
output y(t) of Equ. (2.93) is due to demodulation of the
transmitted carrier wave.
By neglecting the DC term Ac in Equ. (2.93), the remainder
has a form similar to the output of a DSB-SC receiver using
coherent detection.
Accordingly, the output SNR of an AM receiver using an
envelope detector is approximately
(SNR)O,AM = (Ac2ka2P)/2WNo (2.94)

40. 1). The average noise power is small compared to the average
Equ. (2.94) is valid only if the following are satisfied:
1). The average noise power is small compared to the average
carrier power at the envelope detector input.
2). The amplitude sensitivity ka is adjusted for a percentage
modulation < 100 %.
Using Eqs. (2.90) and (2.94), the figure of merit for AM is
[(SNR)O/(SNR)C]AM = ka2P/(1+ka2P) (2.95)
whereas the figure of merit for a DSB receiver or an SSB receiver
using coherent detection is always = 1, the corresponding figure
of merit for AM receiver using envelope detection is always < 1.
In other words, the noise performance of a full AM receiver is
always inferior to that of a DSB-SC receiver.

41. dominates. In this case the narrowband noise n(t) can be represented
THRESHOLD EFFECT
When CNR is small compared with unity, the noise term
dominates. In this case the narrowband noise n(t) can be represented
in terms of its envelope r(t) and phase y(t), as shown by
n(t) = r(t) cos[2pfct + y(t)] (2.97)
The corresponding phasor diagram for the detector input
x(t) = s(t) + n(t) is shown in Figure 2.38b.
To the noise phasor r(t) we have added a signal phasor
Ac[1+ kam(t)], with the angle between them being the phase y(t)
of the noise n(t).
In Figure 2.38b, the CNR is so low that the carrier
amplitude Ac is small compared with the noise envelope r(t),
at least most of the time.

42. we find from Figure 2.38b that the envelope detector output is
Neglect the quadrature component with respect to the noise,
we find from Figure 2.38b that the envelope detector output is
y(t) 〜 r(t) + Accos[y(t)] + Ackam(t) cos[y(t)] (2.98)
This reveals that when the CNR is low, the detector output
has no component strictly proportional to the message signal m(t).
It follows that we have a complete loss of information in that
the detector output does not contain the message signal m(t) at all.
The loss of a message in an envelope detector that operates at a
low CNR is referred to as the threshold effect.
By threshold we mean a value of the CNR below which
the noise performance of a detector deteriorates much more rapidly
than proportionately to the CNR.
Every nonlinear detector (e.g., envelope detector) exhibits a
threshold effect. Such an effect does not arise in a coherent detector.

43. 2.12 Noise in FM Receivers Gaussian noise of zero mean and psd N0/2.
In Figure 2.40, the channel noise w(t) is modeled as white
Gaussian noise of zero mean and psd N0/2.
The received FM signal s(t) has carrier frequency fc and
transmission bandwidth BT.
The BPF has a midband frequency fc and bandwidth BT and
therefore passes the FM signal essentially without distortion.
BT is small compared with fc, so that we may use the narrowband
representation for n(t), the filtered version of w(t),
in terms of its in-phase and quadrature components.
The amplitude limiter, following the BPF in the receiver of
Figure 2.40, is used to remove amplitude variations by clipping
the modulated wave at the filter output.
The resulting rectangular wave is rounded off by another BPF,
thereby suppressing harmonics of the carrier frequency.

44. 1). A slope network or differentiator with a purely imaginary
The discriminator in Figure 2.40 consists of two components:
1). A slope network or differentiator with a purely imaginary
frequency response 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 message signal.
The post-detection filter, labeled "baseband low-pass filter"
in Figure 2.40, has a bandwidth that is large enough to
accommodate the highest frequency component of the
message signal.
This filter removes the out-of-band components of the noise
at the discriminator output and thereby keeps the output noise
to a minimum.

45. Figure 2.40 Model of an FM receiver.

46. defined in terms of its in-phase and quadrature components by
The filtered noise n(t) at the BPF output in Figure 2.40 is
defined in terms of its in-phase and quadrature components by
n(t) = nI(t) cos(2pfct) – nQ(t) sin(2pfct)
Equivalently, in terms of envelope and phase,
n(t) = r(t) cos[2pfct + y(t)] (2.130)
where the envelope is
r(t) = [nI2(t) + nQ2(t)]1/ (2.131)
and the phase is
y(t) = tan-1 [nQ(t)/nI(t)] (2.132)
The envelope r(t) is Rayleigh distributed, and the phase
y(t) is uniformly distributed.

47. where Ac is the carrier amplitude, fc is the carrier frequency,
The incoming FM signal s(t) is defined by
where Ac is the carrier amplitude, fc is the carrier frequency,
kf is the frequency sensitivity, and m(t) is the message signal.
To proceed, we define
We may thus express s(t) in the simple form
s(t) = Ac cos[2pfct + (t)] (2.135)

48. = Accos[2pfct + (t)] + r(t)cos[2pfct + (t)] (2.136)
The noisy signal at the BPF output is therefore
x(t) = s(t) + n(t)
= Accos[2pfct + (t)] + r(t)cos[2pfct + (t)] (2.136)
It is informative to represent x(t) by means of a phasor diagram,
as in Figure 2.41.
The phase (t) of the resultant phasor x(t) is obtained
from Figure 2.41 as
With an ideal discriminator, its output is proportional
to '(t)/2p.

49. Figure 2.41 Phasor diagram for FM wave plus narrowband noise for the case of high carrier-to-noise ratio.

50. Let R denote the random variable obtained by observing
Assume that CNR > 1 at the discriminator input.
Let R denote the random variable obtained by observing
the envelope process with sample function r(t) [due to
the noise n(t)].
At least most of the time, the random variable R
is small compared with the carrier amplitude Ac,
and the phase (t) simplifies as follows:
or, using the expression for (t) given in Equ. (2.134),

51. The discriminator output is therefore
where the noise term nd(t) is defined by
Provided the CNR is high, the discriminator output v(t) consists of the original message m(t) multiplied by the
constant factor kf, plus an additive noise nd(t).
Accordingly, we may use the output SNR to assess
the quality of performance of the FM receiver.

52. Then we may simplify Equ. (2.141) as :
From the phasor diagram of Figure 2.41, the effect of variations in the phase (t) of the narrowband noise appear referred to the signal term (t).
We know that the phase (t) is uniformly distributed over 2p radians. The phase difference (t)-(t) is also uniformly distributed over 2p-radians.
The noise nd(t) at the discriminator output would be independent of the modulating signal and would depend only on the characteristics of the carrier and narrowband noise.
Then we may simplify Equ. (2.141) as :

53. Therefore, we may rewrite Equ. (2.142) as
From the defining equations for r(t) and (t), the quadrature component nQ(t) of the filtered noise n(t) is
nQ(t) = r(t) sin[(t)] (2.143)
Therefore, we may rewrite Equ. (2.142) as
The additive noise nd(t) at the discriminator output is determined by the carrier amplitude Ac and the quadrature component nQ(t) of the narrowband noise n(t).

54. From Equ. (2.140), the message component in the discriminator output, and the LPF output, is kfm(t).
Hence, the average output signal power is kf2P, where P is the average power of the message signal m(t).

55. The noise nd(t) at the discriminator output is proportional to the time derivative of the quadrature noise component nQ(t).
It follows that we may obtain the noise process nd(t) by passing nQ(t) through a linear filter with a frequency response equal to
This means that the psd SNd(f) of the noise nd(t) is related to the psd SNo(f) of the quadrature noise component nQ(t) as follows:

56. The corresponding psd of the noise nd(t) (Figure 2.42b) is
With the receiver BPF in Figure 2.40 having an ideal frequency response characterized by bandwidth BT and midband frequency fc, the narrowband noise n(t) will have a similar psd characteristic.
The quadrature component nQ(t) of the narrowband noise n(t) has the ideal low-pass characteristic shown in Figure 2.42a.
The corresponding psd of the noise nd(t) (Figure 2.42b) is

57. Figure 2. 42 Noise analysis of FM receiver. (a)
Figure 2.42 Noise analysis of FM receiver. (a). PSD of quadrature component nQ(t) (b). PSD of noise nd(t) at the discriminator output. (c). PSD of noise no(t) at the receiver output.

58. In the receiver model of Figure 2
In the receiver model of Figure 2.40, the discriminator output is followed by a LPF band-limited to the message bandwidth W.
For WBFM, W < BT/2, where BT is the FM transmission bandwidth, means that out-of-band components of noise nd(t) will be rejected.
The psd SNo(f) of the noise no(t) at the receiver output is
as shown in Figure 2.42c.

59. The average output noise power is determined by integrating SNo(f) from -W to W:
The average output noise power is inversely
proportional to the average carrier power Ac2/2.
That is, in an FM system, increasing the carrier power
has a noise-quieting effect.
Provided the CNR is high, we may divide the average
output signal power kf2P by the average output noise power
of Equ. (2.148) to obtain the output SNR:
(SNR)O,FM = (3Ac2kf2P)/(2NoW3) (2.149)

60. Dividing the output SNR by the channel SNR,
The average power in the modulated signal s(t) is Ac2/2, and the average noise power in the message bandwidth is WNo.
Thus the channel SNR is
(SNR)C,FM = Ac2/2WNo (2.150)
Dividing the output SNR by the channel SNR,
we get the figure of merit for FM:
[(SNR)O/(SNR)C]FM = 3kf2P/W (2.151)

61. That is, the deviation ratio D is proportional to the ratio kfP1/2/W.
The frequency deviation f is proportional to the frequency sensitivity kf of the modulator.
Also, the deviation ratio D is equal to the frequency deviation f divided by the message bandwidth W.
That is, the deviation ratio D is proportional to the ratio kfP1/2/W.
It follows from Equ. (2.151) that the figure of merit of a WBFM is a quadratic function of the deviation ratio.
In WBFM, the transmission bandwidth BT is approximately proportional to the deviation ratio D.

62. When the CNR is high, an increase in the transmission bandwidth BT will provide a quadratic increase in the output SNR or figure of merit of the FM system.
Unlike AM, FM provides a practical mechanism for the exchange of increased transmission bandwidth for improved noise performance.

63. FM THRESHOLD EFFECT
Equ. (2.149), defining the output SNR of an FM receiver, is valid only if the CNR at the discriminator input is > 1.
As input noise power is increased so that CNR is decreased, the FM receiver breaks.
The threshold is defined as the minimum CNR yielding an FM improvement that is not significantly deteriorated from the value predicted by the usual SNR assuming a small noise power.

64. Then the composite signal at the frequency discriminator input is
For a qualitative discussion of the FM threshold effect, consider the case when there is a no signal present, so that the carrier wave is unmodulated.
Then the composite signal at the frequency discriminator input is
x(t) = [Ac+nI(t)]cos(2pfct) – nQ(t)sin(2pfct) (2.153)
where nI(t) and nQ(t) are the in-phase and quadrature components of the narrowband noise n(t) with respect to the carrier wave.

65. The phasor diagram of Figure 2
The phasor diagram of Figure 2.43 displays the phase relations between the various components of x(t) in Equ. (2.153).
As the amplitudes and phases of nI(t) and nQ(t) change in a random manner, the point P1 [the tip of the phasor representing x(t)] wanders around the point P2 (the tip of the phasor representing the carrier).
When CNR is large, nI(t) & nQ(t) << carrier amplitude Ac, and so the wandering point P1 in Figure 2.43 spends most of its time near point P2.
Thus the angle (t) is approximately nQ(t)/Ac to within a multiple of 2p.
When the CNR is low, the wandering point P1 occasionally sweeps around the origin and (t) increases or decreases by 2p radians.

66. Figure 2.43 Phasor diagram interpretation of Equation (2.153).

67. Figure 2.44 illustrates how the excursions in (t), depicted in Figure 2.44a, produce impulse-like components in '(t) = d/dt.
The discriminator output v(t) is equal to '(t)/2p.
When the signal shown in Figure 2.44b is passed through the post-detection LPF, corresponding but wider impulse-like components are excited in the receiver output.
The clicks are produced only when (t) changes by ±2p radians.

68. Figure 2.44 Illustrating impulse-like components in (t)  dq(t)/dt produced by changes of (t); (a) and (b) are graphs of (t) and (t), respectively.

69. In the Figure 2.43, a positive-going click occurs when the envelope r(t) and phase (t) of the narrowband noise n(t) satisfy the conditions:
The phase (t) of the resultant phasor x(t) changes by 2p in the increment dt, during which the phase of the narrowband noise increases by d(t).

70. These conditions ensure that (t) changes by –2p during
Similarly, the conditions for a negative-going click to occur are as follows:
These conditions ensure that (t) changes by –2p during
the time increment dt.
The CNR is defined by
r = Ac2/(2BTN0) (2.154)
As r is decreased, the average number of clicks per unit time increases. When this number becomes appreciably large, threshold occurs.

71. The output SNR is calculated as follows:
1). In the absence of noise, the average output
signal power is calculated assuming a
sinusoidal modulation that produces a
frequency deviation f = BT/2, so that the
carrier swings back and forth across the
input frequency band.
2). The average output noise power is calculated
when the carrier is unmodulated, with no
restriction on the value r of the CNR.

72. Curve I of Figure 2.45 presents a plot of the output SNR versus the CNR when the ratio BT/2W is equal to 5.
This curve shows that the output SNR deviates appreciably from a linear function of the CNR when r is less than about 10 dB.
Curve II of Figure 2.45 shows the effect of modulation on the output SNR when the modulating signal and the noise are present at the same time.
The average output signal power pertaining to curve II may be taken to be effectively the same as for curve I.

73. Figure 2. 45 Dependence of output SNR on input CNR for FM receiver
Figure 2.45 Dependence of output SNR on input CNR for FM receiver. In curve I, the average output noise power is calculated assuming an unmodulated carrier. In curve II, the average output noise power is calculated assuming a sinusoidally modulated carrier.

74. As  decreases from infinity, the output SNR deviates from a linear function of  when  is about 11 dB.
When the signal is present, the resulting modulation of the carrier tends to increase the average number of clicks per second.
Threshold effects in FM receivers may be avoided if the CNR is  > 20 or, equivalently, 13 dB.
Using Equ. (2.154), the message loss at the discriminator output is negligible if
or, equivalently, if the average transmitted power Ac2/2 satisfies the condition

75. 1). For a specified modulation index b and message
To use this formula, we may proceed as follows:
1). For a specified modulation index b and message
bandwidth W, determine the transmission bandwidth BT
of the FM wave, using the universal curve of Figure 2.26
or Carson's rule.
2). For a specified average noise power per unit bandwidth,
N0, use Equ. (2.155) to determine the minimum value of
the average transmitted power Ac2/2 operating above
threshold.

76. FM THRESHOLD REDUCTION
Threshold reduction in FM receivers may be achieved by using an FM demodulator with negative feedback (FMFB demodulator), or by using a PLL demodulator.
Such devices are referred to as extended-threshold demodulators, the idea of which is illustrated in Figure 2.46.
The block diagram of an FMFB demodulator is shown in Figure Assume a WBFM signal is applied to the receiver input, and a second FM signal, with smaller modulation index, is applied to the VCO.
The output of the mixer would consist of the difference frequency, because the sum frequency is removed by the BPF. The frequency deviation of the mixer output would be small, since the difference between their instantaneous deviations is small.
The modulation indices would subtract and the resulting FM wave at the mixer output would have a smaller modulation index.

77. Figure 2.46 FM threshold extension.

78. Figure 2.47 FM demodulator with negative feedback.

79. by feeding the output of the frequency discriminator back to the VCO.
The second WBFM signal applied to the mixer may be obtained
by feeding the output of the frequency discriminator back to the VCO.
The SNR of an FMFB receiver is the same as that of a conventional
FM receiver with the same input signal and noise power if the CNR
is sufficiently large.
In the combined carrier Accos(2pfct) and narrowband noise
n(t) = nI(t) cos(2pfct) – nQ(t) sin(2pfct)
the phase of the composite signal x(t) at the limiter-discriminator input
is approximately equal to nQ(t)/Ac, assuming that the CNR is high.
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.

80. reduces the PM index of the wave, the quadrature component nQ(t)
When feedback is applied, the VCO generates an FM signal that
reduces the PM index of the wave, the quadrature component nQ(t)
of noise, in the BPF output.
As long as the CNR is sufficiently large, the FMFB receiver
does not respond to the in-phase noise nI(t),
but it would demodulate the quadrature noise nQ(t) in exactly
the same fashion as it would demodulate message signal.
For large CNRs, the baseband SNR of an FMFB receiver is
then the same as that of a conventional FM receiver.
An FMFB demodulator is essentially a tracking filter that
can track only the slowly varying frequency of a WBFM signal,
it responds only to a narrowband of noise centered about
the instantaneous carrier frequency.
The noise bandwidth to which the FMFB receiver responds
is precisely the band of noise that the VCO tracks.

81. dependence on the operating frequency; as illustrated in Figure 2.48a.
PRE-EMPHASIS AND DE-EMPHASIS IN FM
The noise psd at the output of an FM receiver has a square-law
dependence on the operating frequency; as illustrated in Figure 2.48a.
In Figure 2.48b, the message psd falls off appreciably at higher
frequencies, but the noise psd increases rapidly with frequency.
Around f = ± W, the relative spectral density of the message
is quite low, whereas that of the output noise is quite high.
Based on the pre-emphasis and de-emphasis (Figure 2.49),
the high-frequency components of the message signal is emphasized
prior to modulation, before the noise is introduced in the receiver.
At the discriminator output, the inverse operation by de-
emphasizing the high-frequency components will restore
the original signal-power distribution of the message.
In such a process, the high-frequency components of the noise
at the discriminator output are also reduced, thereby effectively
increasing the output SNR of the system.

82. Figure 2. 48 (a) Power spectral density of noise at FM receiver output
Figure 2.48 (a) Power spectral density of noise at FM receiver output. (b) Power spectral density of a typical message signal.

83. Figure 2.49 Use of pre-emphasis and de-emphasis in an FM system.

84. in the receiver must have frequency responses that are the inverse of
The pre-emphasis filter in the transmitter and the de-emphasis filter
in the receiver must have frequency responses that are the inverse of
each other:
Hde(f) = 1 / Hpe(f) , －W < f < W (2.156)
From the noise analysis in FM systems, assuming a high CNR,
the psd nd(t) of the noise at the discriminator output is given by
Equ. (2.146).
The modified psd of the noise at the de-emphasis filter output
is therefore
The post-detection LPF has a bandwidth W < BT/2, the average
power of the modified noise at the receiver output is

85. by the combined pre-emphasis and de-emphasis procedure,
The average message power at the receiver output is unaffected
by the combined pre-emphasis and de-emphasis procedure,
the improvement in output SNR by the pre-emphasis and de-emphasis
is defined by
The average output noise power without pre-emphasis and
de-emphasis is (2N0W3/3Ac2), see Equ. (2.148).
We may therefore express the improvement factor I as
This improvement factor assumes a high CNR at the discriminator input in the receiver.