CHAPTER 4Noise in Frequency Modulation Systems
FM with Sinusoidal modulating signal
The FM modulated signal in time domain
FM with Sinusoidal modulating signal
Spectra of an FM Signal with Sinusoidal Modulation
BT
cA
fS
2
1
)(
f
1.0
The following spectra show the effect of modulation index, , on the bandwidth of an FM signal, and the relative amplitudes of the carrier and sidebands
Noise in FM Receivers
• w(t): zero mean white Gaussian noise with PSD = No/2
• s(t): carrier = fc, BW = BT
• BPF: [fC - BT/2 - fC + BT/2]
• Amplitude limiter: remove amplitude variation.
• Discriminator» Slope network : varies linearly with frequency» Envelope detector
Noise in FM Receivers
• FM signal:
• Filtered noise n(t):
)]t(tf2cos[A)t(s
dt)t(mk2)t(
]dt)t(mk2tf2cos[A)t(s
CC
t0f
t0fCC
φπ
πφ
ππ
∫
∫
)t(n
)t(ntan)t(
))t(n())t(n(r(t)
where
)]t(tf2r(t)cos[
)tf2sin()t(n)tf2cos()t(n)t(n
I
Q1
2Q
2I
C
CQCI
ψ
ψπ
ππ
Noise in FM Receivers• Phasor diagram interpretation of noisy demodulator input:
• Due to the PM generated by the noise, noise appears at the demodulator output.
• But the FM demodulator is sensitive to the instantaneous frequency of the input signal.
• The instantaneous frequency is the first derivative of the phase of input signal.
Noise in FM Receivers• The instantaneous frequency is the first derivative of the
phase of input signal.
• Derivation in the time domain corresponds to multiplication by (j2πf) in the frequency domain.
• Multiplication by (j2πf) means that the frequency response of derivation is:
• Recall, power spectral density of the output process equals to the PSD of the input process multiplied by the squared magnitude of the frequency response H(f) of the LTI two-port.
Noise in FM Receivers• Recall, power spectral density of the output process equals to
the PSD of the input process multiplied by the squared magnitude of the frequency response H(f) of the LTI two-port.
• Therefore, the PSD SN0(f) of noise at an FM receiver output has
a square-law dependence on the operating frequency.
• The high-frequency noise is dominant at the output of an FM receiver
Noise in FM ReceiversDiscriminator output
dt
)t(dn
πA2
1)t(n
)]}t(sin[)t(r{dt
d
πA2
1
)]}t()t(sin[)t(r{dt
d
πA2
1)t(n
where
)t(n)t(mkdt
d
2
1)t(v
Q
Cd
C
Cd
df
ψ
φψ
θ(t)
π
Noise in FM Receivers
Noise in FM Receivers
Noise in FM Receivers
Noise in FM Receivers
Noise in FM ReceiversFigure of merit for frequency modulation• If the (SNR)C is high enough and exceeds the threshold level
then:
• where P denotes the average power of message signal m(t), kf is the frequency sensitivity of FM modulator and W denotes the bandwidth of message signal.
• Since:
• and for wide-band FM we may write:
Noise in FM ReceiversFigure of merit for frequency modulation
• we obtain by substituting:
where β is the Modulation Index.
• Note: An increase in the transmission bandwidth BT provides a corresponding quadratic increase in the output signal-to-noise ratio (or in the figure of merit) of the FM system.
FM threshold effect
• The figure of merit discussed above is valid only if the carrier-to-noise ratio (SNR)C is high compared with unity.
• It has been found experimentally that as (SNR)C is decreased below a threshold, each FM demodulator, either coherent or non-coherent, breaks:
– At first isolated clicks are heard and if the (SNR)C is decreased further, the clicks rapidly merge into a crackling. sound
FM threshold effectA qualitative explanation
• If (SNR)C is small then the noise becomes dominant and the phasor representation and the decomposition of noise into a PM and AM are not valid any more.
• The phase of noise is a random variable and it may take any value.
• Recall, the FM demodulator is sensitive to the derivate of phase.
• When the phase of demodulator input varies suddenly by 2π due to the noise then an impulse, i.e., click appears at the receiver output.
Pre-emphasis and de-emphasis in FM systems
• Recall: The power spectral density SN0(f) of noise at an FM receiver
output has a square law dependence on the operating frequency.
• The high-frequency noise is dominant at the output of an FM receiver.
• The power spectral density of message signals usually falls off at higher frequencies.
• Generally, the most part of a message signal is in the low-frequency region.
• These facts may be exploited to improve the noise performance of FM systems
Pre-emphasis and de-emphasis in FM systems
• Basic idea – Apply a filter at the demodulator output which reduces the high
frequency content of the output spectrum. – To compensate this attenuation, a pre-emphasis must be
applied to the high-frequency signals at the transmitter
• Pre-emphasis at the transmitter:– A filter that artificially emphasize the high-frequency
components of the message signal prior to the modulation.
Pre-emphasis and de-emphasis in FM systems
• De-emphasis at the receiver:– An inverse operation performed by a filter placed after the
demodulation. – The de-emphasis filter restores the original signal by de-
emphasizing the high-frequency components.
• Effects of pre-emphasis and de-emphasis filters cancel each other:
Use of pre-emphasis and de-emphasis in an FM system
Comparison of noise performance of CW systems
• Note: Threshold problem is more serious in FM modulation than in AM. The higher the β, the better the FM noise performance. But the price to be paid is the wider transmission bandwidth
