Noise in FM Systems

Frequency modulation offers a remarkable ability to trade bandwidth for improved noise performance, a property not available in amplitude modulation systems. This exchange is captured quantitatively by the figure of merit for FM, which can exceed unity and grow with the square of the modulation index. However, this advantage is conditional on the input SNR being above a critical threshold, below which FM performance collapses dramatically. Understanding both the noise improvement mechanism and the threshold effect is essential for GATE and for practical system design.

How FM Suppresses Noise: The Discriminator and Parabolic PSD

In FM demodulation, the received signal passes through a bandpass limiter that removes amplitude variations, then through a frequency discriminator that converts instantaneous frequency deviations into amplitude. The AWGN added in the channel has a flat (white) power spectral density at the discriminator input. However, the FM discriminator is equivalent to a differentiator for the noise component. Differentiation in time corresponds to multiplication by j2 pi f in the frequency domain, so the output noise power spectral density becomes proportional to f^2. This is called a parabolic noise PSD.

The lowpass filter following the discriminator limits the noise to the message bandwidth W. The total output noise power is the integral of the parabolic PSD from -W to W, giving N_o proportional to W^3. The signal power at the discriminator output, on the other hand, scales with the square of the frequency deviation delta_f = beta W, where beta is the FM modulation index. This squared dependence of signal power on beta, combined with the constant noise integral shape, creates the beta^2 improvement in SNR.

Figure of Merit for FM

For a sinusoidal (tone) message with unity amplitude and FM modulation index beta = delta_f / W, the figure of merit for FM is derived as: FOM = 3 beta^2 (beta + 1). This expression is valid above the threshold and assumes the message has normalized power P_mn = 1/2 for a sinusoidal tone. For a general message with normalized power P_mn, the expression becomes FOM = 3 beta^2 (beta + 1) P_mn. The key observation is that FOM grows approximately as beta^3 for large beta (since beta + 1 approaches beta), giving cubic growth with modulation index. This is the mechanism behind FM's superior noise performance at large bandwidth occupancy.

The bandwidth of FM signal by Carson's rule is BW_FM = 2 (beta + 1) W. As beta increases, bandwidth grows linearly while SNR improvement grows as beta^3, confirming that FM exploits the bandwidth-SNR exchange more efficiently than AM.

The FM Threshold Effect

The parabolic noise PSD and the corresponding FOM derivation are valid only above a minimum input SNR called the FM threshold. When the input SNR falls below approximately 10 dB (the exact value is beta-dependent), the noise phasor occasionally exceeds the signal phasor in the limiter output. When this happens, the discriminator produces large impulsive noise spikes, causing a catastrophic collapse of output SNR. The output SNR drops far below what the linear analysis would predict. This threshold is at a higher input SNR for larger beta, meaning wideband FM systems are more vulnerable to threshold effects.

A practical consequence is that high-beta FM cannot be used in fading channels or weak signal environments without threshold extension techniques. Below threshold, increasing beta actually worsens the SNR instead of improving it. A system designer must ensure the received signal power keeps the input SNR comfortably above the threshold, typically by at least 3 to 5 dB margin.

Capture Effect in FM

The capture effect is a property unique to FM receivers. When two FM signals are present at the input on the same carrier frequency, the limiter in the FM receiver naturally suppresses the weaker signal and locks onto the stronger one. If the stronger signal is only a few decibels above the weaker one, the weaker signal is almost completely captured and rejected. This is beneficial for rejecting co-channel interference but means an FM receiver cannot receive two signals simultaneously, unlike AM where both contribute to output.

Pre-emphasis and De-emphasis

Because FM discriminator output noise has a parabolic PSD (rising with frequency), high-frequency message components are more degraded than low-frequency ones. To compensate, pre-emphasis is applied at the transmitter to boost high-frequency message components before modulation. A corresponding de-emphasis filter at the receiver output attenuates high frequencies equally, restoring the original message. This also suppresses the high-frequency portion of the parabolic noise, improving SNR by approximately 13 dB in commercial FM broadcasting (75 microsecond pre-emphasis time constant).

Numerical Example

An FM system uses a single-tone message and a modulation index of 5. We compute the figure of merit and compare it with DSB-SC to show the noise advantage gained by wideband FM.

Given:
Modulation type: FM
Message: single tone, P_mn = 0.5
Modulation index: β = Δf/W = 5

Why this formula applies:
Above threshold, FM FOM = 3β²(β+1) × P_mn for tone modulation.

Formula:
FOM = 3 β² (β + 1) × P_mn

Substitution:
FOM = 3 × (5)² × (5 + 1) × 0.5
    = 3 × 25 × 6 × 0.5

Calculation:
FOM = 3 × 25 × 3
    = 225

Final Answer: FOM = 225 (23.5 dB)
Bandwidth: BW = 2(β+1)W = 12W (Carson's rule)
FM is 225 times better than baseband SNR, at the cost of 12× bandwidth.

Exam Tip: For GATE, the FM figure of merit formula is FOM = 3 beta^2 (beta + 1) for tone modulation (with P_mn = 0.5 already included, verify the version given in your source). The threshold effect is not FM being AM-like; it is a catastrophic spike-based SNR collapse. Pre-emphasis improves SNR by reducing effective parabolic noise power, not by changing modulation index.

Mechanism Diagram: Parabolic Noise PSD and Threshold Collapse

• FM discriminator differentiates phase, turning flat input noise PSD into parabolic output noise PSD proportional to f^2.
• Figure of merit for FM (tone): FOM = 3 beta^2 (beta + 1). Grows approximately as beta^3 for large beta.
• FM bandwidth by Carson's rule: BW = 2 (beta + 1) W. Larger beta means larger bandwidth but better noise performance.
• FM threshold: below approximately 10 dB input SNR, noise spikes cause catastrophic SNR collapse.
• Capture effect: FM receiver locks onto strongest co-channel signal, suppressing weaker ones.
• Pre-emphasis and de-emphasis: boosts and then cuts high frequencies to offset parabolic noise, improving SNR by ~13 dB in broadcast FM.

Quick Revision

• FM FOM = 3 beta^2 (beta + 1) for sinusoidal message (above threshold). Grows as beta^3 for large beta.
• FM output noise PSD is parabolic: S_no(f) proportional to f^2. Comes from discriminator differentiation action.
• Threshold effect: input SNR below ~10 dB causes catastrophic FM SNR collapse due to noise spikes from limiter.
• Capture effect: FM locks onto stronger signal; useful for interference rejection.
• Pre-emphasis: boost high-freq at transmitter. De-emphasis: cut high-freq at receiver output. Net gain ~13 dB.
• FM vs AM: FM FOM can far exceed 1 (e.g., beta=5 gives FOM=225). AM FOM is always 1 or less.
• Trap: FM noise advantage requires being above threshold. Below threshold, larger beta makes SNR worse.