Frequency Modulation and FSK

In 1927, regulators at the International Radio Conference faced a growing crisis: too many radio stations competing for limited spectrum. This led to the first international frequency allocations—recognition that the electromagnetic spectrum is a finite resource that must be carefully managed. Nearly a century later, this challenge has only intensified. Every hertz of bandwidth is valuable; every transmission system must justify its spectral footprint.

For Frequency Modulation, bandwidth analysis reveals a fundamental tradeoff that shapes all FM system design: FM's superior noise immunity comes at the cost of increased bandwidth compared to Amplitude Modulation. Understanding exactly how much bandwidth FM requires—and why—is essential for designing efficient communication systems, meeting regulatory requirements, and avoiding interference with adjacent channels.

The bandwidth of an FM signal is fundamentally more complex than AM because FM creates an infinite number of sidebands. Understanding this spectral structure is essential for bandwidth estimation.

Recall: The FM Signal Expression

For single-tone modulation with message signal m(t) = A_m cos(2πf_m t), the FM signal is:

s_FM(t) = A_c cos[2πf_c t + β sin(2πf_m t)]

Where:

• β = Δf/f_m is the modulation index
• Δf is the peak frequency deviation
• f_m is the modulating frequency
• f_c is the carrier frequency

The Bessel Function Expansion

Using the Jacobi-Anger expansion, this can be written as an infinite series:

s_FM(t) = A_c Σ J_n(β) cos[2π(f_c + n·f_m)t]

Where:

• J_n(β) is the Bessel function of the first kind, order n, argument β
• The sum extends from n = -∞ to n = +∞
• Each term represents a spectral component at frequency f_c + n·f_m

Physical Interpretation:

• n = 0: Carrier component at frequency f_c
• n = ±1: First-order sidebands at f_c ± f_m
• n = ±2: Second-order sidebands at f_c ± 2f_m
• And so on, theoretically to infinity

Key Observations from the Bessel Functions:

1. Power Conservation: The total power in all components equals the carrier power: Σ J_n²(β) = 1 (sum from n = -∞ to +∞)

2. Carrier Nulls: For certain values of β (approximately 2.4, 5.5, 8.7, ...), J₀(β) = 0, meaning the carrier disappears entirely.

3. Significant Sidebands: For any β, most power is concentrated in sidebands where |n| ≤ β + 1 (approximately). Higher-order sidebands have negligible amplitude.

4. Asymptotic Behavior: For large β, significant sidebands extend approximately to n ≈ β + √β.

Practical Significance

Since FM theoretically has infinite bandwidth, we must define practical bandwidth—the frequency range containing most of the signal power. The criterion is typically:

• 98% Power Bandwidth: Range containing 98% of total signal power
• 99% Power Bandwidth: Range containing 99% of total signal power
• Significant Sideband Bandwidth: Range containing all sidebands with amplitude ≥ 1% of unmodulated carrier

Exact bandwidth calculation using Bessel functions is mathematically complex. Carson's Rule provides a remarkably accurate approximation that has become the standard for FM bandwidth estimation.

Carson's Rule Statement

The bandwidth of an FM signal is approximately:

B_FM ≈ 2(Δf + f_m) = 2Δf(1 + 1/β)

Or equivalently:

B_FM ≈ 2f_m(β + 1)

Where:

• Δf = Peak frequency deviation
• f_m = Maximum modulating frequency
• β = Modulation index = Δf/f_m

Origin and Validity

John Renshaw Carson, while at AT&T in the 1920s, developed this rule empirically. It estimates the bandwidth containing approximately 98% of the signal power. The rule works remarkably well across a wide range of modulation indices:

For β << 1 (Narrowband FM):

• B_FM ≈ 2f_m (similar to AM bandwidth)
• The (β + 1) factor is dominated by the 1

For β >> 1 (Wideband FM):

• B_FM ≈ 2βf_m = 2Δf
• Bandwidth is approximately twice the frequency deviation

For moderate β (1 < β < 5):

• Both terms contribute significantly
• Carson's Rule provides excellent accuracy

Detailed Carson's Rule Examples

Example 1: Standard FM Broadcasting

FM radio stations use:

• Maximum audio frequency: f_m = 15 kHz
• Maximum frequency deviation: Δf = 75 kHz
• Modulation index: β = 75/15 = 5

Carson's Rule bandwidth: B_FM = 2(75 + 15) = 2 × 90 = 180 kHz

Actual allocated channel spacing: 200 kHz (includes 20 kHz guard bands)

Example 2: Narrowband FM (Public Safety Radio)

Typical narrowband FM parameters:

• Maximum audio frequency: f_m = 3 kHz
• Maximum frequency deviation: Δf = 5 kHz
• Modulation index: β = 5/3 ≈ 1.67

Carson's Rule bandwidth: B_FM = 2(5 + 3) = 16 kHz

Typical channel spacing: 12.5 kHz or 25 kHz

Example 3: Wideband FM Telemetry

A telemetry link using:

• Maximum data frequency: f_m = 100 kHz
• Maximum deviation: Δf = 500 kHz
• Modulation index: β = 5

Carson's Rule bandwidth: B_FM = 2(500 + 100) = 1.2 MHz

The modulation index β determines whether an FM system is classified as narrowband or wideband, with profound implications for bandwidth, performance, and application suitability.

Narrowband FM (NBFM): β ≤ 0.5

When β is small (typically ≤ 0.5), the FM signal approximates AM in bandwidth:

Mathematical Approximation: For small β:

• J₀(β) ≈ 1
• J₁(β) ≈ β/2
• J_n(β) ≈ 0 for n ≥ 2

The FM signal reduces to approximately: s_NBFM(t) ≈ A_c cos(2πf_c t) - (A_c β/2)sin(2π(f_c+f_m)t) + (A_c β/2)sin(2π(f_c-f_m)t)

Bandwidth: B_NBFM ≈ 2f_m (same as AM!)

Characteristics:

• Spectrum similar to AM (carrier + two sidebands)
• Bandwidth-efficient but limited noise improvement
• Used when spectrum is scarce
• SNR improvement factor ≈ 3β² (modest for small β)

Applications:

• Paging systems
• Low-bandwidth voice communications
• Tone signaling systems

Wideband FM (WBFM): β >> 1

When β is large (typically β > 5), FM exhibits its full bandwidth-noise tradeoff:

Mathematical Properties:

• Many significant sidebands (approximately β + √β sidebands contribute)
• Carrier may be small or zero (depending on exact β value)
• Power distributed across wide spectral range

Bandwidth: B_WBFM ≈ 2Δf (frequency deviation dominates)

Characteristics:

• Excellent noise immunity
• High fidelity audio possible
• Requires more spectrum
• Threshold effect becomes important

The FM Threshold Effect

WBFM has a critical characteristic: the threshold effect. Below a certain input SNR (the threshold, typically 10-12 dB), FM demodulator output quality degrades rapidly—much faster than linearly. Above threshold, WBFM provides substantial SNR improvement over the input.

FM Threshold in dB: SNR_threshold ≈ 10-12 dB (carrier-to-noise ratio)

Above threshold: Output SNR ≈ 3β²(1 + β) × Input SNR Below threshold: Spike noise dominates, quality collapses

This threshold behavior means WBFM is unsuitable for very weak signal conditions—an important system design consideration.

FSK bandwidth analysis differs from analog FM because the modulating signal is a digital waveform rather than a continuous analog signal. The spectral characteristics depend on the frequency separation, bit rate, and phase continuity.

Spectral Characteristics of FSK

The bandwidth of an FSK signal depends on several factors:

1. Frequency separation (f₁ - f₀ = 2Δf)
2. Bit rate (R_b = 1/T_b)
3. Phase continuity (continuous vs. discontinuous)
4. Pulse shaping (rectangular vs. shaped pulses)

Discontinuous-Phase FSK Bandwidth

For discontinuous-phase FSK (switching between two oscillators):

B_FSK ≈ 2Δf + 2R_b = (f₁ - f₀) + 2R_b

The 2R_b term accounts for the spectral spreading caused by the rectangular pulse shape of the data (sinc function sidebands).

This can also be written as:

B_FSK ≈ 2R_b(β + 1)

Where β = Δf/R_b is the FSK modulation index. This parallels Carson's Rule for analog FM!

Continuous-Phase FSK (CPFSK) Bandwidth

CPFSK has smoother phase transitions, resulting in narrower spectrum:

For MSK (β = 0.5):

• 99% power bandwidth ≈ 1.2 × R_b
• Main lobe bandwidth = 1.5 × R_b
• First sidelobes are 23 dB below main lobe (quite good)

For general CPFSK:

• Spectral compactness improves as phase continuity is maintained
• Sidelobe levels depend on modulation index

Gaussian FSK (GFSK) Bandwidth

GFSK uses Gaussian filtering of the data before frequency modulation:

B_GFSK ≈ R_b × (1 + BT)

For Bluetooth (BT = 0.5):

• B_GFSK ≈ 1.5 × R_b ≈ 1.5 MHz for 1 Mbps
• But 99% power bandwidth is narrower yet

For GSM (GMSK, BT = 0.3):

• 99% power bandwidth ≈ 0.8 × R_b
• Extremely spectrally efficient

M-ary FSK Bandwidth

For orthogonal M-ary FSK with minimum frequency separation:

B_MFSK = M × R_s = M × R_b / log₂(M)

Where R_s is the symbol rate.

Example: 8-FSK at 9600 bps

• Bits per symbol = log₂(8) = 3
• Symbol rate = 9600/3 = 3200 symbols/sec
• Bandwidth = 8 × 3200 = 25.6 kHz
• Spectral efficiency = 9600/25600 = 0.375 bps/Hz

Observation: As M increases, spectral efficiency decreases! This is opposite to PSK/QAM, where higher M improves spectral efficiency. MFSK trades bandwidth for power efficiency—the opposite of the PSK/QAM tradeoff.

Practical FSK Bandwidth Considerations

1. Regulatory masks: Real signals must fit within specified spectral masks that define maximum out-of-band emissions

2. Adjacent channel interference: Practical bandwidth must account for filter rolloff and sideband leakage

3. Implementation imperfections: Real transmitters may have spectral regrowth from nonlinearities

4. Required margin: Add 10-20% to theoretical bandwidth for practical systems

Spectral efficiency measures how effectively a modulation scheme uses available bandwidth. It's defined as the ratio of data rate to bandwidth:

η = R_b / B (bits per second per Hz, or bps/Hz)

Higher spectral efficiency means more data in less spectrum—increasingly important as spectrum becomes congested.

Spectral Efficiency of FM/FSK vs. Other Modulations

FM and FSK are generally spectrally inefficient compared to phase and amplitude modulations:

FM/FSK Spectral Efficiency:

• Standard BFSK: η ≈ 0.3-0.5 bps/Hz
• MSK: η = 0.67 bps/Hz (best for constant-envelope binary)
• GMSK (BT=0.3): η ≈ 1.0-1.2 bps/Hz

Compare to PSK/QAM:

• BPSK: η = 1 bps/Hz
• QPSK: η = 2 bps/Hz
• 16-QAM: η = 4 bps/Hz
• 64-QAM: η = 6 bps/Hz

Why Use FM/FSK Despite Low Spectral Efficiency?

Given FM/FSK's bandwidth disadvantage, why are they still widely used?

1. Constant Envelope Advantage

FM/FSK signals have constant amplitude, enabling:

• Use of efficient Class C/E amplifiers (up to 70% efficiency vs. 30% for linear)
• Reduced battery drain (critical for mobile/satellite systems)
• Tolerance to amplifier nonlinearity
• Simplified automatic gain control

2. Noise and Interference Immunity

FM's inherent rejection of amplitude noise provides:

• Better performance in high-noise environments
• Resistance to multipath fading
• Capture effect for interference rejection
• Graceful degradation (analog FM) or robust operation (digital FSK)

3. Simpler Non-Coherent Detection

FSK allows excellent non-coherent detection:

• No carrier recovery required
• Robust to phase noise and oscillator drift
• Simpler receiver architecture
• Better performance in fading channels

4. Regulatory and Legacy Reasons

Many frequency bands are allocated specifically for FM/FSK:

• VHF/UHF land mobile channels (12.5/25 kHz channelization)
• Existing infrastructure designed for FM
• Regulatory rules specifying FM for certain services

Real-world FM/FSK systems must operate within allocated channel bandwidths, meeting regulatory requirements while providing desired performance. This section connects theoretical bandwidth to practical system design.

Channel Bandwidth Allocation

Regulatory bodies allocate channels with specific bandwidths:

FM Broadcasting (worldwide):

• Channel bandwidth: 200 kHz (US), 100-200 kHz (Europe varies)
• Guard bands included for adjacent channel protection
• Carson's BW (180 kHz) fits within 200 kHz allocation

Land Mobile Radio (US FCC):

• Wideband: 25 kHz channels
• Narrowband: 12.5 kHz channels
• Ultra-narrowband: 6.25 kHz channels (emerging)

Deviation Rules

Frequency deviation is typically regulated:

FM Broadcasting:

• Maximum deviation: ±75 kHz
• Pre-emphasis: 75 μs (US) or 50 μs (Europe)

Land Mobile:

• 25 kHz channels: ±5 kHz deviation typical
• 12.5 kHz channels: ±2.5 kHz deviation
• 6.25 kHz channels: ±1.25 kHz deviation

Guard Bands and Adjacent Channel Considerations

Adjacent Channel Power Ratio (ACPR): Measures how much power leaks into adjacent channels:

• Typical requirement: -60 to -70 dBc at first adjacent channel
• Requires careful filter design at transmitter
• Receiver selectivity must reject adjacent channel signals

Calculating Required Guard Band:

1. Determine Carson's bandwidth of signal
2. Add margin for filter rolloff (typically 10-20%)
3. Add margin for transmitter frequency uncertainty
4. Add margin for receiver filter skirts

Example: Designing a narrowband FM voice system

• Voice bandwidth: 3 kHz
• Deviation: ±2.5 kHz
• Carson's BW: 2(2.5 + 3) = 11 kHz
• With margins: ≈12.5 kHz channel appropriate

Occupied Bandwidth Measurement

The occupied bandwidth is typically defined as the bandwidth containing 99% of the signal's total power. Regulatory bodies require that occupied bandwidth not exceed a specified limit:

• FM broadcasting: Occupied BW < 200 kHz
• Narrow FM: Occupied BW < 11.25 kHz (for 12.5 kHz channel)
• FSK data: Varies by standard/regulation

When spectrum is precious, several techniques can reduce FM/FSK bandwidth while maintaining acceptable performance.

1. Reducing Modulation Index

Lower β directly reduces bandwidth, but with tradeoffs:

• Lower SNR improvement (noise immunity decreases)
• Must ensure adequate threshold margin
• May require increased transmit power to compensate

Practical approach: Find the minimum β that meets SNR requirements at the weakest signal condition.

2. Pulse Shaping (for FSK)

Shaping the data waveform before frequency modulation reduces spectral spread:

Gaussian Filtering (GMSK/GFSK):

• Filter data with Gaussian LPF before VCO
• Reduces high-frequency components that cause spectral spread
• BT (bandwidth-time product) controls tradeoff:
  • Lower BT = narrower spectrum, more ISI
  • Higher BT = wider spectrum, less ISI
• GSM uses BT = 0.3 for maximum spectral compactness

Raised Cosine Filtering:

• Alternative pulse shape with controlled spectral rolloff
• Nyquist criterion prevents ISI at sample points
• Roll-off factor α controls bandwidth-ISI tradeoff

3. Continuous Phase Modulation

Maintaining phase continuity at symbol transitions eliminates spectral splatter:

Benefits:

• Phase discontinuities create high-frequency energy (spectral spreading)
• CPFSK avoids this by ensuring smooth phase transitions
• MSK and GMSK are optimized forms of CPFSK

4. Pre-emphasis and De-emphasis

For analog FM voice, pre/de-emphasis controls the effective deviation:

Pre-emphasis at transmitter:

• Boosts high-frequency audio before modulation
• Higher frequencies get larger deviation
• Improves SNR at high frequencies (where noise is worse)

De-emphasis at receiver:

• Attenuates high frequencies after demodulation
• Reduces boosted noise along with boosted signal
• Restores flat frequency response

Net effect: Same deviation limit, but effective SNR improvement of 13 dB at high frequencies.

5. Voice Compression (Companders)

Dynamic range compression reduces required deviation:

Compressor at transmitter:

• Amplifies quiet passages, limits loud passages
• Reduces dynamic range from ~40 dB to ~15 dB
• Allows lower deviation while maintaining intelligibility

Expander at receiver:

• Reverses compression, restores original dynamics
• Low-level noise is also reduced

Result: 2:1 or 3:1 compression allows narrower channels.

We have thoroughly analyzed bandwidth requirements for FM and FSK systems—a critical aspect of communication system design. Let's consolidate the essential knowledge:

What's Next: Noise Immunity

With bandwidth requirements understood, we're ready to analyze FM's remarkable noise immunity in detail. The next page examines why FM resists noise better than AM, the mathematical basis for SNR improvement, threshold effects, and how these properties influence system design decisions.