In 1962, the first communications satellite, Telstar 1, relayed telephone calls and television signals across the Atlantic Ocean. At its heart was a simple but revolutionary technique: Frequency Shift Keying (FSK)—the digital descendant of analog FM. By representing binary data as distinct frequency tones, FSK bridged the gap between the digital world of computers and the analog world of radio transmission.

Today, FSK and its variants remain fundamental to digital communications. From the characteristic warbles of dial-up modems to the silent efficiency of RFID tags, from wireless sensor networks to deep-space telemetry, FSK's combination of simplicity and robustness has earned it a permanent place in the communications engineer's toolkit. Understanding FSK means understanding how digital data rides on analog waves—the very essence of modern communication.

Before diving into FSK specifically, let's establish the broader context of digital modulation and understand where FSK fits in the landscape of digital communication techniques.

The Digital Challenge

Digital data exists as discrete symbols—most commonly binary digits (bits) with values 0 or 1. Transmission media (radio waves, cables, optical fibers) carry analog signals—continuously varying quantities like voltage, electromagnetic fields, or light intensity. Digital modulation bridges this gap by encoding digital data onto analog carrier signals.

Key Concepts in Digital Modulation

Signal Space Representation

Digital modulation can be visualized using signal space (constellation) diagrams, where:

• Each possible transmitted symbol corresponds to a point in the space
• The axes represent orthogonal signal components
• Distance between points relates to distinguishability and error probability

Bit Rate vs. Symbol Rate

• Bit Rate (R_b): Number of bits transmitted per second (bps)
• Symbol Rate (R_s or Baud Rate): Number of symbols transmitted per second
• Relationship: R_b = R_s × log₂(M), where M is the number of symbol states

For binary FSK: M = 2, so R_b = R_s (one bit per symbol) For 4-FSK: M = 4, so R_b = 2 × R_s (two bits per symbol)

Why Choose FSK?

In the landscape of digital modulation, each technique offers different tradeoffs. FSK's characteristics make it particularly suited for specific applications:

Advantages of FSK:

• Constant envelope: The signal amplitude doesn't vary, allowing efficient use of nonlinear amplifiers
• Noise immunity: Inherits FM's excellent resistance to amplitude noise
• Simple implementation: Especially for binary FSK, both transmitters and receivers can be straightforward
• Power efficiency: All transmitted power carries information (unlike some ASK schemes)
• Robust in fading: Less susceptible to amplitude fading in wireless channels

Disadvantages of FSK:

• Bandwidth inefficiency: Requires more bandwidth than PSK for the same data rate
• Spectral spillover: Abrupt frequency transitions create spectral sidebands
• Complex receivers at high speeds: High-data-rate FSK requires more sophisticated detection

Binary Frequency Shift Keying (BFSK) is the simplest form of FSK, where binary data is represented by two distinct frequencies. It forms the foundation for understanding all FSK variants.

BFSK Signal Definition

A BFSK signal uses two frequencies:

• Mark frequency (f₁): Represents binary '1'
• Space frequency (f₀): Represents binary '0'

The mathematical representation is:

For binary '1': s₁(t) = A_c · cos(2πf₁t + φ), for 0 ≤ t ≤ T_b

For binary '0': s₀(t) = A_c · cos(2πf₀t + φ), for 0 ≤ t ≤ T_b

Where:

• A_c = carrier amplitude (constant)
• f₁, f₀ = mark and space frequencies
• T_b = bit duration = 1/R_b
• φ = initial phase

Frequency Deviation in BFSK

The frequencies are typically symmetric around a center frequency f_c:

f₁ = f_c + Δf (mark frequency) f₀ = f_c - Δf (space frequency)

Where Δf is the frequency deviation. The total frequency separation is:

Δf_total = f₁ - f₀ = 2Δf

Continuous-Phase vs. Discontinuous-Phase BFSK

An important distinction in BFSK implementation is whether phase continuity is maintained at bit transitions:

Discontinuous-Phase BFSK (Non-Coherent FSK):

• At each bit boundary, the signal may have a phase discontinuity
• Simpler to implement using two separate oscillators
• Results in wider bandwidth due to spectral splatter from discontinuities
• Typically detected non-coherently

Continuous-Phase FSK (CPFSK):

• Phase is continuous at bit transitions
• Requires careful frequency/phase relationships
• Produces narrower spectrum (better bandwidth efficiency)
• Special case: MSK (Minimum Shift Keying) is CPFSK with minimum frequency separation for orthogonality

Orthogonality Condition in BFSK

Two signals are orthogonal if their cross-correlation over the symbol period is zero:

∫₀^(T_b) s₁(t)·s₀(t) dt = 0

For BFSK, this occurs when the frequency separation satisfies:

f₁ - f₀ = n/(2T_b) for some integer n ≥ 1

The minimum separation for orthogonality is:

Δf = 1/(2T_b) = R_b/2

This minimum separation defines Minimum Shift Keying (MSK), the most bandwidth-efficient form of binary FSK with orthogonal signaling.

M-ary FSK (MFSK) extends the BFSK concept to use M distinct frequencies, where M > 2. Each symbol represents log₂(M) bits, enabling higher data rates or improved performance at the cost of bandwidth.

MFSK Signal Definition

An M-ary FSK system uses M frequencies:

s_i(t) = A_c · cos(2πf_i·t + φ), for i = 0, 1, 2, ..., M-1

Where each frequency f_i represents a different symbol. For orthogonal MFSK with minimum frequency separation:

f_i = f_c + (2i - M + 1) · Δf

With Δf = 1/(2T_s) where T_s is the symbol duration.

Relationship Between Bits and Symbols

For M-ary signaling:

• Number of bits per symbol: k = log₂(M)
• Symbol duration: T_s = k × T_b = k/R_b
• Symbol rate: R_s = R_b/k = R_b/log₂(M)

Example Calculations:

• 4-FSK (M=4): 2 bits per symbol, T_s = 2T_b
• 8-FSK (M=8): 3 bits per symbol, T_s = 3T_b
• 16-FSK (M=16): 4 bits per symbol, T_s = 4T_b

Performance Tradeoffs in MFSK

Increasing M provides fascinating tradeoffs that illuminate fundamental communications principles:

Bandwidth Increases with M

For orthogonal MFSK with minimum frequency separation:

• Required bandwidth ≈ M × (1/T_s) = M × R_s = M × R_b/log₂(M)

As M increases, bandwidth per bit increases, making MFSK bandwidth-inefficient.

Error Probability Decreases with M

Somewhat counterintuitively, the probability of symbol error decreases as M increases (for fixed energy per bit). This is because:

• Each symbol carries more energy (longer duration)
• Orthogonal signals become more distinguishable as their number increases
• In the limit as M → ∞, error probability approaches zero (with sufficient bandwidth)

This remarkable property illustrates Shannon's channel capacity theorem: you can trade bandwidth for reliability.

The Bandwidth-Power Tradeoff

MFSK exemplifies a fundamental communications principle:

• Bandwidth-limited systems (limited spectrum): Use small M or other modulation
• Power-limited systems (limited transmit power): Use large M for better error performance

Deep-space communications (extremely power-limited, bandwidth available) often use large M values like 64-FSK or 256-FSK.

Generating FSK signals can be accomplished through several methods, each with distinct advantages for different applications.

Method 1: Switched Oscillator (Direct Method)

The simplest approach uses two independent oscillators (or frequency sources) that are switched by the data signal:

• Data = '1' → Connect f₁ oscillator to output
• Data = '0' → Connect f₀ oscillator to output

Advantages:

• Very simple implementation
• Precise frequency control
• Easy to implement with discrete components

Disadvantages:

• Phase discontinuities at switching points
• Wider bandwidth due to spectral spreading from discontinuities
• Potential switching transients

Method 2: Voltage-Controlled Oscillator (VCO)

The data signal (after level shifting) directly controls a VCO:

• Digital data → Level converter → VCO → FSK output
• '1' produces voltage → VCO outputs f₁
• '0' produces voltage → VCO outputs f₀

Advantages:

• Naturally continuous phase (CPFSK)
• Single oscillator (simpler hardware)
• Smooth spectral characteristics

Disadvantages:

• VCO calibration required
• Frequency drift possible
• May need temperature compensation

Method 3: PLL-Based FSK Generator

A Phase-Locked Loop synthesizer provides both frequency stability and modulation capability:

1. Reference oscillator (crystal) provides stability
2. Divide ratio is switched by data signal
3. PLL locks VCO to modified reference
4. Produces crystal-accurate FSK

Advantages:

• Excellent frequency accuracy and stability
• Phase coherence can be maintained
• Well-suited for integration

Disadvantages:

• More complex than simple methods
• Modulation bandwidth limited by PLL bandwidth
• Transient response affects fast data rates

Method 4: Direct Digital Synthesis (DDS)

Modern digital approach using numerical controlled oscillators:

1. Digital data controls phase accumulator step size
2. Phase accumulator value addresses sine lookup table
3. Digital-to-analog converter produces analog output
4. Low-pass filter removes digital artifacts

Advantages:

• Extremely precise frequency control
• Easy to implement continuous phase
• Highly flexible (software-defined)
• Excellent for MFSK (any M easily programmed)

Disadvantages:

• Maximum frequency limited by DAC and clock speed
• Quantization noise and spurious signals
• Requires digital signal processing knowledge

FSK detection (demodulation) extracts the original digital data from the received FSK signal. Two fundamentally different approaches exist: coherent detection (requires carrier phase knowledge) and non-coherent detection (does not require phase knowledge).

Coherent Detection

Coherent detection exploits knowledge of the carrier phase to achieve optimal performance. The receiver must synchronize its local oscillator to the incoming carrier phase—a process called carrier recovery.

Matched Filter / Correlation Receiver:

The optimal coherent BFSK receiver uses correlation with the two possible signals:

1. Correlate received signal with local copy of s₁(t) → output r₁
2. Correlate received signal with local copy of s₀(t) → output r₀
3. Compare r₁ and r₀
4. If r₁ > r₀, decide '1'; otherwise decide '0'

Implementation:

• Two bandpass filters (or correlators) centered on f₁ and f₀
• Each followed by envelope detector or synchronous detector
• Compare outputs; larger wins

Performance:

• Optimal in Gaussian noise
• Probability of bit error: P_b = Q(√(2E_b/N_0)) for orthogonal FSK
• Requires approximately 2 dB less E_b/N_0 than non-coherent detection

Non-Coherent Detection

Non-coherent detection doesn't require knowledge of the carrier phase, making it simpler and more robust for many applications.

Envelope Detection Method:

1. Bandpass filter centered on f₁ → Envelope detector → output E₁
2. Bandpass filter centered on f₀ → Envelope detector → output E₀
3. Compare E₁ and E₀ at sampling instant
4. If E₁ > E₀, decide '1'; otherwise decide '0'

Zero-Crossing Detection:

1. Limit received signal to constant amplitude
2. Count zero crossings over bit period
3. Higher count → higher frequency (f₁)
4. Lower count → lower frequency (f₀)

Frequency Discriminator Method:

1. Pass FSK through frequency discriminator (FM detector)
2. Output is baseband signal representing frequency deviation
3. Sample and threshold to recover data

Performance:

• Approximately 1-3 dB worse than coherent detection
• But much simpler to implement
• More robust in practical conditions

Differential Detection

A hybrid approach that compares the phase of successive symbols:

1. Compare current bit's phase to previous bit's phase
2. Change in phase indicates frequency
3. No absolute phase reference required

Digital Detection Methods

Modern receivers often digitize the received signal and perform detection algorithmically:

Discrete Fourier Transform (DFT) Detection:

1. Sample received signal over bit period
2. Compute DFT bins at f₁ and f₀
3. Compare magnitudes; larger indicates transmitted frequency

Goertzel Algorithm:

• Efficient computation of single DFT bins
• Ideal for FSK detection (need only 2 frequency bins)
• Computationally efficient for embedded applications

Understanding FSK error performance requires analyzing how noise affects detection decisions. This analysis connects theory to practical system design.

Bit Error Rate (BER) Definition

The Bit Error Rate is the probability that a transmitted bit is received incorrectly:

BER = Number of bit errors / Total bits transmitted

BER depends on the signal energy relative to noise, expressed as E_b/N_0:

• E_b = Energy per bit = Signal power × Bit duration = P × T_b
• N_0 = Noise power spectral density (watts/Hz)
• E_b/N_0 is dimensionless, often expressed in dB

Coherent BFSK Error Probability

For coherent detection of orthogonal BFSK signals:

P_b = Q(√(2E_b/N_0))

Where Q(x) is the Q-function (complementary Gaussian distribution):

Q(x) = (1/√(2π)) ∫_x^∞ exp(-t²/2) dt

Non-Coherent BFSK Error Probability

For non-coherent detection:

P_b = (1/2) × exp(-E_b/(2N_0))

This formula shows exponential decrease in error rate with increasing E_b/N_0—a characteristic of non-coherent detection.

Comparison with Other Modulations

FSK performance can be compared to other modulation schemes at the same E_b/N_0:

For coherent detection:

• BPSK: P_b = Q(√(2E_b/N_0)) — Same as coherent BFSK!
• QPSK: P_b = Q(√(2E_b/N_0)) — Same BER, 2× spectral efficiency
• Coherent BFSK: P_b = Q(√(2E_b/N_0))

For non-coherent detection:

• DPSK: P_b = (1/2)exp(-E_b/N_0) — 3 dB better than non-coherent FSK
• Non-coherent FSK: P_b = (1/2)exp(-E_b/2N_0)

Key Insight: Coherent BFSK and BPSK have identical BER performance! However:

• BPSK achieves this in bandwidth B ≈ R_b (bits/Hz efficiency = 1)
• BFSK requires bandwidth B ≈ 2R_b (bits/Hz efficiency = 0.5)

So BPSK is more bandwidth-efficient, but FSK has other advantages (constant envelope, simpler non-coherent detection).

M-ary FSK Error Performance

For orthogonal M-ary FSK with coherent detection:

P_s ≤ (M-1) × Q(√(E_s/N_0))

Where E_s = Energy per symbol = k × E_b (k = log₂M bits per symbol).

As M increases:

• Symbol error probability for fixed E_b/N_0 decreases
• Bandwidth increases proportionally with M
• This demonstrates the bandwidth-power tradeoff

Several important FSK variants optimize specific aspects of performance, addressing the bandwidth, spectral efficiency, and implementation tradeoffs.

Minimum Shift Keying (MSK)

MSK is a special case of continuous-phase binary FSK with the minimum frequency separation that maintains orthogonality:

Δf = 1/(4T_b) = R_b/4

Key properties:

• Modulation index β = 0.5
• Frequency deviation = 1/4 of bit rate
• Constant envelope
• Phase changes by exactly ±π/2 each bit period
• Main spectral lobe is narrower than conventional FSK

MSK can also be viewed as a form of OQPSK (Offset QPSK) with half-sinusoidal pulse shaping, giving it excellent spectral properties.

Gaussian MSK (GMSK)

GMSK passes the modulating data through a Gaussian low-pass filter before frequency modulating:

1. Binary data → Gaussian filter → VCO → GMSK output

The Gaussian filter smooths the frequency transitions, producing:

• Even narrower spectrum than MSK
• Slight ISI (intersymbol interference) due to filtering
• Extremely compact spectrum
• Used in GSM cellular systems

The key parameter is BT (Bandwidth-Time product):

• BT = 0.3 typical for GSM
• Lower BT → narrower spectrum, more ISI
• Higher BT → wider spectrum, less ISI

Audio Frequency Shift Keying (AFSK)

AFSK uses audio-frequency tones to carry FSK signals over voice-grade channels:

• Standard telephone lines (300-3400 Hz bandwidth)
• Radio links designed for voice
• Amateur radio packet systems

Common AFSK standards:

• Bell 103: 1270/1070 Hz (originate), 2225/2025 Hz (answer)
• V.21: 980/1180 Hz (channel 1), 1650/1850 Hz (channel 2)
• AX.25 (amateur radio): 1200/2200 Hz at 1200 bps

Gaussian Frequency Shift Keying (GFSK)

Generalizes GMSK to arbitrary frequency deviations:

1. Binary data → Gaussian filter → Frequency modulator
2. BT parameter controls filter bandwidth
3. Modulation index can vary (not restricted to 0.5)

Used in:

• Bluetooth (GFSK with BT = 0.5, modulation index 0.28-0.35)
• DECT cordless phones
• Zigbee (in some modes)

Continuous-Phase Modulation (CPM) Family

FSK is part of the broader CPM family, which includes:

• MSK and GMSK
• Multi-h CPM (multiple modulation indices)
• Tamed FM (TFM)
• Shaped CPFSK

These share the constant-envelope and continuous-phase properties that make them attractive for power-efficient amplifiers.

We have thoroughly explored Frequency Shift Keying—the digital embodiment of frequency modulation principles. Let's consolidate the essential knowledge:

What's Next: Bandwidth Requirements

With solid understanding of FM and FSK signal characteristics, we're ready to analyze their bandwidth requirements quantitatively. The next page examines Carson's Rule, the relationship between modulation index and spectral spread, and practical bandwidth allocation for FM and FSK systems. Understanding bandwidth is essential for frequency planning, regulatory compliance, and system optimization.