FSK Modulation

Frequency Shift Keying encodes digital data by switching the carrier between discrete frequency values, leaving the amplitude and phase nominally constant. FSK is far more robust to amplitude disturbances than ASK, making it practical in real wireless and telephony channels. Understanding FSK, particularly its orthogonality condition and bandwidth, is a recurring topic in GATE Digital Communications.

Core Concept: Frequency Switching and Orthogonality

In Binary Frequency Shift Keying (BFSK), one frequency f1 is transmitted for bit 1 and another frequency f2 for bit 0. The receiver must decide which frequency was present during each bit period. The key mathematical requirement is that the two basis functions must be orthogonal so that the correlator receiver can perfectly distinguish them in the absence of noise.

Two sinusoids are orthogonal over the bit interval [0, Tb] if their inner product equals zero: integral from 0 to Tb of cos(2πf1t) × cos(2πf2t) dt = 0. This condition is satisfied when |f1 - f2| = n/(2Tb) for coherent detection (n = 1, 2, ...) and when |f1 - f2| = n/Tb for non-coherent detection. The minimum frequency spacing for coherent BFSK is Rb/2 Hz, while for non-coherent it is Rb Hz. This distinction is critical in GATE problems.

Mathematical Expression

The BFSK signal is: s1(t) = Ac cos(2πf1t) for bit 1 and s2(t) = Ac cos(2πf2t) for bit 0. The probability of error for coherent BFSK is: Pb = Q(sqrt(Eb/N0)). For non-coherent BFSK, the BER is: Pb = (1/2) exp(-Eb / 2N0). This shows that coherent detection is better than non-coherent at moderate to high SNR but requires a more complex receiver with carrier phase synchronization.

The transmission bandwidth of BFSK is BT = |f1 - f2| + 2Rb using the null-to-null criterion. Substituting the minimum coherent separation |f1 - f2| = Rb/2 gives BT = Rb/2 + 2Rb = 2.5Rb. FSK is therefore less bandwidth-efficient than PSK because it requires the additional frequency separation on top of the symbol bandwidth.

Practical Understanding

FSK is preferred over ASK in channels with amplitude fading because the receiver only needs to detect which frequency is present, not measure its magnitude. Early radio modems, paging systems, caller ID, and DTMF telephone signaling all use FSK variants. Bluetooth uses Gaussian FSK (GFSK) where the bit stream is pre-filtered to reduce bandwidth before modulation. The Gaussian filter rounds the sharp transitions and reduces the spectral sidelobes significantly.

M-ary FSK (MFSK) uses M distinct frequencies and transmits log₂(M) bits per symbol. MFSK improves power efficiency (reduces Eb/N0 requirement) at the cost of increased bandwidth. This is the opposite trade-off from M-ary PSK or QAM, making MFSK suitable for power-limited channels such as deep-space communication.

Given:
Bit rate Rb = 4 kbps, BFSK with f1 = 1200 Hz, f2 = 2200 Hz, coherent detection, Eb/N0 = 10 dB

Why this formula applies:
Coherent BFSK BER uses Q(sqrt(Eb/N0)) and bandwidth uses null-to-null formula.

Formula:
Pb = Q(sqrt(Eb/N0))
BT = |f1 - f2| + 2Rb
Orthogonality check: |f1 - f2| >= Rb/2 for coherent

Substitution:
|f1 - f2| = |1200 - 2200| = 1000 Hz
Min separation required = Rb/2 = 4000/2 = 2000 Hz
Check: 1000 < 2000, so the pair is NOT orthogonal for coherent detection at Rb = 4 kbps.

Redo with Rb = 500 bps (so Rb/2 = 250 Hz < 1000 Hz — orthogonal):
BT = 1000 + 2 × 500 = 2000 Hz
Eb/N0 = 10 dB = 10 (linear)
Pb = Q(sqrt(10)) = Q(3.162) ≈ 7.8 × 10^-4

Final Answer with units:
Bandwidth BT = 2000 Hz, BER ≈ 7.8 × 10^-4 (at Rb = 500 bps, Eb/N0 = 10)

Exam Tip: The most common FSK trap in GATE is the orthogonality condition. Coherent FSK requires minimum separation of Rb/2 Hz; non-coherent FSK requires Rb Hz. If the question gives you two frequencies and a bit rate, always verify orthogonality before applying the bandwidth or BER formula.

• The receiver uses two parallel correlators matched to cos(2πf1t) and cos(2πf2t) respectively. The correlator output with the larger value determines the decoded bit.
• Orthogonality ensures that when s1(t) is sent, the output of correlator 2 is zero on average (no cross-interference). Violation of orthogonality causes inter-symbol distortion.
• In 2D signal space, s1 = (sqrt(Eb), 0) and s2 = (0, sqrt(Eb)). The Euclidean distance is sqrt(2Eb), giving BER = Q(sqrt(Eb/N0)) for coherent BFSK.
• This is 3 dB worse than BPSK (distance = 2sqrt(Eb/2) = sqrt(2Eb) — same distance actually, confirmed in theory: coherent BFSK and coherent OOK share the same BER formula Q(sqrt(Eb/N0))).

Quick Revision

• BFSK transmits f1 for bit 1 and f2 for bit 0; amplitude is constant.
• Orthogonality for coherent BFSK: |f1 - f2| = n/(2Tb), minimum = Rb/2 Hz.
• Orthogonality for non-coherent BFSK: |f1 - f2| = n/Tb, minimum = Rb Hz.
• Coherent BFSK BER: Pb = Q(sqrt(Eb/N0)). Non-coherent BFSK: Pb = 0.5 exp(-Eb/2N0).
• Bandwidth: BT = |f1 - f2| + 2Rb. Minimum coherent BT = 2.5Rb.
• M-ary FSK improves power efficiency at cost of bandwidth; opposite trade-off to QAM.
• Common trap: confusing the minimum frequency separation for coherent vs non-coherent detection — non-coherent requires 2x the separation.