Questions: Frequency Shift Keying Modulation

Question 1

A binary FSK system has symbol period T = 2 ms. What is the minimum frequency separation Δf that achieves tone orthogonality for coherent matched-filter demodulation?

• A. Δf = 1/T = 500 Hz — one full cycle difference per symbol period
• B. Δf = 1/(2T) = 250 Hz — minimum separation satisfying the orthogonality integral
• C. Δf = 2/T = 1000 Hz — two cycles needed for reliable tone discrimination
• D. Δf can be arbitrarily small if matched filters are designed with sufficient precision

Answer: (B)

Tone orthogonality requires ∫₀ᵀ cos(2πf₁t)cos(2πf₀t)dt = 0, which holds when Δf = n/(2T) for integer n. The minimum is n = 1, giving Δf = 1/(2T) = 250 Hz for T = 2 ms — this defines Minimum Shift Keying (MSK). Option A (1/T) achieves orthogonality but wastes bandwidth. Option D is wrong: orthogonality is a mathematical condition on Δf determined by the integral, not something achievable through filter design at arbitrary separation.

Question 2

Bluetooth Classic uses GFSK with a bandwidth-time product BT = 0.5 instead of standard binary FSK. What tradeoff does this choice make?

• A. It trades lower bit error rate for higher bandwidth — GFSK is less spectrally efficient but more reliable
• B. It trades a small amount of inter-symbol interference for significantly reduced spectral bandwidth
• C. It gains both lower bit error rate and lower bandwidth — GFSK is a strict improvement over standard FSK
• D. It trades phase continuity for bandwidth — GFSK eliminates carrier phase discontinuities that standard FSK produces

Answer: (B)

The Gaussian pre-filter smooths frequency transitions between symbols, dramatically reducing spectral sidebands caused by abrupt rectangular transitions. The cost is mild inter-symbol interference: the smooth Gaussian tails of adjacent symbols overlap slightly. GFSK is not a strict improvement — standard FSK has zero ISI, while GFSK introduces a small ISI penalty. The tradeoff is worthwhile for Bluetooth because the congested 2.4 GHz ISM band demands spectral containment, and the ISI is manageable without complex equalization.

Question 3

In binary FSK, when the two carrier tones are orthogonal over the symbol period, a matched filter tuned to frequency f₁ produces exactly zero output when frequency f₀ is transmitted.

• T. True
• F. False

Answer: True

This is precisely the value of orthogonality in FSK. The matched filter for f₁ computes the inner product of the received signal with a reference sinusoid at f₁. When the transmitted signal is at f₀ and the tones are orthogonal (their inner product over T is zero), the matched filter output is exactly zero — no cross-talk. This is why orthogonality is the key design criterion: it enables perfect separation by matched filters in the absence of noise.

Question 4

Non-coherent FSK demodulation is more reliable than coherent FSK demodulation because it avoids the carrier phase estimation errors that degrade coherent detection.

• T. True
• F. False

Answer: False

Non-coherent detection is simpler but less reliable. Coherent detection uses the full phase information of the received signal, extracting maximum signal energy per bit. Non-coherent detection uses only amplitude (envelope detection), discarding phase information — this costs approximately 3 dB in required SNR for the same bit error rate. Non-coherent detection is chosen for implementation simplicity and phase noise tolerance, not for reliability. The statement reverses the tradeoff: coherent is more reliable; non-coherent is easier.

Question 5

Explain why the orthogonality condition determines the minimum frequency separation in FSK, and what happens to demodulation performance when the tones are not orthogonal.

Model answer: Matched filter demodulation computes the inner product of the received signal with a reference tone. When f₁ and f₀ are orthogonal over the symbol period, the matched filter for f₁ produces a nonzero output only when f₁ was transmitted, and zero when f₀ was transmitted — clean discrimination with no cross-talk. When tones are not orthogonal, their inner product is nonzero, so the matched filter for f₁ produces a partial output even when f₀ is transmitted. This cross-talk adds to noise and reduces the effective SNR, increasing bit error rate. The minimum separation Δf = 1/(2T) achieves orthogonality with the smallest possible frequency gap, maximizing bandwidth efficiency while preserving clean detection.

Orthogonality gives the matched filter receiver exact zero rejection of the wrong tone — not just small, but exactly zero. Once orthogonality is violated, cross-talk scales with the overlap integral, which grows as Δf decreases below 1/(2T). This is a threshold condition: satisfy orthogonality and get clean detection regardless of filter quality; violate it and face a fundamental sensitivity penalty that no filter design can eliminate.