How does a frequency hopping waveform provide protection against narrowband jamming?

Frequency hopping (FH) spread spectrum provides protection against narrowband jamming by rapidly switching the carrier frequency across a wide band, ensuring that a jammer can only affect a small fraction of the transmitted hops: (1) Principle: the transmitter hops across N frequencies within a total hopping bandwidth W. Each hop occupies a narrow instantaneous bandwidth B_hop for a short dwell time T_hop. The hopping pattern is determined by a pseudorandom sequence known only to the transmitter and receiver. A narrowband jammer (bandwidth B_jam) can only jam the frequencies within B_jam at any given time. The fraction of hops that are jammed: f_jammed = B_jam / W. If B_jam = 1 MHz and W = 100 MHz: only 1% of hops are jammed. The receiver discards the jammed hops (which appear as errors) and reconstructs the message from the unjammed hops. (2) Processing gain: the anti-jam processing gain of FH: PG = 10 × log10(W / B_hop) (in dB). For W = 200 MHz and B_hop = 25 kHz: PG = 10 × log10(8000) = 39 dB. This means the jammer must be 39 dB stronger than the desired signal (within the hopping band) to jam all hops simultaneously. (3) Jammer strategies: spot jammer (narrowband): jams one frequency at a time. Only effective against the fraction f = B_jam/W of hops. Follower jammer: detects the hop frequency and tunes the jammer to that frequency. Must detect, tune, and transmit within the hop dwell time T_hop (typically 1-10 ms for slow hopping, < 100 us for fast hopping). If the follower cannot react within T_hop: it misses the hop. Barrage jammer: spreads its power across the entire hopping band W. The jammer power per hop frequency: P_jam_per_hop = P_jam_total × (B_hop / W). This is W/B_hop times less than a spot jammer of the same total power. The barrage jammer is the least efficient against FH. (4) Error correction: FH systems use forward error correction (FEC) coding to recover from the occasional jammed hops. If 10% of hops are jammed: the FEC can correct the resulting errors. If 30-50% of hops are jammed: stronger FEC (lower code rate) is needed, reducing the data throughput. The system fails when the jammed fraction exceeds the FEC correction capability.