9. End-to-End Stories

9.1 Active noise cancellation (Bose, AirPods)

1. Reference mic (outside the earbud) captures ambient noise.
2. Error mic (inside, near the eardrum) captures the residual at the ear.
3. DSP runs an adaptive FIR (LMS) that learns the transfer function from the reference mic to the ear, in real time.
4. Output = predicted ear-canal noise, inverted in sign.
5. Speaker plays the anti-noise; sound waves cancel acoustically.

The DSP's challenge: total latency (mic → ADC → algorithm → DAC → speaker) must be much less than the 10 ms travel time of sound across the ear, otherwise cancellation fails. This is why ANC chips run at 768 kHz or higher with very short FFT/filter blocks.

9.2 Wi-Fi OFDM modem

1. TX: input bits are mapped to QAM symbols (16-QAM, 64-QAM, 256-QAM, ...). Symbols populate sub-carriers in a vector. IFFT of the vector produces a time-domain OFDM symbol. Cyclic prefix is prepended. Out the antenna.
2. RX: receive samples. Strip cyclic prefix. FFT of the symbol. Each sub-carrier is now an isolated QAM constellation point, possibly rotated by the channel. Per-sub-carrier equalization (one complex multiply each) corrects.
3. Decode bits.

Wi-Fi 6 (802.11ax) uses 4096 sub-carriers and 1024-QAM, 4096-point FFTs at every symbol time (around 12 microseconds). The FFT is so central that an OFDM modem is basically a high-throughput FFT engine.

9.3 Sigma-delta ADC for audio

1. Input audio at, say, 48 kHz Nyquist (24 kHz max signal).
2. Modulator at $128 \times 48 = 6.144$ MHz. A 1-bit feedback loop with high-order noise-shaping, pushing quantization noise out of the audio band into the ultrasonic range.
3. Decimation chain: a CIC filter (coarse, low-cost) gets us partway down; cascaded half-band FIRs (sharp, multirate) take it from there.
4. Output: 24-bit, 48 kHz audio with $\sim 110$ dB SNR.

The sigma-delta modulator is a control loop driving its quantization error away from the audio band; the decimation chain is multirate DSP filtering it out. Together, they produce audio precision from a 1-bit quantizer.

9.4 GPS receiver

1. RF front end mixes the satellite L1 signal (1575.42 MHz) down to baseband.
2. ADC at, say, 2 MS/s.
3. Acquisition: for each candidate satellite (each has a unique pseudorandom code), cross-correlate the received signal against a locally-generated copy of the code, sliding in both time and Doppler frequency. The peak of the 2D correlation surface tells you the satellite is visible and gives you initial timing/frequency estimates.
4. Tracking: narrow phase-locked loops keep the lock.
5. Navigation: decode the satellite's broadcast message; combine timing from $\geq 4$ satellites to triangulate position.

The cross-correlation in step 3 is precisely the matched-filter operation we discussed in Section 4 of Chapter 3 (and its cousin in DPA Section 8.1 here). GPS works because cross-correlation can pull a signal $\sim 30$ dB below the noise floor out of the noise. Same math, different application, same answer: cross-correlation is the optimal detector for a known signal in additive Gaussian noise.

9.5 Software-defined radio

1. Wideband ADC captures, say, 60 MHz of spectrum at 200 MS/s (complex I+Q).
2. Tune to a channel by complex multiplication with a numerically-controlled local oscillator.
3. Decimation filter (polyphase) reduces sample rate to channel bandwidth.
4. Demodulation (whatever the protocol requires: AM, FM, FSK, PSK, OFDM).
5. Decoding.

All in software (CPU, GPU, or FPGA). One radio replaces a thousand specialized analog receivers. SDR has revolutionized amateur radio, signals intelligence, military communications, and academic research.