Prerequisites & Notation
Before You Begin
The previous twenty-five chapters built an elegant machinery for massive MIMO on paper: favorable propagation, channel hardening, tight capacity bounds, closed-form precoders, sharp CRBs on sensing. Every result rested on three silent assumptions — that the symbols arrive perfectly synchronized, that TDD reciprocity is exact, and that the radio hardware is linear, noise-free, and temperature-stable. A testbed is the place where these assumptions collide with physics. In this chapter we assume the reader is fluent enough in the theoretical side to recognize, for each impairment introduced below, exactly which equation of the prior chapters it breaks.
- Massive MIMO TDD operating point: pilot training, channel estimation, reciprocity-based downlink precoding(Review ch03)
Self-check: Can you sketch the TDD frame, identify the pilot phase, and explain why requires calibration of the RF chains?
- Linear detection and precoding complexity for a K-user uplink with N-antenna array(Review ch06)
Self-check: Count the complex multiplications per slot to form the MR, ZF, and MMSE combiners. Which scales like , which like ?
- OFDM waveform and 5G NR numerology: subcarrier spacing, symbol duration, slot length(Review ch10)
Self-check: What is the slot duration at kHz numerology, and how many OFDM symbols does it contain?
- 5G NR MIMO framework: CSI-RS, SRS, beam management procedures(Review ch22)
Self-check: Can you explain how the gNB uses uplink SRS to estimate the downlink channel in TDD, and where reciprocity calibration fits in?
- Cell-free architecture: distributed access points, fronthaul, shared time/frequency reference(Review ch11)
Self-check: Can you explain why distributed APs need a shared timebase much more stringently than a centralized array does?
- Digital signal processing in fixed-point arithmetic: quantization noise, dynamic range, headroom
Self-check: If a signal with 60 dB dynamic range is multiplied in a 16-bit fixed-point accumulator, how many bits of headroom do you need to avoid saturation?
- Thermal noise floor and basic link budget calculations(Review ch02)
Self-check: What is the kTB noise power in dBm for a 100 MHz channel at room temperature, and why does this set a hard floor on calibration accuracy?
Notation for This Chapter
Symbols introduced or specialized in this chapter. Customizable symbols use tokens; the values shown are the defaults from the notation registry. See the NGlobal Notation Table master table.
| Symbol | Meaning | Introduced |
|---|---|---|
| Number of base-station transmit antennas on the testbed array | s01 | |
| Number of single-antenna user-equipments served simultaneously | s01 | |
| Instantaneous RF bandwidth of the testbed (Hz) | s01 | |
| Carrier frequency of the testbed front end | s01 | |
| Uplink channel matrix observed at the BS array, | s02 | |
| Duration of one 5G NR slot (for 30 kHz numerology, s) | s02 | |
| Number of bits in the fixed-point mantissa used by the MAC pipeline | s02 | |
| TDD reciprocity calibration matrix — a diagonal phase/amplitude correction on the BS antennas, | s03 | |
| Diagonal matrices of transmit and receive RF-chain responses (gain and phase per antenna) | s03 | |
| Interval between reciprocity calibration updates | s03 | |
| Residual timing offset between an AP and the reference timebase | s04 | |
| Residual carrier frequency offset after coarse acquisition | s04 | |
| Phase-noise process of the local oscillator (rad) | s04 | |
| Thermal-plus-quantization-plus-EVM noise, | s02 | |
| Effective SNR after all testbed impairments (hardware EVM, quantization, residual offsets) | s02 | |
| Residual calibration error (rms fraction of the signal amplitude) | s03 | |
| Calibration pilot signal injected during the calibration window | s03 |