OFDM and MIMO

In short

OFDM splits one high-rate data stream over many closely spaced, mutually orthogonal narrowband subcarriers generated by an IFFT, so a frequency-selective fading channel looks like a set of flat-fading subchannels and intersymbol interference is removed by a cyclic prefix. MIMO uses multiple transmit and receive antennas to send several data streams in the same time-frequency resource (spatial multiplexing) or the same stream over independent paths (diversity), multiplying capacity or reliability without extra bandwidth. Together, MIMO-OFDM is the physical layer of Wi-Fi (IEEE 802.11n/ac/ax), 4G LTE and 5G NR, while OFDM alone carries DVB-T and digital radio. It is a recurring area in university communications papers, in M.Tech/PhD entrance tests, and in interviews at Qualcomm, Samsung R&D, Nokia, Ericsson, Mediatek, Tejas Networks, Saankhya Labs, Signalchip, HFCL, C-DOT and ISRO/DRDO labs. Exam questions centre on orthogonality conditions, cyclic prefix sizing, PAPR, capacity formulas, Alamouti coding and linear detectors.

Explained

Why OFDM: the ISI problem

• A single-carrier symbol of duration Ts passing through a channel of delay spread τ_max suffers ISI when τ_max > Ts; equalisers then need many taps and become impractical at high rates.
• Coherence bandwidth B_c ≈ 1/(5σ_τ) (σ_τ = rms delay spread). If signal bandwidth B > B_c the channel is frequency selective; if B < B_c it is flat.
• OFDM fix: divide B into N subcarriers of width Δf = B/N so each Δf ≪ B_c. Each subcarrier sees flat fading, needing only a one-tap complex equaliser per subcarrier.
• Symbol duration stretches to T = N·Ts, so τ_max becomes a small fraction of T — robustness grows linearly with N.
• OFDM is a multicarrier scheme: it does not reduce total bandwidth, it reorganises it.

Orthogonality, IFFT/FFT and the OFDM chain

• Subcarriers at f_k = k·Δf with Δf = 1/T are orthogonal over one symbol: ∫₀ᵀ e^{j2πf_k t} e^{−j2πf_m t} dt = 0 for k ≠ m. Spectra overlap, yet each peak falls on the nulls of the others — no guard bands needed between subcarriers.
• Each subcarrier's spectrum is a sinc of first-null width Δf; hence the OFDM spectrum is nearly rectangular for large N.
• Transmitter chain: source bits → channel coding + interleaving → serial-to-parallel → QAM/PSK mapping per subcarrier (X₀…X_{N−1}) → N-point IFFT → add cyclic prefix → parallel-to-serial → DAC → RF up-conversion.
• Receiver chain: RF down-conversion → ADC → time/frequency synchronisation → remove CP → N-point FFT → channel estimation and one-tap equalisation Ŷ_k/Ĥ_k → demapping → de-interleave → decoding.
• The IFFT is the modulator: x[n] = (1/N) Σ X_k e^{j2πkn/N}. Complexity is O(N log₂N) instead of N oscillators and N filters — this is why OFDM became practical only with cheap DSP/VLSI.
• Hermitian symmetry of the IFFT input gives a real output — used in DMT for ADSL and in optical/visible-light OFDM.

Cyclic prefix, guard interval and overhead

• The last N_cp samples of the IFFT output are copied to the front. Two functions: (i) it absorbs the multipath tail, so there is no inter-symbol interference; (ii) it makes linear convolution with the channel look like circular convolution, so the FFT output is exactly Y_k = H_k X_k + W_k — no inter-carrier interference.
• Condition: T_cp ≥ τ_max (CP longer than the maximum excess delay). A CP shorter than the delay spread creates both ISI and ICI and an irreducible error floor.
• Cost: CP carries no new information → power and rate loss factor N/(N+N_cp). Typical CP overhead 1/32 to 1/4.
• Alternatives: zero padding (ZP-OFDM) saves power but needs overlap-add at the receiver; cyclic suffix and windowing reduce out-of-band emission.
• LTE example: Δf = 15 kHz, T = 66.67 µs, normal CP ≈ 4.69 µs (≈7%), extended CP ≈ 16.67 µs (25%) for large cells. 5G NR scales both by numerology µ: Δf = 15·2^µ kHz (15, 30, 60, 120, 240 kHz), slot time halving as Δf doubles.

Impairments: PAPR, synchronisation, Doppler

• PAPR (peak-to-average power ratio): summing N independent subcarriers gives a nearly Gaussian time signal with large peaks; theoretical worst case PAPR = N (10 log₁₀N dB), practical values 8–12 dB. Peaks drive the power amplifier into its nonlinear region, causing spectral regrowth and ICI; the PA must be backed off, cutting efficiency — critical for battery-powered uplinks.
• PAPR reduction: clipping and filtering, companding, selective mapping (SLM), partial transmit sequences (PTS), tone reservation, constellation shaping, and coding. LTE uplink avoids the problem entirely with SC-FDMA (DFT-spread OFDM), which pre-codes with a DFT and yields single-carrier-like PAPR; 5G NR uplink supports both CP-OFDM and DFT-s-OFDM.
• Carrier frequency offset (CFO): a frequency error ε (normalised to Δf) rotates and attenuates the wanted subcarrier and leaks energy into the others → ICI. OFDM is far more sensitive to CFO than single carrier; ε must be kept to a few percent of Δf. Corrected using cyclic-prefix correlation (van de Beek), training symbols (Schmidl–Cox) and pilot tracking.
• Symbol timing offset: tolerable inside the CP slack (T_cp − τ_max); beyond it, ISI appears.
• Phase noise and Doppler spread f_d also cause ICI; the design rule is Δf ≫ f_d (typically f_d/Δf < 1–2%), which is why high-mobility and mmWave systems use larger subcarrier spacing.
• Deep fades: a subcarrier in a notch is lost; hence OFDM is always paired with coding + frequency interleaving (COFDM) or adaptive bit/power loading.

OFDMA, resource allocation and standards

• OFDMA: different users get different subsets of subcarriers (and time slots), giving fine-grained multiple access and multi-user frequency-domain scheduling; the LTE/NR downlink unit is the resource block = 12 subcarriers × 1 slot (180 kHz at 15 kHz spacing). Wi-Fi 6 (802.11ax) introduced OFDMA with resource units (RUs).
• Adaptive modulation and coding (AMC)/link adaptation: CQI feedback selects QPSK/16-QAM/64-QAM/256-QAM and code rate per user per subband. Water-filling is the capacity-optimal power allocation when the transmitter knows the channel.
• Indian context: OFDM/MIMO systems operate in TRAI/DoT-regulated, WPC-assigned spectrum; 5G NR bands n78 (3.3–3.67 GHz), n28 (700 MHz) and n258 (26 GHz mmWave) were auctioned in 2022. Delicensed Wi-Fi bands are 2.4 GHz, 5 GHz and (after the 2025 delicensing of the lower 6 GHz band) 5925–6425 MHz, under DoT/WPC exemption rules; exact band edges and power limits are set by notification and can change. India's 5Gi/LMLC profile, standardised through TSDSI and folded into 3GPP Release 17, extends low-mobility large-cell coverage for rural deployment.
• Standards using OFDM: DVB-T/T2, DAB, DRM, ADSL/VDSL (DMT), 802.11a/g/n/ac/ax/be, WiMAX, LTE, 5G NR, powerline communication, 802.15.4g.

MIMO: channel model, diversity and multiplexing

• With N_t transmit and N_r receive antennas, the narrowband model is y = Hx + n, H an N_r × N_t complex matrix, n ~ CN(0, σ²I).
• Degrees of freedom = rank(H) ≤ min(N_t, N_r): the maximum number of parallel streams. Rich scattering → full rank; line-of-sight or closely spaced antennas → rank deficiency / spatial correlation and loss of multiplexing gain.
• Three gains: array (power) gain from coherent combining; diversity gain (max order N_t·N_r) which steepens the BER curve; multiplexing gain min(N_t,N_r) which scales rate. Diversity and multiplexing trade off (Zheng–Tse DMT).
• Receive diversity combining: selection combining, equal-gain combining, and MRC (maximum ratio combining, weights h_i*, optimal, SNR adds).
• Transmit diversity without channel knowledge — Alamouti 2×1 STBC: over two symbol periods send [s₁, s₂] then [−s₂*, s₁*]. The receiver recovers both symbols by simple linear combining and achieves full diversity order 2 with rate 1 and no CSI at the transmitter. Orthogonal STBCs with rate 1 exist for complex constellations only for N_t = 2; higher N_t needs rate < 1 (e.g. rate 3/4 for 3–4 antennas).
• Space-time trellis codes (STTC) add coding gain over STBC at higher decoding complexity. Space-frequency / space-time-frequency coding spreads across OFDM subcarriers.
• Spatial multiplexing (V-BLAST): independent streams per antenna; detection by ZF (W = (HᴴH)⁻¹Hᴴ, simple but noise-amplifying when H is ill-conditioned), MMSE (W = (HᴴH + σ²I)⁻¹Hᴴ, better at low SNR), SIC/OSIC (ordered successive cancellation, suffers error propagation), sphere decoding (near-ML at reduced complexity) and ML (exhaustive, M^{N_t} hypotheses — optimal but exponential).
• Precoding with CSIT: SVD of H = UΣVᴴ turns the channel into min(N_t,N_r) parallel scalar channels (eigenbeamforming) with water-filled power; codebook-based precoding and PMI/RI/CQI feedback are used in LTE/NR when full CSI is unavailable.
• Beamforming: digital, analogue and hybrid (mmWave, where RF chains are costly). MU-MIMO: one base station serves several users on the same resource, separated spatially (ZF/MMSE precoding, dirty-paper coding is the capacity-optimal but impractical bound).
• Massive MIMO: 64–256 antenna elements; channel hardening makes simple MF/ZF processing near-optimal, pilot contamination in TDD reuse is the main limit. Core to 5G NR mid-band (n78) deployments in India.
• MIMO-OFDM: apply MIMO processing per subcarrier — H_k is flat on each, so the matrix inversion/decoding is per-tone, which is why MIMO and OFDM are nearly always combined.
• Channel estimation: block-type and comb-type pilots, LS and MMSE estimators, DFT interpolation; LTE/NR use cell-specific/demodulation reference signals (CRS, DMRS, CSI-RS), and TDD exploits reciprocity.

Key points

• Orthogonality condition - subcarrier spacing Δf = 1/T, where T is the useful (FFT) symbol duration; any deviation causes ICI.
• IFFT = modulator - subcarrier generation is a frequency-to-time transform, O(N log N), no bank of oscillators.
• Cyclic prefix - a copy of the symbol tail prefixed to the symbol; T_cp ≥ τ_max removes ISI and converts linear to circular convolution, removing ICI.
• Flat per-subcarrier channel - Y_k = H_k X_k + W_k, so equalisation is one complex division per subcarrier.
• CP overhead - rate and power loss N/(N+N_cp); LTE normal CP ≈ 7%, extended CP 25%.
• PAPR - up to 10 log₁₀N dB; forces PA back-off; mitigated by clipping, SLM, PTS or by using SC-FDMA/DFT-s-OFDM.
• CFO sensitivity - frequency offset is the dominant OFDM impairment; keep ε = Δf_offset/Δf within a few percent.
• COFDM - OFDM without coding and interleaving is weak, because faded subcarriers are unrecoverable.
• OFDMA - subcarrier-group multiple access; LTE/NR resource block = 12 subcarriers × 1 slot.
• 5G numerology - Δf = 15 × 2^µ kHz; wider spacing resists Doppler and phase noise, shortens latency, but shortens CP.
• MIMO model - y = Hx + n; rank(H) ≤ min(N_t,N_r) sets the number of usable parallel streams.
• Diversity order - up to N_t·N_r; it changes the slope of the BER-versus-SNR curve, not just its offset.
• Multiplexing gain - capacity grows linearly with min(N_t,N_r) at high SNR, logarithmically with power.
• Alamouti STBC - full diversity 2, rate 1, linear decoding, no transmit CSI; the benchmark 2×1 transmit-diversity scheme.
• Detectors ranked - ML/sphere > OSIC > MMSE > ZF in performance; the reverse in complexity.
• SVD precoding + water-filling - capacity-optimal when the transmitter knows H.
• Massive MIMO/beamforming - many elements give array gain and spatial multiplexing; limited by pilot contamination and correlation.
• MIMO-OFDM - MIMO processing is applied independently on each OFDM subcarrier.

Formulas and examples

Example 1 — OFDM parameter design. Bandwidth B = 20 MHz, FFT size N = 2048, rms delay spread σ_τ = 0.5 µs with τ_max = 2 µs.

• Sampling rate = B = 20 MHz → T_s = 1/20 MHz = 0.05 µs.
• Useful symbol time T = N·T_s = 2048 × 0.05 = 102.4 µs; Δf = 1/T = 9.77 kHz.
• Coherence bandwidth B_c ≈ 1/(5 × 0.5 µs) = 400 kHz. Δf = 9.77 kHz ≪ 400 kHz → each subcarrier is flat. ✔
• CP must exceed 2 µs → N_cp ≥ 2/0.05 = 40 samples; choose N_cp = 64 → T_cp = 3.2 µs, T_sym = 105.6 µs.
• CP efficiency = 2048/2112 = 96.97% (3.03% loss).
• With 1200 used subcarriers, 64-QAM (6 bit/symbol), code rate 3/4:

R = 1200 × 6 × 0.75 / 105.6 µs = 5400/105.6 µs = 51.14 Mbit/s → η = 51.14/20 = 2.56 bit/s/Hz.

Example 2 — LTE resource grid and Doppler check. Δf = 15 kHz, normal CP, carrier 2.6 GHz, user speed 120 km/h.

• T = 1/15 kHz = 66.67 µs; T_cp ≈ 4.69 µs → T_sym ≈ 71.36 µs; 14 symbols per 1 ms subframe (two 0.5 ms slots of 7). CP overhead ≈ 4.69/71.36 = 6.57%.
• One resource block = 12 × 15 kHz = 180 kHz; a 20 MHz carrier has 100 RBs = 1200 subcarriers (18 MHz occupied, rest guard).
• Doppler: v = 120 km/h = 33.33 m/s → f_d = v·f_c/c = 33.33 × 2.6×10⁹/3×10⁸ = 288.9 Hz.
• f_d/Δf = 288.9/15,000 = 1.93% — acceptable; at 500 km/h (f_d ≈ 1204 Hz) the ratio is 8%, so a larger Δf (30 or 60 kHz numerology) is needed.

Example 3 — PAPR and PA back-off. N = 256 subcarriers.

• Theoretical worst-case PAPR = 10 log₁₀256 = 24.1 dB (all subcarriers adding in phase — probability vanishingly small).
• Practical 99.9-percentile PAPR for 256 subcarriers is about 10–11 dB; with a 1 dB margin the PA is backed off ≈ 11 dB from saturation.
• If the PA delivers 20 W (43 dBm) at saturation, the average transmit power is limited to 43 − 11 = 32 dBm ≈ 1.6 W, so PA efficiency drops roughly by the same factor — the reason LTE uplink uses SC-FDMA.

Example 4 — MIMO capacity comparison at ρ = 20 dB (ρ = 100).

• SISO: C = log₂(1 + 100) = 6.66 bit/s/Hz.
• 1×2 MRC (receive diversity, equal average gains): effective SNR doubles → C ≈ log₂(1 + 200) = 7.65 bit/s/Hz (only +1 bit — diversity helps reliability, not rate).
• 2×2 with HHᴴ = 2·I (ideal orthogonal channel): C = 2 log₂(1 + (100/2)×2) = 2 log₂(101) = 13.3 bit/s/Hz ≈ 2 × SISO.
• 4×4 ideal: C = 4 log₂(1 + (100/4)×4) = 4 log₂(101) = 26.6 bit/s/Hz.
• Takeaway: capacity scales with min(N_t,N_r), i.e. linearly in antennas, logarithmically in power.

Example 5 — Alamouti versus MRC and ZF detection.

• Alamouti 2×1: with h₁ = 0.6 + j0.8 (|h₁|² = 1.0) and h₂ = 0.8 − j0.6 (|h₂|² = 1.0), the effective gain after combining is |h₁|² + |h₂|² = 2.0, diversity order 2. Transmit power is split over two antennas, so the array gain is 3 dB less than 1×2 MRC, which has the same diversity order 2 — Alamouti matches MRC's slope but not its SNR offset.
• ZF for a 2×2 channel H = [[1, 0.9],[0.9, 1]]: HᴴH = [[1.81, 1.8],[1.8, 1.81]], det = 1.81² − 1.8² = 3.2761 − 3.24 = 0.0361. Per-stream noise enhancement is [(HᴴH)⁻¹]_ii = 1.81/0.0361 ≈ 50× (≈17.0 dB) — an ill-conditioned (highly correlated) channel cripples ZF, and MMSE or ML should be used.

OFDM transmitter (one MIMO branch)
bits -> FEC+interleave -> S/P -> QAM map -> [ IFFT (N) ] -> add CP -> P/S -> DAC -> RF
RX:  RF -> ADC -> sync (timing, CFO) -> remove CP -> [ FFT (N) ] -> per-tone MIMO
     detection (ZF/MMSE/ML with Ĥ_k from pilots) -> demap -> deinterleave -> FEC decode

Where it is used

• Telecom equipment and 5G R&D: base-station and UE physical-layer design at Nokia (Chennai/Bengaluru), Ericsson, Samsung R&D Bengaluru/Noida, Qualcomm India, Mediatek, and at Indian OEMs Tejas Networks, HFCL, VVDN and Saankhya Labs/Signalchip; C-DOT, IIT-Hyderabad-incubated WiSig Networks and the IIT-Madras-led indigenous 5G test bed work on 5G NR and 5Gi stacks under the DoT's programme.
• Operator network engineering: Jio, Airtel and BSNL/Vi radio-planning and optimisation roles use OFDMA scheduling, numerology choice, MIMO rank/beamforming configuration and massive-MIMO site tuning; TRAI/DoT spectrum constraints drive the parameter choices.
• Wi-Fi and consumer SoCs: 802.11ax/be chipsets and routers (MU-MIMO, OFDMA RUs), smart-TV and set-top tuners, DVB-T2 broadcasting.
• Broadcast and wireline: Doordarshan/Prasar Bharati DVB-T2 trials and digital radio; ADSL/VDSL DMT in BSNL copper broadband; powerline communication in smart meters.
• Defence, space and sensing: DRDO/LRDE software-defined radios and MIMO radar; ISRO/ISTRAC telemetry links; OFDM waveforms in tactical data links.
• Job roles: PHY/DSP algorithm engineer, FPGA/ASIC design (FFT, channel estimator, MIMO detector IP), RF systems engineer, 3GPP conformance and protocol test (Keysight, Rohde & Schwarz, Anritsu labs in India), network optimisation engineer.

Interview questions

• Q: Why are OFDM subcarriers orthogonal even though their spectra overlap? A: Spacing them at Δf = 1/T makes each subcarrier's sinc spectrum have nulls at every other subcarrier's centre frequency, so the inner product over one symbol period is zero.
• Q: What exactly does the cyclic prefix do? A: It absorbs the multipath delay spread so successive symbols do not overlap (no ISI), and it makes the channel convolution circular so the FFT gives Y_k = H_k X_k with no inter-carrier interference.
• Q: What happens if the cyclic prefix is shorter than the channel delay spread? A: Both ISI and ICI appear, orthogonality is lost and the BER hits an irreducible floor that extra transmit power cannot remove.
• Q: Why is PAPR a problem and how is it handled in the LTE uplink? A: High peaks force the power amplifier into back-off, wasting battery and causing spectral regrowth; the LTE uplink uses SC-FDMA (DFT-spread OFDM), which has single-carrier-like PAPR.
• Q: Which impairment is OFDM most sensitive to, and why? A: Carrier frequency offset — it destroys subcarrier orthogonality and leaks power between subcarriers as ICI, so the offset must be kept to a few percent of the subcarrier spacing.
• Q: How is the number of subcarriers chosen? A: Δf must be much smaller than the coherence bandwidth (flat per-tone channel) but much larger than the Doppler spread and phase-noise bandwidth; latency and FFT complexity bound N from above.
• Q: Difference between diversity gain and multiplexing gain? A: Diversity sends the same information over independent paths and steepens the BER-versus-SNR slope (reliability); multiplexing sends different streams in parallel and raises data rate linearly with min(N_t,N_r).
• Q: Why is the Alamouti scheme attractive? A: It gives full diversity order 2 at rate 1 with simple linear decoding and needs no channel knowledge at the transmitter.
• Q: When does MIMO fail to deliver capacity gain? A: When H is rank-deficient or ill-conditioned — strong line-of-sight, closely spaced or correlated antennas, or keyhole channels — and when SNR is too low for multiple streams.
• Q: Compare ZF and MMSE detectors. A: ZF fully nulls inter-stream interference but amplifies noise when HᴴH is ill-conditioned; MMSE balances interference and noise and performs better at low SNR, converging to ZF at high SNR.
• Q: Why is MIMO almost always combined with OFDM? A: Each subcarrier sees a flat channel, so the MIMO detection or precoding reduces to a small per-tone matrix operation instead of complex space-time equalisation over a frequency-selective channel.
• Q: What limits massive MIMO performance? A: Pilot contamination from reuse of the same pilots in neighbouring cells, channel estimation error, spatial correlation, hardware impairments and the cost of many RF chains (hence hybrid beamforming at mmWave).
• Q: What is a 5G NR numerology and why does it exist? A: It is the scalable subcarrier spacing Δf = 15 × 2^µ kHz; wider spacing suits high Doppler, mmWave phase noise and low latency, while narrower spacing suits large delay spread and wide coverage.

Common mistakes

• Saying OFDM "removes" fading — it converts frequency-selective fading into per-subcarrier flat fading; deep-faded tones still need coding and interleaving.
• Confusing the FFT size N with the number of used subcarriers (guard tones and the DC null are unused), which gives a wrong data rate and spectral efficiency.
• Using the total symbol time (T + T_cp) instead of the useful time T when computing Δf = 1/T, or forgetting CP overhead in rate calculations.
• Assuming the cyclic prefix only fights ISI and forgetting the circular-convolution role that prevents ICI.
• Treating ZF as always adequate — ignoring noise amplification when the MIMO channel is correlated or ill-conditioned.
• Claiming MIMO capacity grows with N_t alone; it grows with min(N_t, N_r) and only in a sufficiently scattering, full-rank channel.
• Mixing up diversity and multiplexing: quoting an N_t·N_r "rate gain" for Alamouti (its rate is 1) or a BER-slope improvement for plain spatial multiplexing.
• Unit slips in design problems: µs with MHz, Hz with kHz, and forgetting f_d = v·f_c/c needs v in m/s.

Related topics

Sampling and digital modulation · Information theory and coding · Noise in communication systems · Fourier, Laplace and Z-transforms · Spread spectrum and CDMA · Telecommunication networks and switching