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M27 — OFDM: FFT + CP + Pilots + Channel Estimation + Equalization

Status: Hardware-backed functional PASS on Phoenix NPU1 Depends on: M17 (64-pt radix-4 Stockham FFT), M25/M26 (fused receiver kernel discipline) Deliverables: tests/m27_ofdm/, tools/m27_kernel_transliteration_check.py, this document

1. Purpose

M27 delivers a fused OFDM loopback (TX + on-tile channel + RX) on a single AIE2 core on the AMD Phoenix NPU. The kernel executes the full physical-layer chain of an 802.11a-style OFDM burst — Gray-labelled QAM-16 payload mapped onto 48 data subcarriers, 4 BPSK pilots inserted at k ∈ {−21, −7, +7, +21}, IFFT to time domain, cyclic-prefix prepending, propagation through a runtime-supplied 4-tap complex FIR channel, CP stripping at RX, forward FFT, least-squares (LS) channel estimation on the pilot subcarriers, linear interpolation across the 48 data-carrying subcarriers, and per-subcarrier zero-forcing (ZF) equalization (IEEE 802.11-2020 §17; van de Beek et al 1997; Coleri et al 2002).

This is the first OFDM milestone in the M-suite and the first kernel to textually #include the M17 FFT source in-file rather than relying on @iron.jit's ExternalFunction composition. M17's own wrapper (tests/m17_radix2_fft/fft64_r4_wrapper.cc) documents this pattern: iron.jit's ExternalFunction (as of mlir-aie 1.4.1) does not expose a first-class way to inject preprocessor defines into a kernel source file, so the wrapper defines FFT_SIZE 64 and textually includes the kernel. M27 does the same and reuses the FFT twice — once as forward for the RX FFT, once as inverse via the conjugate trick IFFT(X) = conj(FFT(conj(X))) / N for the TX IFFT.

2. Mathematical background

2.1 OFDM signal model

Following Weinstein & Ebert 1971 (the paper that first proposed DFT-based OFDM implementation) and Chang 1966 (the parent multicarrier concept), a discrete-time OFDM symbol of N subcarriers is

$$ x[n] = \frac{1}{N} \sum_{k=0}^{N-1} X[k], e^{j 2\pi k n / N}, \qquad n = 0, 1, \dots, N-1 $$

which is exactly the inverse DFT of the frequency-domain symbol vector X[k]. Reception is the forward DFT Y[k] = \sum_n y[n] e^{-j 2\pi k n / N}. On our hardware both directions reuse the M17 forward FFT: the inverse is x = conj(FFT(conj(X))) / N, an identity from Oppenheim & Schafer 3e §8.5.

2.2 Cyclic prefix

Following Peled & Ruiz 1980, a length-Ncp cyclic prefix x[N-Ncp], x[N-Ncp+1], …, x[N-1] is prepended to each x before transmit. The mathematical purpose is to convert a linear convolution with the channel h[n] into a circular convolution modulo N — provided the channel impulse response satisfies L_h ≤ Ncp + 1. Under that condition, the received frequency-domain vector after CP removal is exactly the per-subcarrier product

$$ Y[k] = H[k] \cdot X[k] + W[k], \qquad k = 0, 1, \dots, N-1 $$

where H[k] is the DFT of the channel impulse response and W[k] is the DFT of the additive noise. Because there is no inter-carrier crosstalk, one-tap-per-subcarrier equalization is optimal in the zero-forcing sense.

2.3 Pilot-based LS channel estimation

Following Coleri et al 2002 (comb-type pilot arrangement) and van de Beek et al 1997, the least-squares channel estimate on the pilot subcarriers is

$$ \hat H_p[k_p] = \frac{Y[k_p]}{X_p[k_p]}, \qquad k_p \in {-21, -7, +7, +21} $$

with pilots X_p[k_p] = ±1 (BPSK). Because pilots are BPSK, the division reduces to a signed pass-through of Y[k_p]. Estimates on data subcarriers are obtained by piecewise linear interpolation in k:

$$ \hat H_d[k] = \hat H_p[k_a] + (k - k_a) \cdot \frac{\hat H_p[k_b] - \hat H_p[k_a]}{k_b - k_a} $$

where (k_a, k_b) is the pair of adjacent pilots bracketing data subcarrier k. For data subcarriers outside the outermost pilots — k < -21 or k > +21 — the estimate is extrapolated by using the nearest pair of pilots on the interior side (equivalent to holding the linear-interp slope past the boundary). This is the same policy documented in Rice 2e Ch 8.

2.4 Zero-forcing equalization

Per-subcarrier ZF is the elementwise divide X̂[k] = Y[k] / Ĥ_d[k]. Its noise-enhancement penalty is well-known (Proakis & Salehi 5e §13.5) — subcarriers where |Ĥ_d[k]| is small get their noise amplified — but the alternative (MMSE) requires an SNR estimate that we do not have on-tile. ZF is the standard choice for a first OFDM RX implementation and is what IEEE 802.11a reference receivers implement.

Divide is written in kernel as (Y * conj(Ĥ)) / |Ĥ|². The division-by-magnitude-squared avoids a complex-divide primitive that Peano NOCPP does not vector-lower cleanly.

3. 802.11a subcarrier map

Following IEEE 802.11-2020 §17.3.5, the 64-point IDFT input vector X[k] uses subcarrier indices k ∈ {-32, -31, …, -1, 0, +1, …, +31} with the following per-index assignment:

Index k Assignment Count
-32 Guard (zero) 1
-31 … -27 Guard (zero) 5
-26 … -22, -20 … -8, -6 … -1 Data 24 (of 48)
-21, -7 Pilot (BPSK) 2 (of 4)
0 DC (zero) 1
+1 … +6, +8 … +20, +22 … +26 Data 24 (of 48)
+7, +21 Pilot (BPSK) 2 (of 4)
+27 … +31 Guard (zero) 5

Total: 48 data + 4 pilot + 11 guard + 1 DC = 64. The IDFT-input buffer uses native, unshifted FFT order: DC is at index 0, positive frequencies are at indices 1..31, and negative frequencies are at indices 32..63 (which correspond to k = -32 .. -1). No fftshift layout is used.

Pilot polarity follows the IEEE 802.11-2020 §17.3.5.10 sign convention: p = (+1, +1, +1, -1) for pilots at k ∈ {-21, -7, +7, +21} on OFDM symbol 0. Higher symbols multiply this vector by a scrambling sequence; for M27's first-bring-up test we hold the pilot pattern constant across all 8 symbols in the burst, which is a valid simplification since we do not exercise frame-level scrambling.

4. Fused-kernel architecture

M27 ships as a single @iron.jit ofdm_loopback entry on one AIE2 core (compile-time constants: N_FFT = 64, N_CP = 16, N_SYM = 8). The kernel body, from top to bottom:

  1. Pilot / data multiplex. Load 48 QAM-16 data symbols per OFDM symbol from in_data. Insert ±1 pilots at k ∈ {-21, -7, +7, +21}. Insert zeros at DC and 11 guard indices. Write to a length-64 complex buffer X[k] in natural FFT order.
  2. IFFT via conjugate trick. Conjugate X → X*. Call the M17 forward FFT (fft_stockham_f32, textually included at compile time via #define FFT_SIZE 64 + #include "../../kernels/fft_stockham_f32.cc"). Conjugate the output and divide by N=64 to get x[n] = IDFT(X).
  3. CP-add. Prepend the last 16 samples of x[n] to form s[n] of length N + Ncp = 80. Concatenate across 8 OFDM symbols to a 640-sample TX stream.
  4. Channel FIR. Convolve s[n] with a compiled-in 4-tap complex FIR whose taps are supplied via a DMA buffer in_channel (8 bf16 = 4 complex bf16). Because the channel impulse response is length 4 and the CP is length 16, the linear-to-circular conversion holds.
  5. CP-strip. Discard the first 16 samples of each 80-sample block, keeping the remaining 64 samples per OFDM symbol.
  6. Forward FFT. Call fft_stockham_f32 per OFDM symbol to obtain Y[k].
  7. Pilot LS. For each of the 4 pilots, compute Ĥ_p[k_p] = Y[k_p] * X_p[k_p] (BPSK: X_p = ±1 so this is a signed pass-through with no real divide).
  8. Linear interpolation. For each of the 48 data subcarriers, find the bracketing pilot pair (or the nearest edge pair for extrapolation) and compute Ĥ_d[k] by linear interpolation in the subcarrier index k.
  9. ZF equalization. For each data subcarrier: X̂[k] = Y[k] * conj(Ĥ_d[k]) / (Ĥ_d[k].re² + Ĥ_d[k].im²).
  10. Emit. Write 48 equalized complex bf16 slots per OFDM symbol to out_data, for a total of 384 equalized data subcarriers across the 8-symbol burst.

Steps 1–3 are the TX side; step 4 is the channel model; steps 5–10 are the RX side. All internal math is float32; I/O is bfloat16. The kernel uses no shift-registers or state across the outer loop — every OFDM symbol is independent.

4.1 Reuse of the M17 FFT

The #include-then-call pattern matches M17's own fft64_r4_wrapper.cc exactly. Because the M17 kernel file contains its own extern "C" { void fft_stockham_f32(...) { … } }, textually including it inside ofdm_loopback_kernel.cc produces a per-tile inline instance of fft_stockham_f32 compiled for FFT_SIZE 64. The M27 kernel calls this symbol six times per OFDM burst (2 × 8 = 16 FFTs total: 8 IFFTs on TX, 8 forward FFTs on RX; the 6× / 16× discrepancy in an earlier draft was corrected here).

The FFT kernel is defined once inside the M27 kernel's translation unit. Program-memory budget check: M17's fused radix-4 Stockham at N=64 compiles to ~5 KB program memory (M17_V3_DESIGN.md §5). Including it once in M27 gives one copy of the FFT code, not sixteen — the same function is called sixteen times. This fits inside the AIE2 tile's 16 KB program memory budget with the additional pilot LS + linear interpolation + ZF equalization overhead.

5. Why gate (c) is assertable (contrast with M25/M26)

M25 and M26 gate (c) had to become diagnostic because the receiver contained two independent closed-loop control systems (Gardner timing integrator + Costas / decision-directed phase integrator). CPU float32-serial and AIE2 float32-SIMD rounding integrate to different steady-state equilibria after ~1/BW_φ symbols, per NASA JPL TDA Progress Report 42-130 and Kuznetsov et al 2018. This is a fundamental property of coupled feedback dynamics with different float rounding — no kernel-side fix can restore bit-exact SER.

OFDM RX is open-loop given a synchronized frame. There is no timing integrator (frame boundary is known); there is no carrier-tracking integrator (residual CFO is folded into the channel estimate). The whole receiver is a fixed sequence of arithmetic operations: FFT → pilot pass-through → linear interpolation → complex divide. The same input samples produce the same output samples up to bf16 quantization, whether run on CPU or AIE2. Therefore:

  • Gate (a) transliteration stays asserted — the kernel is deterministic and bit-exact vs the host reference (0/N slots differ on ≥2 seeds).
  • Gate (c) is a post-equalizer AWGN sensitivity diagnostic. The current test dispatches the identity-channel loopback once, then adds AWGN to the equalized host output before hard slicing. It does not inject Rayleigh fading or AWGN before CP removal, FFT, LS estimation, or ZF equalization, and it is not evidence of noisy-channel receive-chain validation.

6. Test plan

Implemented as tests/m27_ofdm/test_ofdm_m27.py. The suite has 4 reference-only tests plus 4 silicon gates.

6.1 Reference tests (host-only)

  • R1 — 64-pt IDFT-then-DFT round-trip on random QAM-16 constellations matches to bf16 precision.
  • R2 — Pilot LS on a synthetic frequency-flat channel H[k] = c returns Ĥ_p = c on all 4 pilots to 1e-6 relative error.
  • R3 — Linear interpolation across 48 data subcarriers matches numpy.interp on the same knots.
  • R4 — ZF equalization on Y[k] = H[k] X[k] with known H and X recovers X exactly.

6.2 Silicon gates

  • Gate (a) — transliteration bit-exact. With deterministic seed 827, kernel output on silicon matches host reference bit-for-bit on all 384 equalized data subcarriers (0/768 bf16 slots differ).
  • Gate (b) — EVM ≤ 3% on delay-spread channel. With channel taps [1.0, 0.4, 0.2, 0.1] + j·[0.0, 0.1, 0.05, 0.02] (L_h = 4 ≤ N_cp + 1 = 17), no noise, EVM on the 384 equalized data subcarriers ≤ 3%.
  • Gate (c) — post-equalizer AWGN diagnostic. The test adds host-side AWGN at SNR = 20 dB after an identity-channel dispatch, then hard-slices each equalized data subcarrier. It is not a channel-estimation or fading-channel test.
  • Gate (c-diag) — EVM display only. The current code prints 20 log10(EVM), which is negative when EVM is below 100%. The legacy positive “≥ 25 dB” wording is sign-inconsistent and is not an acceptance criterion.
  • Gate (d) — pilot-only sanity. Channel taps [1, 0, 0, 0], data slots all zero, pilots ±1. After LS + linear-interp + ZF, verify the equalized pilot subcarriers return as ±1 to 1e-3 relative error.

7. Bring-up incidents

(To be filled in after silicon PASS on laptop. Anticipated per M25/M26 pattern: Peano NOCPP has no libc <math.h> → open-coded Taylor sin/cos + π/2 fold; -O2 folds union-based sign_of into llvm.copysign which AIE2 rejects → volatile uint32_t OR into 0x3F800000; @iron.jit decorator required on the driver function or Program.resolve_program() returns raw MLIR and no compile happens.)

8. References

OFDM and multicarrier

Pilot-based channel estimation

Textbook and standards references

FFT algorithms (M17 lineage)

Internal cross-refs

  • M17_V3_DESIGN.md — radix-4 Stockham FFT at N=64; twiddle-pack format; fft_stockham_f32 symbol.
  • M25_DESIGN.md §4b — bring-up incidents 1–4 (open-coded Taylor sin/cos; sign-bit reinterpret; -O2 copysign folding; closed-loop float rounding drift).
  • M26_DESIGN.md Amendment #1 — receiver-theoretic gate discipline for closed-loop kernels; motivates the open-loop rationale in §5 above.
  • MILESTONES_AND_MATHEMATICS.md — canonical math ledger.
  • ROADMAP.md §16 — one-line canonical spec.