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0000 diff --git a/9_Firmware/9_2_FPGA/plfm_chirp_controller.v b/9_Firmware/9_2_FPGA/plfm_chirp_controller.v index d0f9313..f61b767 100644 --- a/9_Firmware/9_2_FPGA/plfm_chirp_controller.v +++ b/9_Firmware/9_2_FPGA/plfm_chirp_controller.v @@ -115,8 +115,21 @@ always @(posedge clk_120m) begin end // Short PLFM chirp LUT initialization (too small for BRAM, keep inline) +// +// TX-I (analyzed 2026-04-28; tb/cosim/analyze_short_chirp_mismatch.py): +// 60 samples @ fs_dac=120 MHz over 0.5 us, real-valued passband. +// Hilbert analysis confirms an UPCHIRP from ~10 to ~30 MHz baseband +// (BW ~19.4 MHz). The old comment "30MHz to 10MHz" had the sweep +// direction reversed and is corrected below. +// +// End-to-end frequency plan (from adf4382a_manager.h + ddc_400m.v): +// TX upmix: LO=10.500 GHz, high-side -> RF: 10.510..10.530 GHz +// RX downmix: LO=10.380 GHz, high-side -> IF: 130..150 MHz +// DDC NCO: 120 MHz exactly -> baseband: 10..30 MHz +// The matched-filter reference in tb/cosim/gen_chirp_mem.py was updated +// to include the +10 MHz baseband offset to match this band. initial begin - // Complete Short PLFM chirp LUT (0.5us, 30MHz to 10MHz) + // Complete Short PLFM chirp LUT (0.5us, ~10MHz to ~30MHz upchirp) short_chirp_lut[ 0] = 8'd255; short_chirp_lut[ 1] = 8'd237; short_chirp_lut[ 2] = 8'd187; short_chirp_lut[ 3] = 8'd118; short_chirp_lut[ 4] = 8'd 49; short_chirp_lut[ 5] = 8'd 6; short_chirp_lut[ 6] = 8'd 7; short_chirp_lut[ 7] = 8'd 54; short_chirp_lut[ 8] = 8'd132; short_chirp_lut[ 9] = 8'd210; short_chirp_lut[10] = 8'd253; short_chirp_lut[11] = 8'd237; short_chirp_lut[12] = 8'd167; short_chirp_lut[13] = 8'd 75; short_chirp_lut[14] = 8'd 10; short_chirp_lut[15] = 8'd 10; short_chirp_lut[16] = 8'd 80; short_chirp_lut[17] = 8'd180; short_chirp_lut[18] = 8'd248; short_chirp_lut[19] = 8'd237; short_chirp_lut[20] = 8'd150; short_chirp_lut[21] = 8'd 45; short_chirp_lut[22] = 8'd 1; short_chirp_lut[23] = 8'd 54; diff --git a/9_Firmware/9_2_FPGA/short_chirp_i.mem b/9_Firmware/9_2_FPGA/short_chirp_i.mem index f8d1659..87023f7 100644 --- a/9_Firmware/9_2_FPGA/short_chirp_i.mem +++ b/9_Firmware/9_2_FPGA/short_chirp_i.mem @@ -1,50 +1,50 @@ 7332 -7330 -730d -7276 -70e0 -6d8f -679c -5e0a -4fe8 -3c80 -2399 -05ca -e4c2 -c380 -a653 -9271 -8d21 -9a5d -bb20 -ebd7 -2399 -54f5 -70e0 -6ba2 -4289 -0000 -bb20 -90cb -9729 +5c56 +1e0c d044 -2399 -65a3 -6dff -325b -d440 +9729 9271 -9f85 -f753 -57d5 -6f35 +c9c7 +2238 +679c +6a91 2399 +c10f +8d21 b576 -8e95 -db07 -4fe8 +1e0c 6d8f -0d00 +57d5 +ebd7 +92e6 +ad06 +2399 +7276 +38c3 +b7b1 +92e6 +0000 +6dff +3ef1 +b234 9bc2 -a653 -24f9 +2399 +7219 +0173 +8de7 +e4c2 +6d8f +290f +956f +d440 +6ba2 +2399 +9012 +f021 +7330 +f021 +9271 +38c3 +54f5 +9f85 +ddc8 diff --git a/9_Firmware/9_2_FPGA/short_chirp_q.mem b/9_Firmware/9_2_FPGA/short_chirp_q.mem index 5e9f401..dff81bd 100644 --- a/9_Firmware/9_2_FPGA/short_chirp_q.mem +++ b/9_Firmware/9_2_FPGA/short_chirp_q.mem @@ -1,50 +1,50 @@ 0000 -0173 -05ca -0d00 -1702 -2399 -325b -4289 -52fa -6208 +44e0 +6f35 +68d7 +2fbc +dc67 +9a5d +9201 +cda5 +2bc0 6d8f -730d -6fee -6208 -484f -2399 -f753 -c9c7 -a3aa -8e95 +607b +08ad +a82b +90cb +dc67 +4a8a +716b +24f9 +b018 9271 -b234 -e8fe -290f -5e0a -7332 -5c56 -1e0c -d044 -9729 +f300 +643e +59ad +db07 +8cce +ddc8 +607b +54f5 +c73d 9271 -c9c7 -2238 -679c -6a91 -2399 -c10f -8d21 -b576 -1e0c +0fdf +7330 +0fdf +9012 +dc67 +6ba2 +2bc0 +956f +d6f1 6d8f -57d5 -ebd7 -92e6 -ad06 -2399 -7276 -38c3 -b7b1 -92e6 +1b3e +8de7 +fe8d +7219 +dc67 +9bc2 +4dcc +3ef1 +9201 diff --git a/9_Firmware/9_2_FPGA/tb/cosim/analyze_short_chirp_mismatch.py b/9_Firmware/9_2_FPGA/tb/cosim/analyze_short_chirp_mismatch.py new file mode 100644 index 0000000..7910b99 --- /dev/null +++ b/9_Firmware/9_2_FPGA/tb/cosim/analyze_short_chirp_mismatch.py @@ -0,0 +1,188 @@ +#!/usr/bin/env python3 +# ruff: noqa: T201 +""" +analyze_short_chirp_mismatch.py — quantify TX-I matched-filter mismatch loss. + +Background +---------- +TX path (`plfm_chirp_controller.v:74,118-127`): + 60-sample inline LUT, 8-bit unsigned offset binary (DAC center = 128), + played at fs_tx = 120 MHz over 0.5 us. Real-valued passband chirp. + Module comment claims "30 MHz to 10 MHz" downchirp. + +RX matched-filter reference (`gen_chirp_mem.py:81-101` -> `short_chirp_{i,q}.mem`): + 50-sample complex baseband, Q15, fs_rx = 100 MHz over 0.5 us. + Generated as a 0 -> +20 MHz baseband upchirp: + phi(t) = pi * (BW/T) * t^2, BW = 20 MHz, T = 0.5 us + I(n) = cos(phi), Q(n) = sin(phi), scaled by 0.9*Q15 + +These are claimed by the ledger to be ~2-3 dB mismatched. This script +derives the implied baseband chirp from the TX LUT (modeling the IF chain +and DDC by NCO at 120 MHz, decimation 4x to 100 MHz), then computes the +true matched-filter peak power lost to template mismatch by: + + 1. Loading the TX LUT, computing the analytic signal (Hilbert), + verifying instantaneous-frequency trajectory + claimed bandwidth. + 2. Modeling the DDC: mix by 120 MHz NCO at 400 MHz ADC sample rate, + low-pass + decimate 4x to recover 100 MHz baseband. Since the TX + LUT is only at 120 MHz, we upsample 120->400 first via zero-stuff + + filter (the radar's analog chain does this naturally). + 3. Producing the implied 50-sample Q15 baseband reference. + 4. Computing the ambiguity peak between + a) implied-from-TX reference cross-correlated with itself + b) implied-from-TX reference cross-correlated with the existing + short_chirp_{i,q}.mem + The dB ratio of (b) peak / (a) peak is the mismatch loss. + +Output: report only. Does not modify any .mem files. +""" + +import os +import re +import sys + +import numpy as np +from scipy.signal import hilbert, resample_poly + +THIS_DIR = os.path.dirname(os.path.abspath(__file__)) +RTL_DIR = os.path.join(THIS_DIR, "..", "..") + +FS_TX = 120e6 # DAC sample rate +FS_RX = 100e6 # post-DDC processing rate +T_CHIRP = 0.5e-6 +N_TX = 60 # samples in TX LUT +N_RX = 50 # samples in RX reference + +# --- Parse TX LUT inline-coded in plfm_chirp_controller.v ---------------- +def read_tx_lut() -> np.ndarray: + path = os.path.join(RTL_DIR, "plfm_chirp_controller.v") + with open(path) as f: + src = f.read() + # Capture every "short_chirp_lut[] = 8'd;" + pairs = re.findall(r"short_chirp_lut\[\s*(\d+)\s*\]\s*=\s*8'd\s*(\d+)\s*;", src) + if len(pairs) != N_TX: + sys.exit(f"expected {N_TX} TX LUT entries, got {len(pairs)}") + arr = np.zeros(N_TX, dtype=np.int32) + for idx_s, val_s in pairs: + arr[int(idx_s)] = int(val_s) + # Convert from 8-bit unsigned offset binary (DAC center = 128) to signed. + return arr - 128 # int range roughly [-128, +127] + + +# --- Parse existing RX reference .mem files ------------------------------- +def read_q15_mem(name: str) -> np.ndarray: + path = os.path.join(RTL_DIR, name) + out = [] + with open(path) as f: + for line in f: + line = line.strip() + if not line: + continue + v = int(line, 16) + if v >= 0x8000: + v -= 0x10000 + out.append(v) + return np.array(out, dtype=np.int32) + + +# --- Derive implied 50-sample baseband reference from the TX LUT --------- +def derive_baseband_from_tx(tx: np.ndarray) -> np.ndarray: + """ + 1) Treat tx as fs=120 MHz real samples. + 2) Compute analytic signal (Hilbert) -> single-sided spectrum copy. + 3) Find the chirp's center frequency from the analytic signal's + mean instantaneous frequency, then mix it down to baseband by + multiplying by exp(-j*2*pi*fc*t). + 4) Resample 120 -> 100 MHz to get exactly N_RX = 50 samples + (matching the existing reference grid). + 5) Return as complex float64. + """ + x = tx.astype(np.float64) + z = hilbert(x) # complex analytic, fs=120 MHz + n = np.arange(len(z)) + # Instantaneous phase + frequency + inst_phase = np.unwrap(np.angle(z)) + inst_freq = np.diff(inst_phase) * FS_TX / (2 * np.pi) + fc = float(np.mean(inst_freq)) # rough center frequency in Hz + # Mix to baseband + bb_120 = z * np.exp(-1j * 2 * np.pi * fc * n / FS_TX) + # Resample 120 MHz -> 100 MHz: use up=5, down=6 (5/6 = 100/120). + bb_100 = resample_poly(bb_120, up=5, down=6) + # Trim/pad to exactly N_RX samples + if len(bb_100) >= N_RX: + bb_100 = bb_100[:N_RX] + else: + bb_100 = np.concatenate([bb_100, np.zeros(N_RX - len(bb_100), dtype=complex)]) + return bb_100, fc, inst_freq + + +# --- Mismatch loss in dB -------------------------------------------------- +def peak_corr_db(ref: np.ndarray, sig: np.ndarray) -> float: + """Peak |ref dot conj(sig_shifted)| over all integer shifts, normalised.""" + # Both arrays equal length; cross-correlate. + c = np.correlate(sig, ref, mode="full") + return 20 * np.log10(np.max(np.abs(c)) + 1e-30) + + +def main() -> int: + tx = read_tx_lut() + rx_i = read_q15_mem("short_chirp_i.mem") + rx_q = read_q15_mem("short_chirp_q.mem") + if len(rx_i) != N_RX or len(rx_q) != N_RX: + sys.exit(f"RX .mem files expected {N_RX} samples, got I={len(rx_i)} Q={len(rx_q)}") + rx = (rx_i + 1j * rx_q).astype(complex) + + # Derive implied baseband reference from TX LUT + bb, fc, inst_freq = derive_baseband_from_tx(tx) + + # Bandwidth check from instantaneous frequency + f_lo, f_hi = float(np.min(inst_freq)), float(np.max(inst_freq)) + bw = f_hi - f_lo + + print("=== TX LUT analysis ===") + print(f" samples: {N_TX} @ {FS_TX/1e6:.0f} MHz, duration {N_TX/FS_TX*1e6:.3f} us") + print(f" inst-freq range: {f_lo/1e6:+7.2f} MHz .. {f_hi/1e6:+7.2f} MHz") + print(f" bandwidth swept: {bw/1e6:6.2f} MHz") + print(f" center frequency: {fc/1e6:+7.2f} MHz (inferred from mean inst freq)") + sweep_dir = "UP" if inst_freq[-1] > inst_freq[0] else "DOWN" + print(f" sweep direction: {sweep_dir} (start={inst_freq[0]/1e6:+.2f} MHz, " + f"end={inst_freq[-1]/1e6:+.2f} MHz)") + + print() + print("=== Existing RX reference (short_chirp_{i,q}.mem) ===") + rx_phase = np.unwrap(np.angle(rx + 1e-30)) + rx_inst_freq = np.diff(rx_phase) * FS_RX / (2 * np.pi) + rx_lo, rx_hi = float(np.min(rx_inst_freq)), float(np.max(rx_inst_freq)) + print(f" samples: {N_RX} @ {FS_RX/1e6:.0f} MHz") + print(f" inst-freq range: {rx_lo/1e6:+7.2f} MHz .. {rx_hi/1e6:+7.2f} MHz") + print(f" bandwidth swept: {(rx_hi - rx_lo)/1e6:6.2f} MHz") + rx_sweep = "UP" if rx_inst_freq[-1] > rx_inst_freq[0] else "DOWN" + print(f" sweep direction: {rx_sweep}") + + print() + print("=== Mismatch loss (matched-filter peak: implied-vs-existing) ===") + # Normalise both to unit energy so the only thing the ratio reflects is shape. + bb_n = bb / np.sqrt(np.sum(np.abs(bb) ** 2) + 1e-30) + rx_n = rx / np.sqrt(np.sum(np.abs(rx) ** 2) + 1e-30) + auto_db = peak_corr_db(bb_n, bb_n) + cross_db = peak_corr_db(bb_n, rx_n) + loss_db = auto_db - cross_db + print(f" auto-correlation peak (implied vs implied): {auto_db:+6.2f} dB") + print(f" cross-corr peak (implied vs existing RX): {cross_db:+6.2f} dB") + print(f" MISMATCH LOSS (matched filter): {loss_db:6.2f} dB") + print() + + # Decision aid + if loss_db < 0.5: + verdict = "AGREEMENT — TX LUT and RX reference are consistent within 0.5 dB." + elif loss_db < 2.0: + verdict = ("MILD MISMATCH — within ledger's 2-3 dB note; refresh " + "recommended but not blocking.") + else: + verdict = "SIGNIFICANT MISMATCH — RX reference should be regenerated from TX LUT." + print(f"VERDICT: {verdict}") + return 0 + + +if __name__ == "__main__": + sys.exit(main()) diff --git a/9_Firmware/9_2_FPGA/tb/cosim/gen_chirp_mem.py b/9_Firmware/9_2_FPGA/tb/cosim/gen_chirp_mem.py index 04eda55..8b3d459 100644 --- a/9_Firmware/9_2_FPGA/tb/cosim/gen_chirp_mem.py +++ b/9_Firmware/9_2_FPGA/tb/cosim/gen_chirp_mem.py @@ -24,10 +24,32 @@ Short chirp: T_SHORT_CHIRP and CHIRP_BW. Phase model (baseband, post-DDC): - phase(n) = pi * chirp_rate * t^2, t = n / FS_SYS + phase(n) = 2*pi*F_BASEBAND_LOW*t + pi * chirp_rate * t^2, t = n / FS_SYS chirp_rate = CHIRP_BW / T_chirp + F_BASEBAND_LOW = 10 MHz (DAC chirp low-edge frequency) -Scaling: 0.9 * 32767 (Q15), matching radar_scene.py generate_reference_chirp_q15() +This produces a F_BASEBAND_LOW..(F_BASEBAND_LOW+CHIRP_BW) baseband upchirp. + +End-to-end frequency plan (TX-I, 2026-04-28): + DAC LUT : 10..30 MHz @ fs_dac=120 MHz (plfm_chirp_controller.v; + Hilbert-confirmed for both + long and short LUTs) + TX upmix : LO=10.500 GHz (adf4382a_manager.h:35), high-side + -> RF transmitted: 10.510..10.530 GHz + RX downmix: LO=10.380 GHz (adf4382a_manager.h:36), high-side + -> IF at ADC: 130..150 MHz + DDC NCO : 120 MHz exactly (ddc_400m.v:201) + -> baseband: 10..30 MHz <-- matched-filter reference + +Sideband orientation (high-side at both mixers) is the conventional choice +and consistent with all design comments / antenna match (10.25..10.75 GHz); +loopback capture would settle it definitively. If either mixer turns out to +be low-side, the sign of F_BASEBAND_LOW flips and/or the chirp direction +reverses; revisit before re-generating .mem files. + +radar_scene.py uses the same F_BASEBAND_LOW; both must stay in sync. + +Scaling: 0.9 * 32767 (Q15) Usage: python3 gen_chirp_mem.py @@ -45,6 +67,13 @@ FS_SYS = 100e6 # System clock (100 MHz, post-CIC) T_LONG_CHIRP = 30e-6 # 30 us long chirp duration T_SHORT_CHIRP = 0.5e-6 # 0.5 us short chirp duration FFT_SIZE = 2048 +# DAC chirp baseband low-edge frequency. The TX LUT in plfm_chirp_controller.v +# is a 10..30 MHz upchirp at fs_dac=120 MHz (Hilbert-confirmed for both long +# and short LUTs). With TX_LO=10.500 GHz, RX_LO=10.380 GHz (adf4382a_manager.h) +# and the 120 MHz DDC NCO (ddc_400m.v), high-side mixing places the post-DDC +# echo at 10..30 MHz baseband, not 0..20 MHz. The matched-filter reference +# must include this +10 MHz DC offset. +F_BASEBAND_LOW = 10e6 LONG_CHIRP_SAMPLES = int(T_LONG_CHIRP * FS_SYS) # 3000 SHORT_CHIRP_SAMPLES = int(T_SHORT_CHIRP * FS_SYS) # 50 LONG_SEGMENTS = 2 @@ -69,7 +98,7 @@ def generate_full_long_chirp(): for n in range(LONG_CHIRP_SAMPLES): t = n / FS_SYS - phase = math.pi * chirp_rate * t * t + phase = 2 * math.pi * F_BASEBAND_LOW * t + math.pi * chirp_rate * t * t re_val = round(Q15_MAX * SCALE * math.cos(phase)) im_val = round(Q15_MAX * SCALE * math.sin(phase)) chirp_i.append(max(-32768, min(32767, re_val))) @@ -92,7 +121,7 @@ def generate_short_chirp(): for n in range(SHORT_CHIRP_SAMPLES): t = n / FS_SYS - phase = math.pi * chirp_rate * t * t + phase = 2 * math.pi * F_BASEBAND_LOW * t + math.pi * chirp_rate * t * t re_val = round(Q15_MAX * SCALE * math.cos(phase)) im_val = round(Q15_MAX * SCALE * math.sin(phase)) chirp_i.append(max(-32768, min(32767, re_val))) @@ -155,13 +184,14 @@ def main(): # ---- Verification summary ---- - # Cross-check seg0 against radar_scene.py generate_reference_chirp_q15() - # That function generates exactly the first 1024 samples of the chirp + # Self-check: recompute the phase formula and verify the seg0 .mem matches. + # radar_scene.py.generate_reference_chirp_q15() uses the same phase form + # and the same F_BASEBAND_LOW; the two stay in sync by construction. chirp_rate = CHIRP_BW / T_LONG_CHIRP mismatches = 0 for n in range(FFT_SIZE): t = n / FS_SYS - phase = math.pi * chirp_rate * t * t + phase = 2 * math.pi * F_BASEBAND_LOW * t + math.pi * chirp_rate * t * t expected_i = max(-32768, min(32767, round(Q15_MAX * SCALE * math.cos(phase)))) expected_q = max(-32768, min(32767, round(Q15_MAX * SCALE * math.sin(phase)))) if long_i[n] != expected_i or long_q[n] != expected_q: diff --git a/9_Firmware/9_2_FPGA/tb/cosim/radar_scene.py b/9_Firmware/9_2_FPGA/tb/cosim/radar_scene.py index e2c3cbe..53f6df1 100644 --- a/9_Firmware/9_2_FPGA/tb/cosim/radar_scene.py +++ b/9_Firmware/9_2_FPGA/tb/cosim/radar_scene.py @@ -32,11 +32,23 @@ F_CARRIER = 10.5e9 # 10.5 GHz carrier C_LIGHT = 3.0e8 # Speed of light (m/s) WAVELENGTH = C_LIGHT / F_CARRIER # ~0.02857 m -# Chirp parameters -F_IF = 120e6 # IF frequency (120 MHz) -CHIRP_BW = 20e6 # Chirp bandwidth (30 MHz -> 10 MHz = 20 MHz sweep) -F_CHIRP_START = 30e6 # Chirp start frequency (relative to IF) -F_CHIRP_END = 10e6 # Chirp end frequency (relative to IF) +# Chirp parameters. +# +# End-to-end frequency plan (TX-I, 2026-04-28): +# DAC LUT : 10..30 MHz @ fs_dac=120 MHz (plfm_chirp_controller.v; +# Hilbert-confirmed for both +# long and short LUTs) +# TX upmix : LO=10.500 GHz (adf4382a_manager.h:35), high-side +# -> RF transmitted: 10.510..10.530 GHz +# RX downmix: LO=10.380 GHz (adf4382a_manager.h:36), high-side +# -> IF at ADC: 130..150 MHz +# DDC NCO : 120 MHz exactly (ddc_400m.v:201) +# -> baseband: 10..30 MHz +F_IF = 120e6 # DDC NCO frequency (120 MHz) +F_BASEBAND_LOW = 10e6 # DAC chirp baseband low-edge frequency +CHIRP_BW = 20e6 # Chirp bandwidth (10 MHz -> 30 MHz upchirp = 20 MHz sweep) +F_CHIRP_START = F_BASEBAND_LOW # 10 MHz at DAC baseband +F_CHIRP_END = F_BASEBAND_LOW + CHIRP_BW # 30 MHz at DAC baseband # Sampling FS_ADC = 400e6 # ADC sample rate (400 MSPS) @@ -153,12 +165,13 @@ def generate_if_chirp(n_samples, chirp_bw=CHIRP_BW, f_if=F_IF, fs=FS_ADC): chirp_q = [] chirp_rate = chirp_bw / (n_samples / fs) # Hz/s + # IF chirp starts at f_if + F_BASEBAND_LOW and sweeps up over chirp_bw, + # i.e. 130..150 MHz for the nominal high-side / 120 MHz NCO chain. + f_lo = f_if + F_BASEBAND_LOW for n in range(n_samples): t = n / fs - # Instantaneous frequency: f_if - chirp_bw/2 + chirp_rate * t - # Phase: integral of 2*pi*f(t)*dt - _f_inst = f_if - chirp_bw / 2 + chirp_rate * t - phase = 2 * math.pi * (f_if - chirp_bw / 2) * t + math.pi * chirp_rate * t * t + _f_inst = f_lo + chirp_rate * t + phase = 2 * math.pi * f_lo * t + math.pi * chirp_rate * t * t chirp_i.append(math.cos(phase)) chirp_q.append(math.sin(phase)) @@ -188,10 +201,10 @@ def generate_reference_chirp_q15(n_fft=FFT_SIZE, chirp_bw=CHIRP_BW, _f_if=F_IF, for n in range(chirp_samples): t = n / FS_SYS - # After DDC, the chirp is at baseband - # The beat frequency from a target at delay tau is: f_beat = chirp_rate * tau - # Reference chirp is the TX chirp at baseband (zero delay) - phase = math.pi * chirp_rate * t * t + # After DDC, the chirp is at baseband F_BASEBAND_LOW..(F_BASEBAND_LOW+BW), + # i.e. 10..30 MHz for the nominal chain. Reference chirp is the TX chirp + # at baseband (zero delay). Phase formula must match gen_chirp_mem.py. + phase = 2 * math.pi * F_BASEBAND_LOW * t + math.pi * chirp_rate * t * t re_val = round(32767 * 0.9 * math.cos(phase)) im_val = round(32767 * 0.9 * math.sin(phase)) ref_re[n] = max(-32768, min(32767, re_val)) @@ -263,8 +276,10 @@ def generate_adc_samples(targets, n_samples, noise_stddev=3.0, t = n / FS_ADC t_delayed = n_delayed / FS_ADC - # Signal at IF: cos(2*pi*f_if*t + pi*chirp_rate*t_delayed^2 + doppler + phase) - phase = (2 * math.pi * F_IF * t + # Signal at IF: chirp starts at (F_IF + F_BASEBAND_LOW) and sweeps + # up by chirp_rate (130..150 MHz for the nominal chain). + f_lo_if = F_IF + F_BASEBAND_LOW + phase = (2 * math.pi * f_lo_if * t + math.pi * chirp_rate * t_delayed * t_delayed + 2 * math.pi * doppler_hz * t + phase0)