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    "# FMCW Radar 203- SNR\n",
    "[![](https://colab.research.google.com/assets/colab-badge.svg)](https://colab.research.google.com/github/matt-chv/mmWrt/blob/main/docs/FMCW-Radar-203_SNR.ipynb)"
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    "## SNR\n",
    "\n",
    "For a mono-static radar, the `Signal to Noise Ratio` or `SNR` is the ratio of the received power to the power of the noise.\n"
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    "### Received Power\n",
    "\n",
    "The received power, or `Pr` is defined by the radar equation\n",
    "\n",
    "$$P_R = \\frac{P_T \\cdot G_T \\cdot G_R \\cdot \\lambda^2 \\cdot \\sigma}{(4 \\pi)^3 \\cdot R^4}$$\n",
    "\n",
    "Where:\n",
    "\n",
    "* $ P_T $ is the transmitted power from the input of the antenna\n",
    "* $ P_R $ is the received power at the antenna\n",
    "* $ G_T  $ is the gain of the transmitting antenna (so the power transmitted to the medium is $P_t \\cdot G_T$ )\n",
    "* $ G_R $ is the gain of receiving antenna\n",
    "* $ \\lambda $ is carrier wavelength, often the center frequency of the chirp for a FMCW modulation\n",
    "* $ \\sigma $ is the equivalent surface of the scatterer, also sometimes called `Radar Cross Section`, aka `RCS`\n",
    "* $ R $ is the range or the distance between the radar and the scatterer"
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    "### Noise\n",
    "\n",
    "A resistor $ R$ at temperature $T$ can be represented as either:\n",
    "\n",
    "* a series voltage source $ v_n(t) $ with with mean-square spectral density $S_v = 4 \\cdot k_B \\cdot T \\cdot R $\n",
    "* an equivalent current source $ i_n(t) $ with with mean-square spectral density $S_i = \\frac{4 \\cdot k_B \\cdot T }{ R}$\n",
    "\n",
    "Assuming we have a perfectly matched antenna port to $50 \\Omega$ (i.e. both antenann and receiver input have $R= 50 \\Omega$), the power delivered to the matched load (i.e. the LNA input) is:\n",
    "\n",
    "$$P_{LNA} (t)= \\frac{v_{LNA}^2 (t)}{R} = \\frac{\\frac{v_n}{2}^2}{R} = \\frac{v_n^2}{4 R} $$\n",
    "Now, take the mean-square (per unit bandwidth):\n",
    "$$ \\langle P_{LNA} \\rangle = \\frac{\\langle v_n^2 \\rangle}{4R} = \\frac{S_v \\Delta f}{4R}$$\n",
    "\n",
    "Substitute $S_v = 4 \\cdot k_B \\cdot T \\cdot R $:\n",
    "\n",
    "$$ \\langle P_{LNA} \\rangle = k_B \\cdot T \\Delta f $$\n",
    "\n",
    "### instantaneous SNR (before processing)\n",
    "\n",
    "Receiver noise power over an effective noise bandwidth $B_n$ is\n",
    "\n",
    "$ N = k T_0 \\cdot \text{NF} \\cdot B_n $\n",
    "\n",
    "Common choice: take $B_n$ as the IF / instantaneous noise bandwidth (often approximately the chirp bandwidth B if the receiver front-end passes the entire beat spectrum). For the moment keep it general as `Bn`\n",
    "\n",
    "Instantaneous SNR (linear) at the ADC / detector prior to any coherent processing:\n",
    "\n",
    "$ SNR_0 = \\frac{Pr}{N} = \\frac{P_R}{k T_0 \\cdot text{NF} \\cdot B_n} $\n",
    "\n",
    "This is the baseline SNR per unit bandwidth (no matched-filter gain yet)."
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    "## Integration time\n",
    "\n",
    "If the receiver noise bandwidth $ B_n$ equals the chirp bandwidth $ B$, the processing gain (linear) from the matched filter is:\n",
    "\n",
    "$ G_{range,linear} = B T_c $\n",
    "\n",
    "(If sampling and DFT are used, $G_{range,linear} \\approx N_{eff} $ — the number of coherent samples that the matched filter coherently sums — same order as $ B T_c$)\n",
    "\n",
    "So after the range/matched filter (or equivalently after the first FFT across fast-time) the SNR becomes:\n",
    "\n",
    "$ SNR_{afterRangeFFT} = SNR_0 \\cdot BTc $\n",
    "\n",
    "Notes:\n",
    "\n",
    "1. if the FFT length (N_FFT) is larger than the number of samples (like in the case of zero-padding), the processing gain does not increase beyond the matched-filter $ BT_c $; zero-padding simply interpolates the spectrum.\n",
    "2. Windowing (Hanning, Hamming, etc.) reduces coherent gain (typical loss a few dB, called processing/window loss).\n"
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