FMCW Radar 203- SNR

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SNR

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.

Received Power

The received power, or Pr is defined by the radar equation

\[P_R = \frac{P_T \cdot G_T \cdot G_R \cdot \lambda^2 \cdot \sigma}{(4 \pi)^3 \cdot R^4}\]

Where:

  • $ P_T $ is the transmitted power from the input of the antenna

  • $ P_R $ is the received power at the antenna

  • $ G_T $ is the gain of the transmitting antenna (so the power transmitted to the medium is \(P_t \cdot G_T\) )

  • $ G_R $ is the gain of receiving antenna

  • $ :nbsphinx-math:`lambda `$ is carrier wavelength, often the center frequency of the chirp for a FMCW modulation

  • $ \sigma `$ is the equivalent surface of the scatterer, also sometimes called ``Radar Cross Section`, aka RCS

  • $ R $ is the range or the distance between the radar and the scatterer

Noise

A resistor $ R$ at temperature \(T\) can be represented as either:

  • a series voltage source $ v_n(t) $ with with mean-square spectral density $S_v = 4 \cdot `k_B :nbsphinx-math:cdot T :nbsphinx-math:cdot `R $

  • an equivalent current source $ i_n(t) $ with with mean-square spectral density \(S_i = \frac{4 \cdot k_B \cdot T }{ R}\)

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:

\[P_{LNA} (t)= \frac{v_{LNA}^2 (t)}{R} = \frac{\frac{v_n}{2}^2}{R} = \frac{v_n^2}{4 R}\]

Now, take the mean-square (per unit bandwidth):

\[\langle P_{LNA} \rangle = \frac{\langle v_n^2 \rangle}{4R} = \frac{S_v \Delta f}{4R}\]

Substitute $S_v = 4 \cdot `k_B :nbsphinx-math:cdot T :nbsphinx-math:cdot `R $:

\[\langle P_{LNA} \rangle = k_B \cdot T \Delta f\]

instantaneous SNR (before processing)

Receiver noise power over an effective noise bandwidth \(B_n\) is

$ N = k T_0 \cdot `ext{NF} :nbsphinx-math:cdot `B_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

Instantaneous SNR (linear) at the ADC / detector prior to any coherent processing:

$ SNR_0 = \frac{Pr}{N} = \frac{P_R}{k T_0 \cdot text{NF} \cdot B_n} $

This is the baseline SNR per unit bandwidth (no matched-filter gain yet).

Integration time

If the receiver noise bandwidth $ B_n$ equals the chirp bandwidth $ B$, the processing gain (linear) from the matched filter is:

$ G_{range,linear} = B T_c $

(If sampling and DFT are used, $G_{range,linear} :nbsphinx-math:`approx `N_{eff} $ — the number of coherent samples that the matched filter coherently sums — same order as $ B T_c$)

So after the range/matched filter (or equivalently after the first FFT across fast-time) the SNR becomes:

$ SNR_{afterRangeFFT} = SNR_0 :nbsphinx-math:`cdot `BTc $

Notes:

  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.

  2. Windowing (Hanning, Hamming, etc.) reduces coherent gain (typical loss a few dB, called processing/window loss).

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