LOFAR2.0 Station SDP Firmware quantization model

Author: Eric Kooistra, Aug - Dec 2022

Purpose: Model the expected signal levels in the SDP firmware and clarify calculations in [1].

Usage:

References:

  1. https://support.astron.nl/confluence/display/L2M/L4+SDPFW+Decision%3A+LOFAR2.0+SDP+Firmware+Quantization+Model
  2. Understanding digital signal processing, R.G. Lyons

1 SDP Parameters

1.1 Subband gain factor

1.2 Beamlet gain factor

1.3 Maximum input level for beamlet_sum and BST

2 Quantization noise

2.1 dB full scale (dBFS)

2.2 Signal to noise ratio (SNR)

2.3 Impact of quantization on the system noise

3 Expected signal levels in the SDP FW

3.1 ADC Statistics (AST)

From measured P_ast and DC_ast to signal input sigma in q units:

3.2 Subband statistics (SST)

For a complex signal (like subbands and beamlets), assume mean complex = 0 so rms = std and power = var (= std^2):

3.2.1 Coherent, narrow band, sine input

3.2.2 Incoherent, wide band, noise input

Conclusion (for W_fft_proc = 5 bits):

3.2.3 From SST level to input level

3.3 Crosslet statistics (XST)

The crosslet statistics have W_crosslet = 16b, but use the same LSbit level as the subbands. The subbands have W_subband = 18b and the maximum subband sine amplitude is 17b (for W_fft_proc = 5 bits). Therefore the maximum sine input for no XST overflow is A = 0.25. If subband_weight = 1.0 then the auto correlations of the XST are equal to the SST. Hence the crosslets have the same sensitivity as the subbands, but less dynamic range.

3.4 Beamlet statistics (BST)

3.4.1 Coherent, narrow band, sine input

3.4.2 Icoherent, wide band, noise input

If subband_weight = 1.0 and beamlet_weight = 1.0 and N_ant = 1 then the BST are equal to the SST

3.5 Beamlet output

The beamlet output is W_beamlet = 8 bit. The beamlet has a sign bit, about 1 bit for the sigma and about 2 bits to fit a range of 4 sigma, so about 4 bits can carry the noise signal. The extra 4 bits are for some RFI and to fit differences in subband noise level due to the antenna and RCU2 band filter shape, in case these differences are not fully equalized by the subband weights. The subband noise level can be equalized using the subband weights, to have more dynamic range for RFI in the beamlet output.

Choosing FPGA_beamlet_output_scale_RW = 1 / sqrt(N_ant) makes that the beamlet output level for noise input is equal to that of N_ant = 1 for all N_ant. The BST can be used to check whether the beamlet output will fit.

Appendix 1: DFT of real input DC, sine, block and noise

Conclusion: