Barker decoding using both matched & unmatched

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Barker_Codes_v3.2_13-169baud.nb

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Rotate1D[array_] := Module{len = Length[array]}, IfEvenQ[len], RotateRightarray, len  2, RotateRightarray, len - 1  2; (* function needed below *)

__________________________ **Some basic definitions ��

(* Note: sampleinterval = deltatime - units are μsec *) nsamples = 182; (*182, 202*) nsamplesbaud = 3; (* we use bauds of integer sample length *) zeropad = Table[0, nsamplesbaud - 1]; padding = {1, 1}; codelength = 13; plotxvalues = Range- nsamples  2, nsamples  2 - 1; (* Needed for plotting below *)

____________________________ **A single sampled 1 baud “code”. ��

pulsebaud = Table[1, nsamplesbaud]; (* our representation of single baud of the code it must be nsamplesbaud long *) plotpulsebaud = PadRight[pulsebaud, codelength + 3]; ListPlot[plotpulsebaud, PlotRange → {{0, codelength}, {- 1.2, 1.2}}, Filling → Axis, PlotLabel → "�� "] �� 1.0

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______________________________________ **Get the impulse response of the coding filter. ��

barker13base = {1, 1, 1, 1, 1, - 1, - 1, 1, 1, - 1, 1, - 1, 1}; (* template bit pattern for the 13 baud Barker code *) (* build the coding filter -*) step1 = Flatten[Riffle[barker13base, {zeropad}]]; barker13h = Join[zeropad, step1, zeropad]; (* impulse response of coding filter - eqn 3 in �� *) ListPlot[barker13h, PlotRange → {{0, nsamplesbaud * codelength + 1}, {- 1.2, 1.2}}, Filling → Axis, PlotLabel → "�� "] (* hc (n) - eqn 3, �� *) �� 1.0

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______________________________________ **Get the sampled Barker code. ��

barker13 = ListConvolve[pulsebaud, barker13h]; (* sampled code *) ListPlot[barker13, PlotRange → {{0, nsamplesbaud * codelength + 1}, {- 1.2, 1.2}}, Filling → Axis, PlotLabel → "�� "] �� 1.0

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______________________________________ **Decode the sampled Barker code. ��

samplecount = nsamplesbaud + 1 * codelength; (*barker13p=PadLeftbarker13,nsamples2;*) barker13p = PadRight[barker13, nsamples]; (* *) barker13plot = Rotate1D[ListCorrelate[barker13, Conjugate[barker13p], {1, 1}]]; ListPlot[Transpose[{plotxvalues, barker13plot}], Joined → True, Mesh → All, PlotRange → {{- 50, 50}, {0, 40}}, PlotLabel → "�� "] �� 40

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totalsamples = nsamples; (* Note: we repeat some of the above for simplicity. *) sampledBaudTD = PadRight[pulsebaud, totalsamples]; (* Establish the single baud "code" with nsamples length *) sampledBaudFD = Fourier[sampledBaudTD]; (* frequency domain version of single baud code: Q(ω) - eqn (15) in Lehtinen et al. *) pwrspecBaud = Sqrt[Abs[sampledBaudFD * Conjugate[sampledBaudFD]]]; (* Get the amplitude spectrum *) ListPlot[Rotate1D[pwrspecBaud], PlotLabel → "sampledBaudFD: Shown is Sqrt[power spectrum] of single baud code"] barker13TD = PadRight[barker13h, totalsamples]; (* impulse response hc (n) of the Barker 13 coding filter *) ListPlot[barker13TD]; (* Enable plotting barker13TD by removing ";" *) barker13FD = Fourier[barker13TD]; (* Hc (ω) - eqns (14) & 16 in Reference *) pwrspecB13Hc = Sqrt[Abs[barker13FD * Conjugate[barker13FD]]]; ListPlot[Rotate1D[pwrspecB13Hc], PlotRange → All, Joined → True, PlotLabel → "barker13FD: Shown is Sqrt[power spectrum] of 13 baud code"] sampledBaudFD: Shown is Sqrt[power spectrum] of single baud code

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barker13FD: Shown is Sqrt[power spectrum] of 13 baud code

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gamma = barker13FD  sampledBaudFD; (*decodeFilter=InverseFouriersampledBaudFDbarker13FD; eqn 14 Lehtinen et al. *) decodeFilter = InverseFourier[1 / gamma]; decodeFilter = RotateRightdecodeFilter, totalsamples  2; ListPlot[Re[decodeFilter], PlotRange → All, Joined → True, PlotLabel → "Truncated zero-sidelobe compression filter"] (* see middle-right panel, Figure 3 of Lehtinen et al. *) decodeFilterTD = InverseFourier[gamma]; (* time domain*) Truncated zero-sidelobe compression filter 1.0

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(* Equation-17, Lehtinen et al *) MisMatchFilterOut = Rotate1DInverseFourierFourier[PadRight[barker13, totalsamples]]  Fourier[decodeFilterTD] Sqrt[Dimensions[decodeFilterTD][[1]]]; ListLinePlotTransposeplotxvalues, Abs[MisMatchFilterOut]  nsamplesbaud, PlotRange → All, PlotLabel → "Mis-matched compression filter decoding of sampled 13 baud Barker code" ListLinePlot Transposeplotxvalues, Log10Abs[MisMatchFilterOut]  nsamplesbaud, PlotRange → All, PlotLabel → "Log10 of the mis-matched compression filter decoding of sampled 13 baud Barker code" (*Matched filter*) MatchFilterOut = Rotate1D[ ListCorrelate[Conjugate[barker13], PadRight[barker13, totalsamples], {1, 1}]]; ListLinePlotTransposeplotxvalues, MatchFilterOut  nsamplesbaud, PlotRange → All, PlotLabel → "Matched filter decoding of sampled 13 baud Barker code" Mis-matched compression filter decoding of sampled 13 baud Barker code 14 12 10 8

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Log10 of the mis-matched compression filter decoding of sampled 13 baud Barker code

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Matched filter decoding of sampled 13 baud Barker code 12 10 8 ��

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_________________________________________________________ 2. 169 Bit Barker decoding using mis-matched, sidelobe-free filtering... This directly follows the solution path outlined above. Based on Kesaraju et al. (2017). ��

(*sampleinterval=1; (* the same as above with possibility of changing it - units are μsec *)*) nsamples169 = 28 589; (* Magnitude of mismatch filter depends on this*) nsamplesbaud169 = 10; (* we use bauds of integer sample length *) codelength169 = 169;

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plotxvalues1 = Range[0, codelength169 + 2]; plotxvalues169 = Range- nsamples169  2, nsamples169  2 - 1; (* Needed for plotting below *) pulsebaud169 = ConstantArray[1, nsamplesbaud169]; (* our representation of single, sampled baud of the code nsamplesbaud169 long *) (*pulsebaud=PadRight[pulsebaud,nsamples]; *) barker13base = {1, 1, 1, 1, 1, - 1, - 1, 1, 1, - 1, 1, - 1, 1}; (* template bit pattern for 13 baud Barker code - same as above for clarity *) barker169base = Flatten[KroneckerProduct[Reverse[barker13base], Reverse[barker13base]]]; (* template bit pattern for 169-baud Barker code *)

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(* fill in necessary 0's between current elements *) barker169 = Partition[barker169base, 1]; (* an elligant alternative to what we did above! *) barker169 = Flatten[ Table[PadRight[barker169[[i]], nsamplesbaud169, 0], {i, 1, codelength169}]]; barker169h = barker169; (* impulse response of coding filter eqn 3 in reference *) barker169 = PadLeftbarker169, nsamplesbaud169 * codelength169 + 1 - 1; (* add 0's on left *) barker169 = ListCorrelate[pulsebaud169, barker169]; (* impulse response of the coding filter hc (n)which is zero for ncodelength-1; n is an integer *)

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plotpulsebaud169 = PadRight[pulsebaud169, codelength169 + 3]; ListPlot[Transpose[{plotxvalues1, plotpulsebaud169}], PlotRange → {{0, codelength169}, {- 1.2, 1.2}}, Filling → Axis, PlotLabel → "Sampled single baud"] Sampled single baud 1.0

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ListPlot[barker169h, PlotRange → {{0, nsamplesbaud169 * codelength169 + 1}, {- 1.2, 1.2}}, Filling → Axis, PlotLabel → "Sampled Barker169 coding filter impulse response"] (* hc (n) - reference eqn 3 *) Sampled Barker169 coding filter impulse response 1.0

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ListPlot[barker169, PlotRange → {{0, nsamplesbaud169 * codelength169 + 1}, {- 1.2, 1.2}}, Filling → Axis, PlotLabel → "Sampled 169-baud Barker code "] Sampled 169-baud Barker code 1.0

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sampledBaud169TD = PadRight[pulsebaud169, nsamples169]; (* pulse baud- baud length with 1's*) sampledBaud169FD = Fourier[sampledBaud169TD]; (* Q(ω) - eqn (15) in Reference *) pwrspecBaud169 = Sqrt[Abs[sampledBaud169FD * Conjugate[sampledBaud169FD]]]; ListLinePlot[Rotate1D[pwrspecBaud169], PlotRange → All, PlotLabel → "sampledbaud169FD: Shown is Sqrt[power spectrum] of the sampled single baud code"] barker169hTD = PadRight[barker169h, nsamples169]; (* barker 169h with zero weight*) (* impulse response hc (n) of the Barker 169 coding filter *) barker169FD = Fourier[barker169hTD]; (* Hc (ω) - eqns (14) & 16 in Reference *) pwrspecB169Hc = Sqrt[Abs[barker169FD * Conjugate[barker169FD]]]; ListLinePlot[Abs[barker169FD], PlotRange → All, PlotLabel → "barker169FD: Shown is Sqrt[power spectrum] of the sampled 169 baud code"] sampledbaud169FD: Shown is Sqrt[power spectrum] of the sampled single baud code 0.06

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barker169FD: Shown is Sqrt[power spectrum] of the sampled 169 baud code 0.14 0.12 0.10 ��

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hc169 = barker169FD  sampledBaud169FD; decodeFilter169TDa = InverseFourier1  hc169; decodeFilter169TD = decodeFilter169TDa; decodeFilter169TD = RotateRightdecodeFilter169TD, IntegerPartnsamples169  2; x = Re[decodeFilter169TD]; ListLinePlot[x, PlotRange → All] 1.5 1.0 0.5 ��

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(* Equation-17, Lehtinen et al *) MisMatchFilter169Out = InverseFourier[ Fourier[PadRight[barker169, nsamples169]] * Fourier[decodeFilter169TD]] * Sqrt[nsamples169]  nsamplesbaud169; (* Note the normalization for the Fourier transforms & samplesbaud *) ListLinePlot[Transpose[{plotxvalues169, Abs[MisMatchFilter169Out]}], PlotRange → All, PlotLabel → "Mis-matched compression filter decoding of sampled 169-baud Barker code"] ListLinePlot[Transpose[{plotxvalues169, Log[Abs[MisMatchFilter169Out]]}], PlotRange → All, PlotLabel → "Log10 of the Mis-matched compression filter decoding of sampled 169-baud Barker code"] (* Decoding using the matched filter. *) MatchFilter169Out = Rotate1D[ ListCorrelate[Conjugate[barker169], PadRight[barker169, nsamples169], {1, 1}]]; ListLinePlotTransposeplotxvalues169, Abs[MatchFilter169Out]  nsamplesbaud169, PlotRange → All, PlotLabel → "Matched filter decoding of sampled 169-baud Barker code"

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Mis-matched compression filter decoding of sampled 169-baud Barker code

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Log10 of the Mis-matched compression filter decoding of sampled 169-baud Barker code

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Barker decoding using both matched & unmatched

Barker decoding using both matched & unmatched

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