Transcript
V8: DIRECT SEQUENCE SPREAD SPECTRUM MODULATION AND DEMODULATION with MULTIPLE BIT STREAMS By Laurence G. Hassebrook April 24, 2013
Table of Contents V8: DIRECT SEQUENCE SPREAD SPECTRUM MODULATION AND DEMODULATION with MULTIPLE BIT STREAMS ....................................................................................................................................................... 1 0.
Overview ........................................................................................................................................... 1
1.
SYSTEM ARCHITECTURE .................................................................................................................... 2
2.
MESSAGE SIGNAL GENERATION ....................................................................................................... 4 2.1 Group Name, Number of Bits and Number of Bit Streams ............................................................. 4 2.2 Bit Sequence Generation ................................................................................................................ 5 2.3 Spread Code Generation ................................................................................................................. 5
3.
DSSS MODULATION .......................................................................................................................... 5
4.
CHANNEL MODEL .............................................................................................................................. 9
5.
DSSS DEMODULATION .................................................................................................................... 11
6.
PERFORMANCE VERIFICATION ........................................................................................................ 12
7.
REFERENCES .................................................................................................................................... 14
A. APPENDIX: INSTRUCTOR PROGRAMS ............................................................................................. 14 A.1 BGEN ................................................................................................................................................. 14 A.2 SPREAD CODE GENERATION ............................................................................................................. 15 A.3 CHANNEL ........................................................................................................................................... 17 A.4 BITCHECK .......................................................................................................................................... 18
0. Overview In this visualization we use our MATLAB based communication simulation system. The Direct Sequence Spread Specturm (DSSS) modulation scheme is implemented with the simulation and the student is asked to maximize the noise variance in the channel without receiving an error.
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Baseline parameter values are given and the modulation and demodulation technique is given mathematically. The student is required to implement the mathematics of the modulator and demodulator to MATLAB code. In this visualization the student only needs to submit their simulation .m files to the instructor. No written report is necessary. The student is required to formulate an alias name, send the instructor their modified createBsize.m, modulator.m and demodulator.m and Key.mat files along with their alias and the noise coefficient to be used for the channel.
1. SYSTEM ARCHITECTURE The system architecture includes programs that the student modifies and programs that the instructor uses to evaluate their system with. The student has access to the instructor programs so that they can pre-evaluate their systems performance. Given the student programs, the students are required to comply with the File and Data formatting specifications. The student programs have the I/O functionality already built in and specify sections where the students can insert their code. The students may use the DSSS_dsssKeygenV8.m program to generate two spreading codes (“Keys”), one for each bit sequence. Referring to Figs. 1-1 and 1-2, the procedure for running the system is as follows: 1. (student) Edit in the alias name and use the given number of bits (Nbit) and given number of sequences (Nseq) into name_createBsize.m and change name to name_createBsize.m. 2. (student) Run name_createBsize.m. This stores Nseq, Nbit and aliasname. 3. (instructor) Edit in alias name into Bgenxx.m and run program. This stores two files, one with the random bit sequence matrix of Nseq by Nbit bits and the other file contains the active alias being processed. 4. (student) Run name_dsssKeygenV8.m to generate 2 spread codes stored in name_Key.mat. These codes only need to be generated when there is a change in the number of bits thereby changing the number of samples per bit. The student only needs to send the Key.mat file to the instructor with his 3 other programs. 5. (student) Run name_modulator.m. The input is the Message signal is the random bit sequence. The bit sequences are scaled to be bipolar. Using a kronecker product operation. Each bipolar message sequence is upsampled and multiplied with a tiled spreadcode to length N. Then those two DSSS signals are added together to form one signal. The modulator outputs two files, the first with the modulated signal exactly N real elements long and the other a special trinary signal matrix where 1 indicates time location of a “1” bit and -1 indicates the location of a “0” bit. Each row of the bitcheck matrix is associated with a bit sequence in pseudo-continuous time. This trinary signal is NOT passed to the demodulator but ONLY ACCESSED by the bitcheckxx.m program. 6. (instructor) Run channelxx.m. The channel filters the spectrum of the modulated signal and adds Gaussian white noise to the signal based on the minimum and maximum values in the signal. The instructor determines the channel transfer function and the amount of 2
noise added. In this visualization the student tells the instructor how much noise is to be added. The filtered noisy output is stored in name_r.mat as the real vector “r” that is N elements in length. Channel will also test for errors in the input data format. 7. (student) Run name_demodulator.m. The student may use any parameters used in the modulator, including the number of bits, and will process the input “r” signal vector from the Channel. However, there is NO KNOWLEDGE of the random message bit sequences allowed. The output of the demodulator, real matrix Bs, is stored in name_Bs.mat. 8. (instructor) Run bitcheckxx.m. Bitcheck uses the Bcheck signal generated in the Modulator to test for 1 or 0 bits. It prints out any errors in format but most importantly how many false alarms and misses occur in the detection process. If there are errors in detection then a figure will be generated revealing the local signal characteristics that led to the error.
Figure 1‐1: Simulation Flow Chart with associated MATLAB programs and data variables and vectors.
The flowchart in Fig. 1-1 shows the relationship of the MATLAB functions with respect to a standard communications system flowchart. Notice that “name_modulator.m” both encodes by upsampling the bit sequence and modulates. The function Bitcheckxx.m both decodes by thresholding the Bs output vector from the demodulator as well as tests for detection errors based on the Bcheck vector.
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Figure 1‐2: Communications Simulation System Architecture.
DSSS requires the generation of Spread Codes to be provided to the modulator and demodulator. These are stored in a Key matrix in MATLAB. The simulation architecture is shown in Fig. 1-2. The left group corresponds to the student controlled MATLAB programs, the center group represents the data storage and the right group shows the instructor controlled MATLAB programs.
2. MESSAGE SIGNAL GENERATION The student renames and uses the “name_Bsize.m” file to establish their alias and use the number of sequences and the number bits specified to be transmitted through the communications simulator. Once that is run, it stores these two parameters name_Bsize.mat. The next step is to run Bgenxx.m, given in Appendix: Bgen. But before running Bgenxx.m, the aliasname variable needs to be manually changed to match which student alias name that should be run. This way the instructor can choose which students’ program set is to be run. If the students are using Bgen in development, then they just need to edit in their name one time. 2.1 Group Name, Number of Bits and Number of Bit Streams
Below is the source code for the name_createBsize.m program. For this example, “DSSS” has been entered as the group name. We will be using 8192 bits per Nseq=2 bit streams. % generate groupname_Bsize.mat clear; % INSERT GROUP NAME AND NUMBER OF BITS groupname='DSSS' % name of group Nbit=8192 % number of bits to transmit per sequence Nseq=2 % number of sequences % END OF INSERT % name of output file that stores Nbit and filename filename=sprintf('%s_Bsize.mat',groupname)
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save(filename); % stores groupname, Nbit to DSSS_Bsize.mat
2.2 Bit Sequence Generation
From Appendix: Bgen, the only line of code that would need to be changed is groupname='DSSS' % instructor enters this name to select student project
In this case ‘DSSS’ is entered but should be whatever the group name is. Bgenxx.m generates a pseudo-random set of bits, Nbit long, to be transmitted through the communication system. 2.3 Spread Code Generation
A function set is provided to the student to generate two spread codes. It is not an ideal generator but simply generates an excess of pseudo-random uniformly distributed sequences, partitions that into a large set of spread codes and then selects the two codes that are this least correlated. All the code is given in Appendix A.2 but the first part of the function is Nkeys=2; groupname='DSSS' % instructor enters this name to select student project filename=sprintf('%s_Bsize.mat',groupname); load (filename) % retrieve data filename Nbit N=131072*8 % N is set by instructor and cannot be changed Nsample=floor(N/Nbit) % number of samples per Key Ntiles=8; % must be greater than Nkeys Key=GenDsssKeys(Nkeys,Nsample,Ntiles); % same the Key filename=sprintf('%s_Key.mat',groupname); save(filename,'Key'); % save the random
The student should change the “DSSS” to their group name. The idea is that you let Nkeys=2 to get two spread codes, Nsample is determined by the number of samples per bit and Ntiles is chosen to be much larger than Nkeys so that there is a good selection for the two most orthogonal sequences. The two sequences are stored in Key matrix which is Nkeys x Nsample. The spread codes only need to be regenerated when the number of bits changes such that the number of samples per bit changes. The spread codes are needed by both the modulator and the demodulator.
3. DSSS MODULATION The modulator inputs the M=2 bit sequences and knows the full number of samples N to be used in the system. It first determines the number of samples, Nsample, per bit. Then it creates a positive and negative bit complimentary sequence, b1 and b0, for each bit sequence from the original B sequence. Then using the kronecker product to upsample the bit sequence and simultaneously 5
encode with one spread code per bit sequence, such that the full length sequences sm1 and sm0 are created. Then sm0 and sm1 are simply added together to create “seach” signal.
Figure 3‐1: M bipolar leg configuration for encoding the upsampled bit sequence with the spread codes.
The M bipolar legs DSSS modulation is shown in Fig. 3-1. There are other ways to encode the same signal but this is more representative of what is being done in the program. Note that in the figure, bu(t) is up sampled whereas in the program, the upsampling and encoding are all done at once using the kron() function. % INSERT MODULATION EQUATION: % INSERT MODULATION EQUATION: % INSERT MODULATION EQUATION: Inputs sm vector, Key matrix, t and N % create DSSS modulation signal s b0=zeros(Nseq,Nbit); b1=zeros(Nseq,Nbit); sm0=zeros(Nseq,Nbit*Nsample); sm1=zeros(Nseq,Nbit*Nsample); bmA=zeros(Nseq,Nbit*Nsample); bm=zeros(Nseq,N); seach=zeros(Nseq,1,N); s=zeros(1,N); unitv=ones(1,Nsample); for nseq=1:Nseq b1(nseq,1:Nbit)=B(nseq,1:Nbit); % encode 1 bit locations b0(nseq,1:Nbit)=-(1-B(nseq,1:Nbit)); % encode 0 bit locations sm1(nseq,:)=kron(b1(nseq,:),Key(nseq,:)); % form continuous time dsss signal for ones sm0(nseq,:)=kron(b0(nseq,:),Key(nseq,:)); % form continuous time dsss signal for ones bmA(nseq,:)=kron(b0(nseq,:),unitv)+kron(b1(nseq,:),unitv); % combine 1s and 0s by adding the two signals sm1 and sm0 if N > (Nsample*Nbit) seach(nseq,1:(Nsample*Nbit))=sm0(nseq,1:(Nsample*Nbit))+sm1(nseq,1:(Nsample*Nbit)); bm(nseq,1:(Nsample*Nbit))=bmA(nseq,1:(Nsample*Nbit)); else seach(nseq,:)=sm1(nseq,:)+sm0(nseq,:); % number of bits * Nsample is equal to N bm(nseq,:)=bmA(nseq,:); end; % normalize seach(nseq,:)=seach(nseq,:)/max(max(abs(seach(nseq,:)))); % form composite s=s+seach(nseq,:); end; % normalize s=s/max(max(abs(s))); size(s) % verify shape % END OF MODULATION INSERT % END OF MODULATION INSERT % END OF MODULATION INSERT
A variable number of subplots is outputted with the composite signal plot using the following code: 6
figure(3); if Nbit<(Nshowbits+1) for nseq=1:Nseq subplot((Nseq+1),1,nseq); plot(t,seach(nseq,:)) axis([1,N,-1.1,1.1]); title('Message Signal'); end; subplot((Nseq+1),1,(Nseq+1)); plot(t,s); title('Composite Message Signal'); axis([1,N,-1.1,1.1]); else ts=0:(Nsample*Nshowbits-1); for nseq=1:Nseq subplot((Nseq+1),1,nseq); plot(ts,seach(nseq,1:(Nsample*Nshowbits)),ts,bm(nseq,1:(Nsample*Nshowbits))); axis([0,(Nsample*Nshowbits-1),-1.1,1.1]); title('Sample section of Message Signal'); end; subplot((Nseq+1),1,(Nseq+1)); plot(ts,s(1:(Nsample*Nshowbits))); title('Composite Message Signal'); axis([0,(Nsample*Nshowbits-1),-1.1,1.1]); end; print -djpeg Modulator_figure3
Figure 3‐2: Sample section of Message sequences after upsampling and encoding with spread codes. Bottom subplot is the composite signal.
The spectrum of the composite signal is shown in Fig. 3-3. % FT of modulated waveform Sall=abs(fftshift(fft(s))); figure(4); k=0:(N-1); k=k-N/2; plot(k,Sall); title('Spectrum of DSSS Signal'); print -djpeg Modulator_figure4
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Figure 3‐3: DSSS modulated signal spectra.
It can be seen from Figs. 3-2 and 3-3 that the modulated signal and associated spectrum look like noise. In this regard DSSS gives a certain amount of security and to demodulate it, one must know the spread codes. % create the bit check matrix to only be used by the Bcheckxx.m file % YOU CANNOT PASS THIS INFORMATION TO YOUR DEMODULATOR!! samplepulse=zeros(1,Nsample); samplepulse(floor(Nsample/2))=1; Bcheck=zeros(Nseq,N); b1check=zeros(Nseq,Nbit); bchecktemp=zeros(1,Nsample*Nbit); % modulate first sequence to either +1 and -1 values for nseq=1:Nseq b1check(nseq,1:Nbit)=2*B(nseq,1:Nbit)-1; bchecktemp=kron(b1check(nseq,:),samplepulse); if N > (Nsample*Nbit) Bcheck(nseq,1:(Nsample*Nbit))=bchecktemp(1:(Nsample*Nbit)); else Bcheck(nseq,:)=bchecktemp; end; end; figure(5); if Nbit<(Nshowbits+1) for nseq=1:Nseq subplot((Nseq+1),1,nseq); plot(t,Bcheck(nseq,:)) axis([1,N,-1.1,1.1]); title('Bit Check Signal'); end; else ts=0:(Nsample*Nshowbits-1); for nseq=1:Nseq subplot((Nseq+1),1,nseq); plot(ts,Bcheck(nseq,1:(Nsample*Nshowbits)),ts,bm(nseq,1:(Nsample*Nshowbits))); axis([0,(Nsample*Nshowbits-1),-1.1,1.1]); title('Bit Check Signal'); end; end; print -djpeg Modulator_figure5 save 'DSSS_signal' s; save 'DSSS_Bcheck' Bcheck;
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Figure 3‐4: Sample section of the two bit check signal subplots.
There are two signals shown in Fig. 3-4, the bipolar binary signal in blue and the bitcheck signal in green. The bitcheck has been slightly attenuated from its true +/- 1 value to disconnect from the binary signal curve but it really has 3 values +1, 0 and -1. A “0” indicates no bit value in that time location, a “1” indicates there should be a high bit value and -1 indicates a low bit value for that time location. The student is not allowed to pass this information to the demodulator so there is no prior knowledge of the random bit sequence being used in the demodulator. However, bitcheckxx.m uses this information to test the demodulator output.
4. CHANNEL MODEL See Appendix Channel code. The channel filters and adds noise to the input signal. The mathematical representation of the channel output is r t s t h t ~ t
(4.1)
where s(t) is the input signal, h(t) is the channel impulse response and ~t is a white Gaussian noise process.
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Figure 4‐1: Log Magnitude Spectrum of input signal.
Fig. 4-1 is obtained by taking the log of the input spectrum such that log S f 0.1
Figure 4‐2: Channel filter response.
The channel uses a butterworth frequency response as shown in Fig. 4-2.
Figure 4‐3: Input response after filtering.
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5. DSSS DEMODULATION The DSSS demodulation is accomplished with a M leg correlation process. A flow diagram is shown in Fig. 5-1. The idea is to design a matched filter that multiplies the incoming signal with each of the spread codes followed by correlation with the pulse shape. Depending on whether the bits stream value is 1 or 0, the output is 1 or -1.
Figure 5‐1: DSSS correlation based demodulator.
The signal r(t) comes from the channel model and includes filtering and noise addition. The spread codes are tiled into a vector of length N. These vectors are elementwise multiplied by the input signal, one spread code for each leg of the demodulator. The multiplication is followed by a convolution with a rectangle function the length of the spread codes. This is effectively a moving average filter and the filter is centered such that its peak response occurs at the bit centers. The last operation can be to binarize each of the rn signals or in the case below, store the bipolar signals into matrix Bs. % INSERT DEMODULATION CODE: % INSERT DEMODULATION CODE: % INSERT DEMODULATION CODE: input Keys, Nbit and r S=fft(r); % create the M-array bipolar detectors t=0:(N-1); u=ones(1,Nbit); Ntemp=Nsample*Nbit; % create detection filter h=irect(1,Nsample,1,N); H=fft(h); % for nseq=1:Nseq Akey=Key(nseq,:);
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stemp=kron(u,Akey); % form continuous time dsss signal for ones sm=zeros(1,N); sm(1:Ntemp)=stemp(1:Ntemp); sm=sm.*r; % convolve with moving average rectangle Sm=fft(sm); rn=real(ifft(Sm.*H)); rn=rn-min(rn); rn=rn/max(rn); Bs(nseq,1:N)=rn; end; % END OF DEMODULATION INSERT: output real vector rn that is N long % END OF DEMODULATION INSERT: % END OF DEMODULATION INSERT:
The final step is scaling the signal to be between 0 and 1 for the bit check system. A section of bit reconstruction shown in Fig. 5-4.
Figure 5‐2: Reconstruction of first few bits of two bit stream signals.
6. PERFORMANCE VERIFICATION The detection error is determined by bitcheckxx.m See Appendix A.3 Bitcheck for source code. The bitcheck program checks for error in formatting and provides the false alarm and miss count. If a miss or false alarm occurs then the program will create a figure of the local region where the error occurred.
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Figure 6‐1: Estimate of 1 and 0 bit signal pdf.
Bitcheck also yields the “1” and “0” mean values {0, 1}, their Standard Deviations (STDs) {0, 1}, their discrimination measure and their probability density function estimates shown in Fig. 6-1. The discrimination measure is given by J 01
1 0 2
2 1
02
(6.1)
The specific bitcheck code for these performance values is % STATISTICAL ANALYSIS OF BINARY SIGNAL Bit1index=find(Bcheck>0.5); Bit0index=find(Bcheck<-0.5); [MBit1 NBit1]=size(Bit1index); [MBit0 NBit0]=size(Bit0index); Bits1=zeros(1,NBit1); Bits0=zeros(1,NBit0); Bits1=Bs(Bit1index); Bits0=Bs(Bit0index); mu1=mean(Bits1,2) mu0=mean(Bits0,2) var1=var(Bits1); var0=var(Bits0); STD1=sqrt(var1) STD0=sqrt(var0) SQRTofSNR1=mu1/STD1 SQRTofSNR0=mu0/STD0 Discriminate=abs(mu1-mu0)/sqrt(var0+var1)
The pdf estimation is performed by the following histogram operations: % HISTOGRAM OF BITS W=100; w=1:W; maxhist=1.5; minhist=-.5;
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% find coefficients to map from the received values to the histogram index % W=a*maxhist+b, 1=a*minhist+b, a=(W-1)/(maxhist-minhist) b=1-a*minhist acoef=(W-1)/(maxhist-minhist);bcoef=1-acoef*minhist; h1=zeros(1,W); for n=1:NBit1 m=floor(acoef*Bits1(n)+bcoef); if m>0 if m<(W+1) h1(m)=h1(m)+1; end; end; % if m>0 end; % for n h1=h1/NBit1; h0=zeros(1,W); for n=1:NBit0 m=floor(acoef*Bits0(n)+bcoef); if m>0 if m<(W+1) h0(m)=h0(m)+1; end; end; % if m>0 end; % for n h0=h0/NBit0; maxhisto=max(h0); if maxhisto2 % this is not optimized yet % just grab keys with first key as first tile % other keys chosen to have minimum correlation with 1st one Row=1;Col=1; km0=1+(Row-1)*Nsample;km1=Row*Nsample; A=w(km0:km1); A=A-(sum(A)/Nsample); A=sqrt(Nsample)*A/sqrt(A*A'); Keys(1,1:Nsample)=A; for nkey=2:Nkeys; Col=nkey; kn0=1+(Col-1)*Nsample;kn1=Col*Nsample; B=w(kn0:kn1); B=B-(sum(B)/Nsample); B=sqrt(Nsample)*B/sqrt(B*B'); Keys(nkey,1:Nsample)=B; end; % for nkeys else % first and second keys with minimum correlation J=find(R==min(min(R))); % J = m * N2 + n Col=floor((J(1)-1)/N2); Row=floor((((J(1)-1)/N2)-Col)*N2); Row=Row+1; % convert to Matlab index Col=Col+1; % convert to Matlab index km0=1+(Row-1)*Nsample;km1=Row*Nsample;
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A=w(km0:km1); A=A-(sum(A)/Nsample); % A=sqrt(Nsample)*A/sqrt(A*A'); Keys(1,1:Nsample)=A; kn0=1+(Col-1)*Nsample;kn1=Col*Nsample; B=w(kn0:kn1); B=B-(sum(B)/Nsample); % B=sqrt(Nsample)*B/sqrt(B*B'); Keys(2,1:Nsample)=B; end; % end of Nkeys end
A.3 CHANNEL % channel function clear all; noiseCoef=0.01 % input active group load 'activegroup' groupname; groupname % input groupname_signal.mat filename=sprintf('%s_signal.mat',groupname); load(filename); % make sure s is real signal=real(s); [M,N]=size(signal) if N~= 1048576 'Incorrect vector length, should be 1048576' end; % % form filter fc=N/4; Norder=6; fmax=N/2; n=1:N; K=1; % filter gain % low pass filter [f HLP]=lp_butterworth_oN_dft(fc,K,fmax,N,Norder); % filter signal through channel S=fft(signal); H=HLP; R=S.*H; sn=real(ifft(R)); k=n;k=k-N/2; figure(1); plot(k,log(abs(fftshift(S))+.1)); xlabel('Log Magnitude Spectrum of Input Signal'); print -djpeg Channel_figure1 figure(2); plot(k,abs(fftshift(H))); axis([k(1),k(N),-.1, 1.1]); xlabel('Spectrum of Channel'); print -djpeg Channel_figure2 figure(3); plot(k,fftshift(log(abs(R)+.1)));
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xlabel('Log Spectrum of Output Signal, No Noise'); print -djpeg Channel_figure3 % find noise deviation sigma=noiseCoef*(max(signal)-min(signal)) % add noise w=sigma*randn(1,N); r=sn+w; % store result in groupname_r.mat filename=sprintf('%s_r.mat',groupname); save(filename,'r'); % PLOT spectrum and sample sections of the signal figure(4); Nsamplesection=20; Nsamples=floor(N/Nsamplesection); if N 0.5 % "1" should be present in check signal if Bs(m,n)>0.5 Btest(m,nb)=1; else 'ERROR, missing 1' 'Bcheck' Bcheck(m,n) 'Bs' Bs(m,n) m n nb if Nerror<10 figure(2+1+Nerror);
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istart=n-(2*N/Nbit); istop=n+(2*N/Nbit); if istart<1 istart=1 end; if istop>N istop=N end; clear x; x=1:(1+istop-istart); btemp=x; bchecktemp=x; btemp(1:(1+istop-istart))=Bs(m,istart:istop); bchecktemp(1:(1+istop-istart))=Bcheck(m,istart:istop); plot(x,btemp,x,bchecktemp); %clear x,btemp,bchecktemp; end; miss=miss+1; Nerror=Nerror+1; end; nb=nb+1; nbreceived=nbreceived+1; end; if Bcheck(m,n) < -0.5 % "-1" should be present in check signal if Bs(m,n) < 0.5 % "0" is present demodulated/binarized signal Btest(m,nb)=0; else 'ERROR, missing 0' 'Bcheck' Bcheck(m,n) 'Bs' Bs(m,n) m n nb if Nerror<10 figure(2+1+Nerror); istart=n-(2*N/Nbit); istop=n+(2*N/Nbit); if istart<1 istart=1 end; if istop>N istop=N end; clear x; x=1:(1+istop-istart); btemp=x; bchecktemp=btemp; btemp(1:(1+istop-istart))=Bs(m,istart:istop); bchecktemp(1:(1+istop-istart))=Bcheck(m,istart:istop); istart istop size(x) size(btemp) size(bchecktemp) plot(x,btemp,x,bchecktemp);
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end; false=false+1; Nerror=Nerror+1; end; nb=nb+1; nbreceived=nbreceived+1; end; end; nbsent=Nbit nbreceived miss false Nerror if nbsent~=nbreceived 'Error between sent and recieved' 'Number of ones and zeros sent' Nones=sum(sum(B)) Nzeros=nbsent-Nones end; % STATISTICAL ANALYSIS OF BINARY SIGNAL Bit1index=find(Bcheck>0.5); Bit0index=find(Bcheck<-0.5); [MBit1 NBit1]=size(Bit1index); [MBit0 NBit0]=size(Bit0index); Bits1=zeros(1,NBit1); Bits0=zeros(1,NBit0); Bits1=Bs(Bit1index); Bits0=Bs(Bit0index); mu1=mean(Bits1,2) mu0=mean(Bits0,2) var1=var(Bits1); var0=var(Bits0); STD1=sqrt(var1) STD0=sqrt(var0) SQRTofSNR1=mu1/STD1 SQRTofSNR0=mu0/STD0 Discriminate=abs(mu1-mu0)/sqrt(var0+var1) % HISTOGRAM OF BITS W=100; w=1:W; maxhist=1.5; minhist=-.5; % find coefficients to map from the received values to the histogram index % W=a*maxhist+b, 1=a*minhist+b, a=(W-1)/(maxhist-minhist) b=1-a*minhist acoef=(W-1)/(maxhist-minhist);bcoef=1-acoef*minhist; h1=zeros(1,W); for n=1:NBit1 m=floor(acoef*Bits1(n)+bcoef); if m>0 if m<(W+1) h1(m)=h1(m)+1; end; end; % if m>0 end; % for n h1=h1/NBit1; h0=zeros(1,W); for n=1:NBit0
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m=floor(acoef*Bits0(n)+bcoef); if m>0 if m<(W+1) h0(m)=h0(m)+1; end; end; % if m>0 end; % for n h0=h0/NBit0; maxhisto=max(h0); if maxhisto
