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Binary Phase Shift Keying with

BandPass Channel and Additive White

Gaussian Noise : Modulation andDemodulation

by Laurence G. Hassebrook

2-27-2013

We simulate Binary Phase Shift Keying (BPSK) with amplitude modulation and mixer based

demodulation. The modulated signal is synthesized by using an upsampled random bipolar bit

stream, modulated by a carrier wave and then sent through a bandpass channel and corrupted byAdditive White Gaussian Noise (AWGN). Assuming no phase error, the modulated signal is

demodulated using a mixer configuration. Because the binary signal is bipolar, the resulting

modulation is equivalent to phase modulation by 0 or . Both input and output signals areanalyzed for bandwidth and noise distribution. The goal is to reproduce the figures and

processing presented in this document using MATLAB. The signal length is N; number of bits isNbit; Standard Deviation is STD and carrier frequency is kc.

cl ear al l ; N=10000; Nbi t =8; %300; Nsampl e=f l oor ( N/ Nbi t ) ; STD=0. 1; %0. 5; kc=4*Nbi t ; kcut of f =kc/ 4; Nbi n=100; n=1: Nbi n;

Note that there are Nsample sample values for each bit. Hence the system is upsampled byNsample to simulate continuous time or what we call pseudo-continuous time. The noise level is

controlled by STD which is effectively the standard deviation of the Gaussian noise. The carrier

frequency is kc and kcutoff is used for both the channel bandwidth and the demodulatorlowpass filter cutoff frequency. We separate the received 1 and 0 values and estimate their

probability density functions for two experiments:

Case 1: Use Nbit=8 and STD=0.1.Case 2: Use Nbit=300 and STD=0.5.

1.BinarySequenceSynthesisGenerateNbitrandom bits using a the pseudo-random generater rand() such that

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% Random Bi nar y Si gnal wb=r and(1, Nbi t ) ; bi t s=bi nar i ze( wb) ;

The vector wbis has only one sample per bit which is not a good model for continuous time so

we upsample using a kronecker product such that.

ub=ones( 1, Nsampl e) ; bk=kron( bi t s, ub) ;

where bk is length Nb = NsamplexNbit which might be less than N. So to make sure we have a

signal N long we first generate a zero vector N long and then we move bk into it whicheffectively zero pads any mismatch in length.b=zeros( 1, N) ;Nb=Nsampl e*Nbi t ; t =1: N; b( 1: Nb) =bk( 1: Nb) ; %Force si gnal t o be N sampl es l ong

% t ur n si gnal i nt o a bi pol ar non- r et ur n t o zer o si gnal b=2*( b- 0. 5) ;

The result along with the BPSK modulation is shown in the next section.

2.BPSKModulationfor shift.BPSK modulation with phase shift is achieved by first generating a discrete cosine wave andthen elementwise multiplying it by the upsampled bipolar binary signal sequence bsuch that

% modul atesc=cos( 2*pi *kc*t / N) ; % car r i er s i gnal s=b. *sc;

To plot out both the binary signal and the modulated signal in the same plot, and store a jpegimage of this result, the matlab code is:

f i gur e( 1) ; pl ot ( t , b, t , s) ; t i t l e( ' BPSK Modul at i on' ) ; xl abel ( ' n' ) ; yl abel ( ' s( n) and b( n) ' ) ; axi s([ 1, N, - 1. 5, 1. 5] ) ;

l egend( ' bi nar y message' , ' BPSK' ) ; pr i nt - dj peg Fi g1_BPSKSi gnal ;

The resulting figures for is

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Figure 2.1: Case 1 binary signal and modulated signal.

The mathematical representation of the BPSK modulated signal is

tftbts c2cos (1)

where tis an integer index for sequence of lengthNand carrier frequencyfc= kc/N.

3. ChannelModelandSignalAnalysisThe channel is a lowpass model with Additive White Gaussian Noise (AWGN) on its output.

The lowpass component forces a bandlimited baseband onto the otherwise infinite bandwidth

BPSK signal. The model is shown in Fig. 3.1.

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Figure 3.1: Bandpass AWGN channel model.

The MATLAB code to perform the bandpass component is:

% f or m bandpass f i l t er Nor der =10; % f i l t er or der f max=N/ 2; % r equi r ed var i abl eK=1; % f i l t er gai n% kcut of f i s t he bandwi dt h of t he bandpass f i l t er% wher e kc i s cent er f r equency% band pass f i l t er [ f Hchannel ] =bp_but t er wor t h_oN_df t ( kc, kcut of f , K, f max, N, Nor der ) ; % f i l t er si gnal t hr ough channel S=f f t ( s); R=S. *Hchannel ; r0=real ( i f f t (R) ) ;

k=t ; f i gur e( 2) ; SUBPLOT( 2, 1, 1) ; pl ot ( k, Hchannel , k, abs( S) / ( 2*max( abs( S) ) ) ) ; t i t l e( ' BPSK and Channel Spect r a' ) ; xl abel ( ' k' ) ; yl abel ( ' H( f ) and S( f ) ' ) ; SUBPLOT( 2, 1, 2) ; pl ot ( k, Hchannel , k, abs( R) / ( 2*max( abs( R) ) ) ) ; t i t l e( ' Channel Fi l t ered BPSK and Channel Spect r a' ) ; xl abel ( ' k' ) ; yl abel ( ' H( f ) and R( f ) ' ) ; pr i nt - dj peg Fi g2_BPSKSpect r a;

For a low number of bits, the input and output spectra are difficult to see because they are so lowfrequency as shown in Fig. 3.2.

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Figure 3.2: (top) Channel input BPSK spectrum. (bottom) Bandlimited output spectrum of BPSK.

To generate the pseudo-random AWGN sequence we use randn() in MATLAB and add to thesignal r0(t) such that

twtstr ~0 (2)

The MATLAB code for the AWGN is

%% Gaussi an di st r i but ed noi sew=STD*r andn( 1, N) ; r =r 0+w; f i gur e( 3) pl ot ( t , r ) ; t i t l e( ' BPSK Modul at i on wi t h Noi se' ) ; xl abel ( ' n' ) ; yl abel ( ' r ( n) =s( n) +w( n) ' ) ; pr i nt - dj peg Fi g3_BPSKPl usNoi se;

The noisey signal is shown in Fig. 3.3.

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Figure 3.3: Signal output of channel with noise.

4.DemodulationandSignalAnalysisWe will use a mixer followed by a low pass filter to demodulate the BPSK signal as shown in

Fig. 4.1. Only one mixer is needed because the phase modulation is which means that a 1 bitwill demodulate to a 1 and a 0 bit will demodulate to -1. A mixer multiplies the received signal

with a replica of the carrier signal which must be in phase with the modulated carrier. The

multiplication creates what is known as a baseband and two frequency translated replicas of the

baseband centered at +/- 2kcfrequencies. So to reconstruct the signal we simply low pass filterthe multiplier output with a cutoff around kcutoff. Mathematically this is

thNtktrtr LPcb *2cos

(3)

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Figure 4.1: Mixer based demodulator.

The MATLAB code for the demodulator is

%% DEMODULATI ON USI NG A MI XERsr ef =sc; % r ef er ence si gnal % mi x t he r ef er ence wi t h t he i nput r 1=r . *sr ef ;

% f orm reconstr uct i on f i l t er % f i l t er wi t h some r ecommended paramet ersNor der=8; f max=N/ 2; K=8; % f i l t er gai n[ f H] =l p_but t er wort h_oN_df t ( kcut of f , K, f max, N, Nor der ) ; % f i l t er si gnal t hr ough channel vi a f r equency domai nR1=f f t ( r 1) ; R=R1. *H; rb=real ( i f f t (R) ) ;

To detect 0 and 1 bits separately, we do not want to sample the bit boundaries else we wouldget errors from the transitions. So we define binary windows within the bit boundaries, one for

the 1 detection and the other for the 0 detection.

% def i ne sampl i ng r egi on f or det ect or ueye=zer os( 1, Nsampl e) ; ueye( 1, f l oor ( Nsampl e/ 4) : f l oor ( 3*Nsampl e/ 4) ) =1; bkeye=kr on( bi t s, ueye) ; bkeyenot=kron( ( 1- bi t s) , ueye) ; beye=zer os( 1, N) ;beyenot =zeros( 1, N) ;beye(1: Nb)=bkeye( 1: Nb) ; %Force si gnal t o be N sampl es l ongbeyenot ( 1: Nb)=bkeyenot ( 1: Nb) ; %Force si gnal t o be N sampl es l ong

where beye is within the one boundaries and beyenot is within the zero boundaries. All these

signals are shown in Fig. 4.1

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Figure 4.2: Demodulated signal with the detection windows superimposed.

Using the window sequences we estimate pdfs for the 0 and 1 signals separately as

%% seper at e out t he two noi se di st r i but i onsJ 0=f i nd(beyenot>0. 5) ; J 1=f i nd(beye>0. 5) ; r b0=r b( J 0) ; r b1=r b( J 1) ; %

[ f 0 n0] = PDFest i mat or ( r b0, Nbi n) ; [ f 1 n1] = PDFest i mat or ( r b1, Nbi n) ; %f i gur e( 5) ; pl ot ( n0, f 0, n1, f 1) ; t i t l e( ' Demodul ated PDF Est i mat i on' ) ; xl abel ( ' r ' ) ; yl abel ( ' f ( r ) ' ) ; l egend( ' Detected Zeros' , ' Detect ed Ones' ) ; pr i nt - dj peg Fi g5_Demodul at edSi gnal PDFest i mat i on;

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Figure 4.3: Estimated pdfs within the "0" and "1" detection windows with case 1

5. Case2ResultsRerun your simulation with Nbit=300 and STD=0.5 and re-plot Figs. 3.2 and 4.3 as Figs. 5.1 and

5.2, respectively.

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Figure 5.1: (top) Channel input BPSK spectrum. (bottom) Bandlimited output spectrum of BPSK. Blue line is bandpass

filter envelope.

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Figure 5.2: Estimated pdfs within the "0" and "1" detection windows with case 2.