1. DIGITAL COMMUNICATION USING MATLAB
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MULTI FEEDBACK SIGMA DELTA MODULATOR
This sample assignment shows A fairly hardware exact simulation of a multi-feedback SDM
MFB_SDM_4.m
%Multi-feedback SDM
%this is a simulation of a multi feedback SDM where we look at the noise
%generated after a lengthy simulation time. The inputs are the fraction we
%are trying to synthesize, and the number of bits for the accumulators.
%this is a case with fixed feedback gains of 1.
%The values in the accumulators are all positive, there is a translation
%from the inputs and outputs so we do not deal with negative values in
%these registers.
%The output is from -4 to +4, for example an accumulator value from 0 to
%(1/9)*2^AccumulatorSize would become -4 and from (8/9)*2^AccumulatorSize to the
%max value would be +4
%Adding dither fixes the issues at fractional amounts such as 0.5 but it
%appears if you simply do 0.50001 the thing works ok, simplest is to program in
%a little offset to any frequency you want, then you don't have to have a
%hardware solution. Should check with Monte Carlo simulation.
%The system is as follows;
% Fraction => Subtract => Accumulate1 => Subtract => Accumulate2 => Subtract => Accumulate3
=> Quantize => Divider
% ^ ^ ^ |
%
|_________________________|___________________________|_________________________|
%In steady state the outputs of the accumulators should equal the fraction
%value. In this simulation the quantizer does not provide feedback until a limit
%is reached, so the error output of the accumulators grows before feedback
%contains it.
%The error from accumulator 1 changes at a roughly linear rate, accumulator
%2 error changes at roughly an error^2 rate, and accumulator 3 is roughly an error^3 rate. The
%large swing this causes in the error out of accumulator 3 causes the

2. %quantizer to cover most of it's output values. The low gain quantized feedback to
%the 3 subtraction points keeps the system constrained.
%The number on the busses that connect the accumulators represents the
%fraction we want. Whatever number we enter for the fraction into the first
%accumulator is the number that should occur on all the busses. This is
%what would happen if there were continuous feedback, but in this case the
%quantizer is a gateway that allows feedback only when specific points are reached.
%The -4 to +4 on the output is mapped into equal regions of the number on
%the bus. For example if we have a bus of 27 bits, then the possible
%values on the bus are 0 to 2^27-1. We break this into 9 equal regions of
%width BinSize, calculated below. 0 to BinSize represents -4, BinSize to
%2*BinSize = -3, and so on. We calculate the desired fraction in terms of 2^27
% as shown for FractionalInternal. This is our desired fraction converted
% to our Bus value.
%My hypothesis is that the 3rd order SDM allows for finer resolution of
%small errors. The larger swings gives more granularity for discerning
%small differences in error. Simulations show no real improvement in close
%in noise sidebands if the order is increased higher than 3rd order.
%
%The accumulators are made 1 bit larger than the bus to ensure no overflow
%occurs, in any case there is a check for overflow.
%1/29/2013 added better plot system and description
%1/29/2013 changed the quantization mapping
clear
BusSize=28; %bits
NumberSamples=2^16;
BinSize=floor(2^BusSize/9);
Fraction=.5501; %usable -2.5 to +2.5
FractionInternal=2^BusSize*4.5/9 + floor(BinSize*Fraction);
AccumulatorBits=BusSize+1 ; %bits
AccumulatorSize=2^AccumulatorBits;
Y1(1:NumberSamples)=0;%feedback internally
U1_1(1:NumberSamples)=0;%First accumulator output
Y2(1:NumberSamples)=0;%feedback internally
U1_2(1:NumberSamples)=0;%First accumulator output
U2_2(1:NumberSamples)=0;%Second accumulator output
Y3(1:NumberSamples)=0;%feedback internally
U1_3(1:NumberSamples)=0;%First accumulator output
U2_3(1:NumberSamples)=0;%Second accumulator output
U3_3(1:NumberSamples)=0;%Third accumulator output

3. Yout1(1:NumberSamples)=0;%output to the divider for 1 stage SDM
Yout2(1:NumberSamples)=0;%output to the divider for 2 stage SDM
Yout3(1:NumberSamples)=0;%output to the divider for 3 stage SDM
for index=2:NumberSamples
if U1_1(index-1)>=8*BinSize
Yout1(index)=4;
elseif U1_1(index-1)>=7*BinSize
Yout1(index)=3;
elseif U1_1(index-1)>=6*BinSize
Yout1(index)=2;
elseif U1_1(index-1)>=5*BinSize
Yout1(index)=1;
elseif U1_1(index-1)>=4*BinSize
Yout1(index)=0;
elseif U1_1(index-1)>=3*BinSize
Yout1(index)=-1;
elseif U1_1(index-1)>=2*BinSize
Yout1(index)=-2;
elseif U1_1(index-1)>=1*BinSize
Yout1(index)=-3;
else
Yout1(index)=-4;
end
Y1(index)=(Yout1(index)+4.5)*BinSize;
U1_1(index)=FractionInternal-Y1(index)+U1_1(index-1);
if U2_2(index-1)>=8*BinSize
Yout2(index)=4;
elseif U2_2(index-1)>=7*BinSize
Yout2(index)=3;
elseif U2_2(index-1)>=6*BinSize
Yout2(index)=2;
elseif U2_2(index-1)>=5*BinSize
Yout2(index)=1;
elseif U2_2(index-1)>=4*BinSize
Yout2(index)=0;
elseif U2_2(index-1)>=3*BinSize
Yout2(index)=-1;
elseif U2_2(index-1)>=2*BinSize
Yout2(index)=-2;
elseif U2_2(index-1)>=1*BinSize
Yout2(index)=-3;
else
Yout2(index)=-4;
end
Y2(index)=(Yout2(index)+4.5)*BinSize;
U1_2(index)=FractionInternal-Y2(index)+U1_2(index-1);

4. U2_2(index)=U1_2(index) -Y2(index)+U2_2(index-1);
if U2_3(index-1)>=8*BinSize
Yout3(index)=4;
elseif U3_3(index-1)>=7*BinSize
Yout3(index)=3;
elseif U3_3(index-1)>=6*BinSize
Yout3(index)=2;
elseif U3_3(index-1)>=5*BinSize
Yout3(index)=1;
elseif U3_3(index-1)>=4*BinSize
Yout3(index)=0;
elseif U3_3(index-1)>=3*BinSize
Yout3(index)=-1;
elseif U3_3(index-1)>=2*BinSize
Yout3(index)=-2;
elseif U3_3(index-1)>=1*BinSize
Yout3(index)=-3;
else
Yout3(index)=-4;
end
Y3(index)=(Yout3(index)+4.5)*BinSize;
U1_3(index)=FractionInternal-Y3(index)+U1_3(index-1);
U2_3(index)=U1_3(index) -Y3(index)+U2_3(index-1);
U3_3(index)=U2_3(index) -Y3(index)+U3_3(index-1);
end
if max(U1_1)>AccumulatorSize
fprintf('nerror in U1_1n')
end
if max(U1_2)>AccumulatorSize
fprintf('nerror in U1_2n')
end
if max(U1_3)>AccumulatorSize
fprintf('nerror in U1_3n')
end
if max(U2_2)>AccumulatorSize
fprintf('nerror in U2_2n')
end
if max(U2_3)>AccumulatorSize
fprintf('nerror in U2_3n')
end
if max(U3_3)>AccumulatorSize
fprintf('nerror in U3_3n')
end
MeanFrac=mean(Yout3);
fprintf('nMeanFracMFB= %1.4fn',MeanFrac)
figure(2)
SignalFreq1=20*log10(abs(fft(Yout1)));

5. plot(fftshift(SignalFreq1)-max(SignalFreq1),'g')
hold on
grid on
axis([0 NumberSamples -150 0]);
SignalFreq2=20*log10(abs(fft(Yout2)));
SignalFreq3=20*log10(abs(fft(Yout3)));
plot(fftshift(SignalFreq2)-max(SignalFreq2),'r')
plot(fftshift(SignalFreq3)-max(SignalFreq3),'b')
legend('1 stage','2 stage','3 stage')
title('MFB SDM Noise')
hold off
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