-
Notifications
You must be signed in to change notification settings - Fork 0
/
channel.m
190 lines (168 loc) · 8.22 KB
/
channel.m
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
function [rpOut] = channel(rp)
%% This function computes a seriesRLC-M-parallelRLC antenna channel model
% and plots its frequency response.
%% Symbolically derive transfer function of resonator channel
%Define symbolic values for circuit elements
syms s R1 R2 C1 C2 L1 L2 M;
%Compute circuit impedances symbolically in Matlab
Z1 = 1/(1/R2 + s*C2);
Z2 = Z1 + s*(L2-M);
Z3 = 1/(1/Z2 + 1/(s*M));
Z4 = Z3 + 1/(s*C1) + s*(L1-M) + R1;
%Compute the transfer function symbolically using voltage divider rule
Vz3_Vpcd = Z3/Z4;
Vpicc_Vz3 = Z1/Z2;
Vpicc_Vpcd = Vpicc_Vz3*Vz3_Vpcd;
Vpicc_Vpcd = simplifyFraction(Vpicc_Vpcd); %simplify symbolic fraction
%print the symbolic channel transfer function
%{
fprintf('Symbolic channel transfer function =\n');
Vpicc_Vpcd
pretty(Vpicc_Vpcd)
%}
%% Enter actual circuit values, and compute resonant frequency, and Q factor
%Enter actual values for circuit elements
Cr = 2.86402e-10; %pre-set capacitance values
Cc = 5.58015e-11; %pre-set capacitance values
k = rp.k; %0<=k<=1; k is usually comprised between 0.03 and 0.3
%Compute circuit parameters of series RLC circuit
%Resonant frequency fc
%2*pi*fres = 1/sqrt(LC)
%1/(2*pi*fres) = sqrt(LC)
%1/(2*pi*fres)^2 = LC
%1/(C*(2*pi*fres)^2) = L
Lr = 1/(Cr*(2*pi*rp.fresr)^2);
%Q factor
%Q = 1/(2*pi*fres*R*C)
%1/Q = 2*pi*fres*R*C
%1/(2*pi*fres*C*Q) = R
Rr = 1/(2*pi*rp.fresr*Cr*rp.Qr);
%transfer function of VL/Vin in series circuit
numHseriesS = [Lr*Cr 0 0];
denHseriesS = [Lr*Cr Rr*Cr 1];
HseriesS = tf(numHseriesS,denHseriesS); %Series RLC transfer function in s domain
%Compute circuit parameters of parallel RLC circuit
%Resonant frequency fc
%2*pi*fres = 1/sqrt(LC)
%1/(2*pi*fres) = sqrt(LC)
%1/(2*pi*fres)^2 = LC
%1/(C*(2*pi*fres)^2) = L
Lc = 1/(Cc*(2*pi*rp.fresc)^2);
%Q factor
%Q = 2*pi*fres*C*R
%Q/(2*pi*fres*C) = R
Rc = rp.Qc/(2*pi*rp.fresc*Cc);
%transfer function of IR/Iin in parallel circuit
numHparallelS = [Lc 0];
denHparallelS = [Rc*Lc*Cc Lc Rc];
HparallelS = tf(numHparallelS,denHparallelS); %Parallel RLC transfer function in s domain
%Print resonant frequency and Q factor of reader and card
fprintf('Reader: fresr=%5.2e, Qr=%3.1f\n', 1/(2*pi*sqrt(Lr*Cr)), 1/(2*pi*rp.fresr*Cr*Rr));
fprintf('Card : fresc=%5.2e, Qc=%3.1f\n', 1/(2*pi*sqrt(Lc*Cc)), 2*pi*rp.fresc*Cc*Rc);
fprintf('Coupling factor: k=%2.2f\n', k)
%% Compute actual discrete time transfer function
%Substitue symbolic expressions with real values
Mval = k*sqrt(Lr*Lc);
Vpicc_Vpcd_sub = subs(Vpicc_Vpcd, [R1,R2,C1,C2,L1,L2,M], [Rr,Rc,Cr,Cc,Lr,Lc,Mval]);
%Convert the symbolic expression to Matlab transfer function by obtaining
%the numerator and denominator coefficients of the symbolic expression
[num, den] = numden(Vpicc_Vpcd_sub);
numChannelS = sym2poly(num);
denChannelS = sym2poly(den);
HchannelS = tf(numChannelS,denChannelS); %Channel transfer function in s domain
%Discretize the numerator and denominator coefficients of channel
HchannelZ = c2d(HchannelS,rp.Ts,'matched');
[numChannelZ, denChannelZ] = tfdata(HchannelZ,'V');
%% Map bandpass channel transfer function into its lowpass equivalent
% Method 1 ----------------------------------------------------------------
%Follows the method described on page 143 of "Simulation of Communication
%Systems" by Jeruchim, 2nd ed, 2000
%Decompose transfer function using partial fraction
[r,p,k] = residue(numChannelS,denChannelS);
if ~isempty(k)
error('Error in partial fraction expansion of Hchannel');
end
%Discard the poles that are located on the negative-frequency half-plane
r_pos = r(imag(p)>=0);
p_pos = p(imag(p)>=0);
%Shift poles located on the positive-frequency half-plane to the zero axis
%by substituting s -> s+jw
w0 = 2*pi*rp.fc;
p_pos_baseband = p_pos - 1j*w0;
%The lowpass equivalent transfer function
[numChannelBbS,denChannelBbS] = residue(r_pos,p_pos_baseband,k);
HchannelBbS = tf(numChannelBbS,denChannelBbS);
%Discretize the numerator and denominator coefficients of lowpass equivalent channel
HchannelBbZ_Sshift = c2d(HchannelBbS,rp.Ts,'matched');
[numChannelBbZ_Sshift,denChannelBbZ_Sshift] = tfdata(HchannelBbZ_Sshift,'V');
%--------------------------------------------------------------------------
% Method 2 ----------------------------------------------------------------
%Follows the method described on page 50 of "Signal Processing techniques
%for software radios" by Farhang, 2nd ed, 2010. Example of Matlab
%implementation is given on page 364 of this book by line
%"c=c.*exp(-j*2*pi*[0:length(c)-1]’*Ts*fc);" where c is the channel, Ts is
%the sample time, and fc is the carrier frequency.
%Substitute z^x -> z^x*e^(j*2*pi*fc*Ts*x) in the z-domain transfer function
%of the passband channel. Here, fc is the carrier frequency to donwconvert
%and Ts is the sampling time
numChannelBbZ_Zshift = numChannelZ.*exp(1j*2*pi*rp.fc*rp.Ts*(length(numChannelZ)-1:-1:0));
denChannelBbZ_Zshift = denChannelZ.*exp(1j*2*pi*rp.fc*rp.Ts*(length(denChannelZ)-1:-1:0));
HchannelBbZ_Zshift = tf(numChannelBbZ_Zshift,denChannelBbZ_Zshift,rp.Ts);
%--------------------------------------------------------------------------
%% Plot the (i) pole-zero plot of bandpass and lowpass equivalent channels,
% and (ii) frequency response of bandpass and lowpass equivalent channels
if rp.plotTrue == 1
% Specify position of figure on screen. rect = [left, bottom, width, height]
scrsz = get(0,'screensize'); %Get the screensize to specify figure size and location later
figure('OuterPosition',[1 40 scrsz(3) scrsz(4)-40])
%----------------------------------------------------------------------
handle1 = subaxis(2,2,1, 'Spacing', 0.06, 'Padding', 0.01, 'Margin', 0.03);
ss = 1; legStr = {};
hold on
pzmap(HchannelS,'-b'); legStr{ss} = 'HchannelS'; ss=ss+1;
pzmap(HchannelBbS,'-g'); legStr{ss} = 'HchannelBbS'; ss=ss+1;
hold off
legend(legStr,'location','northeast','interpreter','none');
%----------------------------------------------------------------------
handle2 = subaxis(2,2,2, 'Spacing', 0.06, 'Padding', 0.01, 'Margin', 0.03);
ss = 1; legStr = {};
hold on
pzplot(HchannelZ,'b'); legStr{ss} = 'HchannelZ'; ss=ss+1;
pzplot(HchannelBbZ_Sshift,'g'); legStr{ss} = 'HchannelBbZ_Sshift'; ss=ss+1;
pzplot(HchannelBbZ_Zshift,'r'); legStr{ss} = 'HchannelBbZ_Zshift'; ss=ss+1;
hold off
legend(legStr,'location','southwest','interpreter','none');
%----------------------------------------------------------------------
handle3 = subaxis(2,2,3, 'Spacing', 0.06, 'Padding', 0.01, 'Margin', 0.03);
fs = 1/rp.Ts;
freqAxis = linspace(-fs,fs,200);
[freqResHseriesS,~] = freqs(numHseriesS,denHseriesS,2*pi*freqAxis);
[freqResHparallelS,~] = freqs(numHparallelS,denHparallelS,2*pi*freqAxis);
[freqResChS,~] = freqs(numChannelS,denChannelS,2*pi*freqAxis);
[freqResChZ,~] = freqz(numChannelZ,denChannelZ,2*pi*freqAxis/fs);
[freqResChBbS,~] = freqs(numChannelBbS,denChannelBbS,2*pi*freqAxis);
[freqResChBbZ_Sshift,~] = freqz(numChannelBbZ_Sshift,denChannelBbZ_Sshift,2*pi*freqAxis/fs);
[freqResChBbZ_Zshift,~] = freqz(numChannelBbZ_Zshift,denChannelBbZ_Zshift,2*pi*freqAxis/fs);
ss = 1; legStr = {};
hold on
plot(freqAxis,20*log10(abs(freqResHseriesS)),'-c'); legStr{ss} = 'HseriesS'; ss=ss+1;
plot(freqAxis,20*log10(abs(freqResHparallelS)),'-m'); legStr{ss} = 'HparallelS'; ss=ss+1;
plot(freqAxis,20*log10(abs(freqResChS)),'-b'); legStr{ss} = 'ChS'; ss=ss+1;
plot(freqAxis,20*log10(abs(freqResChZ)),'--b'); legStr{ss} = 'ChZ'; ss=ss+1;
plot(freqAxis,20*log10(abs(freqResChBbS)),'-g'); legStr{ss} = 'ChBbS'; ss=ss+1;
plot(freqAxis,20*log10(abs(freqResChBbZ_Sshift)),'--g'); legStr{ss} = 'ChBbZ_Sshift'; ss=ss+1;
plot(freqAxis,20*log10(abs(freqResChBbZ_Zshift)),'--r'); legStr{ss} = 'ChBbZ_Zshift'; ss=ss+1;
hold off
legend(legStr,'location','southwest','interpreter','none');
xlabel('Frequency (Hz)'); ylabel('Magnitude (dB)'); grid;
ylim([-140 20]) %[xmin xmax ymin y max] = Set x and y axis limits
%----------------------------------------------------------------------
end
%% Function output
rpOut.numChannelZ = numChannelZ;
rpOut.denChannelZ = denChannelZ;
rpOut.numChannelBbZ_Sshift = numChannelBbZ_Sshift;
rpOut.denChannelBbZ_Sshift = denChannelBbZ_Sshift;
rpOut.numChannelBbZ_Zshift = numChannelBbZ_Zshift;
rpOut.denChannelBbZ_Zshift = denChannelBbZ_Zshift;
end