776final/hw12.m at master · mitchburnett/776final · GitHub

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

Tfsamp = 10; % samples/symbol
figC =1;
SPAN = 12;
SL = -SPAN/2;
SR = SPAN/2;
% pulse shape
g = rcosdesign(.5, 12, 10).'; % 50% excess bandwidth, spanning 12 symbols, 10 samples/symbol.
gn = g(1:Tfsamp:end);
g_span = linspace(SL, SR, length(g));
gn_span = linspace(SL, SR, length(gn));

% plot of the pulse shape and downsampled pulse shape gn = g(nT).
figure(figC); figC=figC+1; clf;
plot(g_span,g); grid on; hold on;
stem(gn_span,gn); hold off;
title('pulse shape - g(t)');
legend('g(t)', 'gn = g(nT)');

% channel model
c = zeros(Tfsamp+1,1);
c(1) = 1;
c(Tfsamp+1) = -.9;

% plot the channel model.
figure(figC); figC=figC+1;
stem((0:length(c)-1)/Tfsamp, c); grid on;
xlabel('t/T (symbols)');
title('c(t)');

% h(t) = g(t)*c(t);
h = conv(g,c);
h_span = linspace(-6,7,length(h));

% plot h(t)
figure(figC); figC=figC+1;
plot(h_span, abs(h)); grid on;
title('|h(t)|');

% x(t) = h(t)*h(-t)
x = conv(h,flip(h));
x_span = linspace(-13,13,length(x));

% plot x(t)
figure(figC); figC=figC+1;
plot(x_span,abs(x)); grid on;
title('|x(t)|');

% compute frequency domain for each signal.
nfft = 1024;
G = fftshift(fft(g,nfft));
C = fftshift(fft(c,nfft));
H = fftshift(fft(h,nfft));
X = fftshift(fft(x,nfft));

% plot G(f) and C(f)
figure(figC); figC=figC+1;
freq_span = linspace(-5,5,nfft);
plot(freq_span, 20*log10(abs(G))); grid on; hold on;
plot(freq_span, 20*log10(abs(C))); grid on; hold off;
ylim([-40, 20]);
legend('G(f)', 'C(f)');
ylabel('Magnitude (dB)');

% plot G(f) and H(f)
figure(figC); figC=figC+1;
plot(freq_span, 20*log10(abs(G))); grid on; hold on;
plot(freq_span, 20*log10(abs(H))); grid on; hold off;
ylim([-40, 20]);
xlim([-5, 5]);
legend('G(f)', 'H(f)');
ylabel('Magnitude (dB)');

% plot G(f) and X(f)
figure(figC); figC=figC+1;
plot(freq_span, 20*log10(abs(G))); grid on; hold on;
plot(freq_span, 20*log10(abs(X))); grid on; hold off;
ylim([-20, 40]);
xlim([-5, 5]);
legend('G(f)', 'X(f)');
ylabel('Magnitude (dB)');

% downsampled xn = x(nT)
xn = x(1:Tfsamp:end);
figure(figC); clf; figC=figC+1;
plot(x_span,abs(x)); grid on; hold on;
xn_span = linspace(-13,13, length(xn));
plot(xn_span, abs(xn), 'o'); hold off;
title('x(t) and xn=x(nT)');
legend('|x(t)|', 'xn');

% shorten xn to 99.99% energy
xn_short = xn((abs(xn) > .1));
figure(figC); clf; figC=figC+1;
stem([-1, 0, 1], abs(xn_short)); grid on;
title('Shorten xn');
legend('|xn|');

% compute DTFT of xn
XN = fftshift(fft(xn_short,nfft));
xn_span = linspace(-.5, .5, nfft);
figure(figC); clf; figC=figC+1;
plot(xn_span, 20*log10(abs(XN))); grid on;
ylim([-50, 50]);
title('DTFT of xn');
xlabel('frequency (cycles/symbol)');
ylabel('Magnitude (dB)');

% perform spectral factorization
xn_roots = roots(xn_short);
figure(figC); clf; figC=figC+1;
zplane(xn_roots); grid on;
title('Spectral Factorization of X(z)');

f_n = [1, -xn_roots(2)];

% normalize f_n to unit energy.
sigma_sq = sum(abs(f_n).^2);
sigma = sqrt(sigma_sq);

f_n = f_n/sigma;

% plot |f_n|
figure(figC); clf; figC=figC+1;
stem([0, 1], abs(f_n)); grid on;
title('|f_n|');

% compute DTFT of F(z)
F_z = fftshift(fft(f_n, nfft));
F_z_span = linspace(-0.5, 0.5, nfft);

% plot F(z)
figure(figC); clf; figC=figC+1;
plot(F_z_span, 20*log10(abs(F_z))); grid on;
ylim([-40, 20]);
title('DTFT of F(z)');
ylabel('Magnitude (dB)');