Confusion in MATLAB filter roll off - matlab

Question

I have the following code in matlab:
L = 10000;
Power = -100;
A = 10^0.5;
Fs = 25e6;
fc1 = 100;
fc2 = 1e3;
fc3 = 10e3;
fc4 = 100e3;
fc5 = 1e6;
a1 = 2*pi*fc1/Fs;
a2 = 2*pi*fc2/Fs;
a3 = 2*pi*fc3/Fs;
a4 = 2*pi*fc4/Fs;
a5 = 2*pi*fc5/Fs;
x = wgn(1,L,Power);
y1 = zeros(1,L);
y2 = zeros(1,L);
y3 = zeros(1,L);
y4 = zeros(1,L);
y5 = zeros(1,L);
y = zeros(1,L);
for i = 2:L,
y1(i) = (1-a1)*y1(i-1) + a1*x(i);
y2(i) = (1-a2)*y2(i-1) + a2*x(i)/A;
y3(i) = (1-a2)*y3(i-1) + a3*x(i)/A^2;
y4(i) = (1-a2)*y4(i-1) + a4*x(i)/A^3;
y5(i) = (1-a2)*y5(i-1) + a5*x(i)/A^4;
y(i) = y1(i) + y2(i) + y3(i) + y4(i) + y5(i);
end
fft1 = fft(y);
fft1 = fft1(1:length(y)/2+1);
psd1 = (1/(Fs*length(y)))*abs(fft1).^2;
psd1(2:end-1) = 2*psd1(2:end-1);
freq = 0:Fs/length(y):Fs/2;
figure(3);
semilogx(freq,10*log10(psd1))
grid on
Ts = 40e-9;
z = tf('z',Ts);
H1 = a1/(1-(1-a1)*z^-1);
H2 = (a2/A)/(1-(1-a2)*z^-1);
H3 = (a3/A^2)/(1-(1-a3)*z^-1);
H4 = (a4/A^3)/(1-(1-a4)*z^-1);
H5 = (a5/A^4)/(1-(1-a5)*z^-1);
H = (H1 + H2 + H3 + H4 + H5);
figure(5);
bode(H),grid
The intent of this code is to model flicker noise, which has 10dB/dec of slope in its power spectral density.
To model that there is a filter whose output is y(i) and input x(i) which is white guassian noise in this case. In the bode plot of the filter (labelled as figure 5) I can see that it has 10dB/dec of roll-off as I intended.
But when I check the output noise (y(i) in this case) power spectral density (labelled as figure 3) I am seeing 20dB/dec of roll-off.
Could someone please explain what I did wrong and why I am not able to get the 10dB roll-off in power spectral density?

Answer 1

Your bode plot uses a transfer function which is the sum of 5 similar subfilters, each using a coefficient ai. In your implementation, you should similarly have 5 subfilters with a regular structure. Howerver, while copying each line you have left some of the coefficients for y3, y4 and y5 to use a2. You should get your expected result by simply substituting the respective a3, a4 and a5 like so:
for i = 2:L,
y1(i) = (1-a1)*y1(i-1) + a1*x(i);
y2(i) = (1-a2)*y2(i-1) + a2*x(i)/A;
y3(i) = (1-a3)*y3(i-1) + a3*x(i)/A^2;
y4(i) = (1-a4)*y4(i-1) + a4*x(i)/A^3;
y5(i) = (1-a5)*y5(i-1) + a5*x(i)/A^4;
y(i) = y1(i) + y2(i) + y3(i) + y4(i) + y5(i);
end