add lab6-filters

2024-02-03 23:35:38 +01:00
1 parent 4d5693aa1d
commit aa399dcc93
@@ -161,6 +161,7 @@

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@@ -17,6 +17,7 @@ In according to the sample theory, we have to choose at least 2 times the maximu
 In this case, the maximum frequency is 1 kHz, so we have to choose a sample frequency at least at 2 kHz.
 We take 10 time $f_c$ for the sample frequency, so we choose 20 kHz.

+
 ## Transfer function
 For calculate the transfer function, we have to calculate the poles and zeros of the filter.

@@ -29,64 +30,104 @@ Be careful, $k_a$ is not the DC gain.
 For a low pass filter without zeros, the "digital" gain is :
 $$k_{d} = k_{a}\cdot \prod_{i} \frac{1-e^{P_{ai}\cdot T}}{P_{ai}}$$

----------
+We can apply the bilinear transformation to the poles and zeros of the analog filter to obtain the poles and zeros of the digital filter.
+$$
+p_{di} = \frac{1+p_{ai}\cdot \frac{T}{2}}{1-p_{ai}\cdot \frac{T}{2}}
+\qquad \qquad
+z_{di} = \frac{1+z_{ai}\cdot \frac{T}{2}}{1-z_{ai}\cdot \frac{T}{2}}
+$$
+And for the low pass filter, the "bilinear" gain is :
+$$k_{d} = k_{a}\cdot \frac{1}{\prod_{i}\left( \frac{2}{T}-p_{ai} \right)}$$

-fc = 1000;
-wc = 2*pi*fc;
-fs = 10*fc;
-ts = 1/fs;
-
-num = [0 0 1];
-den = [1/(wc*wc) 1.414/wc 1];
-sysa = tf(num, den)
-[za,pa,ka] = tf2zp(num, den);
-%fvtool(num,den,'polezero')
-
-
-kd = ka .* ((1-exp(pa(1).*ts))./pa(1)) .* ((1-exp(pa(2).*ts))./pa(2));
-kd = real(kd);
-pd = (1+pa.*(ts./2)) ./ (1-pa.*(ts./2));
-zd = (1+za.*(ts/2)) ./ (1-za.*(ts/2));
-sysd = zpk(zd, pd, kd, ts)
-[zdb, pdb, kdb] = bilinear(za, pa, ka, fs);
-sysdb = zpk(zdb, pdb, kdb, ts)
-
-bode(sysa, sysd, sysdb);
+On the following diagram, we can see the bode diagram of the analog, digital filter and the filter with the bilinear transformation.

 ![bode_design_filter.svg](bode_design_filter.svg)

-----------
-clear;
-close all;
-clc;
+# IIR filter design

-H0 = 0;      % dB
-H1 = -3;     % dB
-H2 = -40;    % dB
-fn.p = 7000;   % Hz
-fn.s = 9000   % Hz
-fs = 48000;  % Hz
+We must design an IIR filter with the following specifications :

-fa.p = (fs/pi) * tan((pi*fn.p)/fs);
-fa.s = (fs/pi) * tan((pi*fn.s)/fs)
+- $H_0 = 0 [dB]$
+- $H_1 = -3 [dB]$
+- $H_2 = -40 [dB]$
+- $f_{pass} = 7 [kHz]$
+- $f_{stop} = 9 [kHz]$
+- $f_s = 48 [kHz]$

-wp = 2*pi*fa.p;
-ws = 2*pi*fa.s;
-ts = 1/fs;
-
-[n, wp] = ellipord(wp, ws, 3, 40, 's')
-[z, p, k] = ellip(n, 3, 40, wp, 's');
-sys = zpk(z, p, k);
-[zd, pd, kd] = bilinear(z, p, k, fs);
-sysd = zpk(zd, pd, kd, ts)
-bode(sys, sysd)
-title('Bode Diagram', 'Color', '#808080');
-xlabel('Frequency (rad/s)', 'Color', '#808080');
-ylabel('Magnitude (dB)', 'Color', '#808080');
-ylabel('Phase (deg)', 'Color', '#808080');
-legend('digital','bilinear', 'Color', 'white')
-sos = zp2sos(z,p,k);
+We first calculate the analog filter for search the order with the function *ellipord*.
+Then, we calculate the poles and zeros with the function *ellip*.
+We apply the bilinear transformation to the poles and zeros to obtain the poles and zeros of the digital filter.
+Finally, we can plot the bode diagram of the digital filter and the filter with the bilinear transformation.

 ![iir_filter.svg](iir_filter.svg)


+# DSP implementation
+
+We will implement the previously designed digital filter using a C program that will run in real-time on a fixed-point DSP.
+We will employ a cascade structure, consisting of 2 second-order stages.
+We will use the TMS320C6211 DSK from Texas Instruments.
+
+The following equation represent an universal transfer function of a second-order filter :
+$$H(z) = \frac{b_{0}+b_{1}\cdot z^{-1}+b_{2}\cdot z^{-2}}{a_{0}+a_{1}\cdot z^{-1}+a_{2}\cdot z^{-2}}$$
+
+The following figure depicts the block diagram of this implementation in the case where a0 = 1
+This structure is commonly referred to as "Direct Form I." It requires the storage of past values of both the input and output.
+
+![direct_form_1.png](direct_form_1.svg)
+
+So we have the difference equation :
+$$y(n) = b_{0}\cdot x(n)+b_{1}\cdot x(n-1)+b_{2}\cdot x(n-2)-a_{1}\cdot y(n-1)-a_{2}\cdot y(n-2)$$
+
+It exists a second structure called "Direct Form II" which is more efficient in terms of computation and memory usage.
+
+![direct_form_2.png](direct_form_2.svg)
+
+The difference equation is :
+$$\begin{align}
+y(n) &= tmp\cdot b_{0}+w[0]\cdot b_{1}+w[1]\cdot b_{2} \\
+\\
+tmp &= x(n)-a_{1}\cdot w[0]-a_{2}\cdot w[1] \\
+\\
+w[0] &= tmp_{prev} \\
+\\
+w[1] &= w[0]_{prev}
+\end{align}$$
+
+The variables can be translated into code by shifting the coefficients 14 positions to the left to enhance precision
+after the decimal point, as the DSP only works with integers.
+
+Structures for the coefficients and the state variables are defined as follows :
+```c
+typedef struct {
+    short b0;
+    short b1;
+    short b2;
+    short a0;
+    short a1;
+    short a2;
+} coeff_sec_order;
+
+short w3[2];
+short w4[2];
+
+coeff_sec_order stage1 = {  469,    349,   469, 16384, -23699, 10692};
+coeff_sec_order stage2 = {16384, -11187, 16384, 16384, -19657, 14890};
+```
+
+The following code is the implementation of the Direct Form II function :
+```c
+void iir_sec_order_dir_s(coeff_sec_order Coeff, short*w, short* Input, short* Output) {
+    register short tmp;
+    
+    tmp = (long)* Input - (( (Long)Coeff.a1 * (long)w[0]) >> SHIFT) - (( (long)Coeff.a2 * (long)w[1]) >> SHIFT);
+    *Output = (( (long)Coeff.b0 * (long)tmp >> SHIFT) + (( (long)Coeff.b1 * (long)w[0]) >> SHIFT) + (( (long)Coeff.b2 * (long)w[1]) >> SHIFT);
+    
+    w[1] = w[0];
+    w[0] = tmp;
+}
+```
+
+For a better effect of the filter, the function can be called several times.
+
+
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