Discrete filter bode shows wrong results

Question

Vorname 2023-12-4

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编辑： Paul 2023-12-4

I have a discrete Filter consisting of the difference of two FIR filters. One filter is multiplied with 0.5, the other is delayed by z^-1. But at the end the bode diagram seems not correct and also when I let myself display the resulting function, it is not what I have calculated. Where is my fault?

Ts = 1;
FIR1 = tf([-0.1675, 0.1750, 0.3500, -0.2250, -0.1675, -0.1250], 1, Ts);  
FIR2 = tf([0.1250, -0.5850, 0.8500, -0.3000, 0.0500, -0.5850, -0.1250], 1, Ts);
H_1 = FIR1 * tf(1, [1 0], Ts); 
H_2 = 0.5 * FIR2;
H_diff = H_1 - H_2;
figure;
bode(H_diff)

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Answer 1

Star Strider 2023-12-4

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在 MATLAB Online 中打开

The fault, if there is one, is in not understanding the Control System Toolbox conventions. Addition creates a system with the two systems in parallel (multiplication creates them in series). If you want to subtract the bode results from each other, it is necessary to get the actual results first.

I believe this is what you want.

If it is not, please describe it in detail —

Ts = 1;

FIR1 = tf([-0.1675, 0.1750, 0.3500, -0.2250, -0.1675, -0.1250], 1, Ts);

FIR2 = tf([0.1250, -0.5850, 0.8500, -0.3000, 0.0500, -0.5850, -0.1250], 1, Ts);

H_1 = FIR1 * tf(1, [1 0], Ts);

H_2 = 0.5 * FIR2;

% H_diff = H_1 - H_2;

[mag2,phase2,wout2] = bode(H_2);

[mag1,phase1,wout1] = bode(H_1, wout2);

H_1c = squeeze(mag1) .* exp(1j*deg2rad(squeeze(phase1)));

H_2c = squeeze(mag2) .* exp(1j*deg2rad(squeeze(phase2)));

H_diff = H_1c - H_2c;

figure

tiledlayout(2,1)

nexttile

semilogx(wout1, mag2db(abs(H_diff)))

grid

xlabel('Frequency (rad/s)')

ylabel('Magnitude (dB)')

nexttile

semilogx(wout1, rad2deg(unwrap(angle(H_diff))))

grid

xlabel('Frequency (rad/s)')

ylabel('Phasee (deg)')

sgtitle('Bode Plot of ‘H\_diff’')

.

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Star Strider 2023-12-4

在 MATLAB Online 中打开

MATLAB is consistent.

Checking with the Symbolic Math Toolbox —

Ts = 1;

FIR1 = tf([-0.1675, 0.1750, 0.3500, -0.2250, -0.1675, -0.1250], 1, Ts);

FIR2 = tf([0.1250, -0.5850, 0.8500, -0.3000, 0.0500, -0.5850, -0.1250], 1, Ts);

H_1 = FIR1 * tf(1, [1 0], Ts)

H_1 = -0.1675 z^5 + 0.175 z^4 + 0.35 z^3 - 0.225 z^2 - 0.1675 z - 0.125 ----------------------------------------------------------------- z Sample time: 1 seconds Discrete-time transfer function.

H_2 = 0.5 * FIR2

H_2 = 0.0625 z^6 - 0.2925 z^5 + 0.425 z^4 - 0.15 z^3 + 0.025 z^2 - 0.2925 z - 0.0625 Sample time: 1 seconds Discrete-time transfer function.

% H_diff = H_1 - H_2

H_1.Numerator

ans = 1×1 cell array

{[-0.1675 0.1750 0.3500 -0.2250 -0.1675 -0.1250]}

H_2.Numerator

ans = 1×1 cell array

{[0.0625 -0.2925 0.4250 -0.1500 0.0250 -0.2925 -0.0625]}

syms z

H_1s = vpa(poly2sym(H_1.Numerator{:},z), 5)

H_1s =

H_2s = vpa(poly2sym(H_2.Numerator{:},z), 5)

H_2s =

H_diffs = vpa(H_1s - H_2s, 5)

H_diffs =

I trust these results.

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Answer 2

Paul 2023-12-4

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FIR filters are typically specified with coefficients in ascending powers of z^-1, but the the default for tf in the Control System Toolbox specifies the transfer function numerator and denominator in terms of descending powers of z.

Ts = 1;
FIR1 = tf([-0.1675, 0.1750, 0.3500, -0.2250, -0.1675, -0.1250], 1, Ts)
FIR1 =
 
  -0.1675 z^5 + 0.175 z^4 + 0.35 z^3 - 0.225 z^2 - 0.1675 z - 0.125
 
Sample time: 1 seconds
Discrete-time transfer function.
FIR2 = tf([0.1250, -0.5850, 0.8500, -0.3000, 0.0500, -0.5850, -0.1250], 1, Ts)
FIR2 =
 
  0.125 z^6 - 0.585 z^5 + 0.85 z^4 - 0.3 z^3 + 0.05 z^2 - 0.585 z - 0.125
 
Sample time: 1 seconds
Discrete-time transfer function.

Perhaps you really meant to specify FIR1 and FIR2 in terms of z^-1, like so

FIR1 = tf([-0.1675, 0.1750, 0.3500, -0.2250, -0.1675, -0.1250], 1, Ts,'Variable','z^-1')
FIR1 =
 
  -0.1675 + 0.175 z^-1 + 0.35 z^-2 - 0.225 z^-3 - 0.1675 z^-4 - 0.125 z^-5
 
Sample time: 1 seconds
Discrete-time transfer function.
FIR2 = tf([0.1250, -0.5850, 0.8500, -0.3000, 0.0500, -0.5850, -0.1250], 1, Ts,'Variable','z^-1')
FIR2 =
 
  0.125 - 0.585 z^-1 + 0.85 z^-2 - 0.3 z^-3 + 0.05 z^-4 - 0.585 z^-5 - 0.125 z^-6
 
Sample time: 1 seconds
Discrete-time transfer function.

And then

H_1 = FIR1 * tf(1, [1 0], Ts);

H_2 = 0.5 * FIR2;

H_diff = H_1 - H_2;

figure;

bode(H_diff)

If that's not the issue, it would be helpful to know why you think your Bode diagram is not correct and/or what the difference is between the resulting function (is that H_diff?) and what you calculated.

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Vorname 2023-12-4

编辑：Vorname 2023-12-4

Thank you a lot for this answer! Yes, I really mean a form like this: FIR1 = 0.125 - 0.585 z^-1 + 0.85 z^-2 - 0.3 z^-3 + 0.05 z^-4 - 0.585 z^-5 - 0.125 z^-6.

But I am still struggeling. as I calculated by hand, the resulting transfer function should be

(Fir1 delated by z^-1, FIR2 multiplied with 0.5, both get subtracted)

Paul 2023-12-4

编辑：Paul 2023-12-4

That actually looks like the definition of FIR2, not FIR1. I showed you how to enter that correctly. Presumably you can do the same for FIR1, which i also showed how to enter correctly.

Whatever you wanted to show for the "resulting transfer function" did not show up in your comment, so I don't know what you think the resulting transfer function should be. If it would help, you can insert an image file (.png, .jpeg, etc.) into a comment using the Image (leftmost) icon on the Insert menu.

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