butterworth filter gain higher than zero

Matlab R2021b

 fs = 30000; %sampling freq
 fn=fs/2; 
 n = 10; %order
 Wn_Low = 3000;
 Wn_High = 300;
 
 [a,b,c] = butter(n, Wn_Low/fn, 'low'); 
 [soslow,glow] = zp2sos(a,b,c); 
 fvtool(soslow, Analysis="freq")

This functionality is not available on remote platforms.

 [d,e,f] = butter(n, Wn_High/fn, 'high'); 
 [soshigh,ghigh] = zp2sos(d,e,f); 
 fvtool(soshigh, Analysis="freq")
 

Running these codes, which is sos filter, at low pass filter, the gain (db) of the filter at 0~0.2 (normalized) is much higher than zero (above 100).

Running high pass filter, it also shows higher than zero (about 1).

Could you help us to solve this problem?

(This phenomena happens even though when we lowered the filter order 'n')

However, when using Tf filter code (same parameters used at sos),

 fs = 30000; %sampling freq
 fn=fs/2; 
 n = 10; %order
 Wn_Low = 3000;
 Wn_High = 300;
 
 [p,q] = butter(n, Wn_Low/fn, 'low'); 
 fvtool(p,q, Analysis="freq")
 
 [r,s] = butter(n, Wn_High/fn, 'high'); 
 fvtool(r,s, Analysis="freq")
 

the gain was all below zero...

Thanks in advance!

Answer 1

[Star Strider]

The problem is that the plots provided by either fvtool or freqz do not use the scaling factor (‘glow’ here).

Plot them separately and incorporate that, and it works correctly —

fs = 30000; %sampling freq
fn=fs/2; 
n = 10; %order
Wn_Low = 3000;
Wn_High = 300;

[a,b,c] = butter(n, Wn_Low/fn, 'low'); 
[soslow,glow] = zp2sos(a,b,c);

[h,fv] = freqz(soslow, 2^16, fs);

figure
subplot(2,1,1)
plot(fv, mag2db(abs(h)*glow))                               % Multiply By 'g'
grid
xlabel('Frequency (Hz)')
ylabel('Magnitude (dB)')
subplot(2,1,2)
plot(fv, rad2deg(unwrap(angle(h))))
grid
xlabel('Frequency (Hz)')
ylabel('Phase (degrees)')

The approach in your code is correct, and preferred over the transfer function realisation. It is simply necessary to adjust the plots provided by freqz to get the correct result.

.

[Chocola]

Thank you very much! It helped a lot!

If it is okay for you, may I ask one more question?

If the set parameters are the same for both sos and TF, (like the code above)

will the performance of the sos code and TF code be different?

(e.g., applying to filtfilt)

It is much simpler to just use

[p,q] = butter(n, Wn_Low/fn, 'low')

rather than

[a,b,c] = butter(n, Wn_Low/fn, 'low'); 
[soslow,glow] = zp2sos(a,b,c);

When I searched for this question in google and Matlab, I found the answers: "transforming to sos is much stabler than TF in high order filtering."

If sos not only has the same filtering performance as TF and is much more stable, then why did Matlab also make a TF code? Don't we need only sos?

Thanks in advance again!

[Star Strider]

As always, my pleasure!

‘will the performance of the sos code and TF code be different?’

Perhaps not with this filter, however certainly for others. See the documentation section on Limitations for an extended discussion as to the reason the second-order-section implementation is preferable.

.

[Chocola]

Thanks again!

[Star Strider]

As always, my pleasure!