How to get the value of X-axis at the -3dB

I have a custom transfer function and i would like to get the -3dB BW with great accuracy.

I tried the find function over the frequency response (with some tolerance) and get the corresponding index,

but the return value isn't exactly -3dB. Is there any way to get freq_resp(index) = -3dB, without increasing the number of

total x-axis samples?

Thanks in advance

Answer 1

[William Rose]

@Chris, There is no guarantee that H(f) will = -3 dB at one of the frequencies at which you evaluate H(f). Therefore, since it seems you don't want to increase the number of samples, you can do linear interpolation to estimate the frequency at which H(f) crosses -3 dB. I will add an example. By the way, since you are trying to be highly accurate, you might want to find the -3.01 dB frequency, since that is the frequency at which the power is attenuated by 50%.

[William Rose]

@Chris,

A simple bandpass:

w=logspace(-2,2,39); 
s=1i*w;
T1=.1; T2=10;
H=s*T2./((1+s*T1).*(1+s*T2));
Hdb=20*log10(abs(H));
thresh=-3.01;
% plot results so far
semilogx(w,Hdb,'-b.');
hold on; yline(thresh,'--r')
grid on; xlabel('Frequency (rad/s)'); ylabel('|H| (dB)')

Find indices where threshold is crossed:

idxUp=find(diff(Hdb>=thresh)==1)  % up-crossing index

idxUp = 10

idxDn=find(diff(Hdb>=thresh)==-1)  % down-crossing index

idxDn = 29

Find frequencies by linear interpolation

w1=w(idxUp); w2=w(idxUp+1);  
h1=Hdb(idxUp); h2=Hdb(idxUp+1);  
wUp=w1+(w2-w1)*(thresh-h1)/(h2-h1);
w1=w(idxDn); w2=w(idxDn+1);  
h1=Hdb(idxDn); h2=Hdb(idxDn+1);  
wDn=w1+(w2-w1)*(thresh-h1)/(h2-h1);

display results on console and on plot

plot(wUp,thresh,'r*',wDn,thresh,'r*');
xline(wUp,'--g'); xline(wDn,'--g');

fprintf('wUp=%.3f, wDn=%.3f, passband width=%.3f rad/s\n',wUp,wDn,wDn-wUp)

wUp=0.101, wDn=9.998, passband width=9.897 rad/s

It is not perfect, since we're using linear interpolation on a curve that's not linear. If you sample H(w) more frequently than on the plot above, the accuracy will improve.