DC gain discrete-time transfer function error

Hello everyone,

I'm trying to plot the trasfer function of a CIC filter and I think I'm getting an error in DC gain. The code I'm using is shown here:

M = 8; %Decimation factor
N = 1; %Order of the filter

fs = 25e6; %Sampling frequency
T  = 1/fs; %Sampling period

z = tf('z',T);
gain = 1/(M);
num_tf = 1-z^(-M);
den_tf = 1-z^(-1);
tf_tf  = (gain*num_tf/den_tf)^N;
figure(1)
plot_tf=bodeplot(tf_tf);
setoptions(plot_tf,'FreqUnits','Hz');
dcgain(tf_tf)

The printed bodeplot is:

that shows a DC gain grather than 1 while the line

dcgain(tf_tf)

gives 1 as result (which should be the right one). What am I doing wrong? Why am I finding thi misbehaviour og bodeplot function?

Thanks in advance

Answer 1

[Paul]

I think the problem is just inaccuracy of the numerical computation of the freuqency response at very small frequencies. Keep in mind that tf_tf, as it's defined, would evaluate to 0/0 at z = exp(j*w) = 1 and that tf_tf as a pole/zero cancellation at z = 1. I don't think (don't know for sure) that bode() tries to deal with pole/zero cancellation. However, i think dcgain does try to deal with that, so it gives the expected answer. The desired, or expected, Bode plot can be obtained by using minreal() on tf_tf to remove the common pole and zero at z = 1

M = 8; %Decimation factor
N = 1; %Order of the filter
fs = 25e6; %Sampling frequency
T  = 1/fs; %Sampling period
z = tf('z',T);
gain = 1/(M);
num_tf = 1-z^(-M);
den_tf = 1-z^(-1);
tf_tf  = (gain*num_tf/den_tf)^N

tf_tf =
 
  0.125 z^9 - 0.125 z
  -------------------
       z^9 - z^8
 
Sample time: 4e-08 seconds
Discrete-time transfer function.

mrtf_tf = minreal(tf_tf)

mrtf_tf =
 
  0.125 z^7 + 0.125 z^6 + 0.125 z^5 + 0.125 z^4 + 0.125 z^3 + 0.125 z^2 + 0.125 z + 0.125
  ---------------------------------------------------------------------------------------
                                            z^7
 
Sample time: 4e-08 seconds
Discrete-time transfer function.

bode(tf_tf,mrtf_tf)

[Francesco Laezza]

Thank you very much for your explanation