# Stringency in Digital Filter Implementation

#### J_Rod

Joined Nov 4, 2014
109
Hey there,
I'm using MATLAB to implement some digital filters. I need to determine the filter attenuation, ATT, and transition width, TW. ATT = -20log_10(gain), with gain between 0 and 1, and TW = fs -fp, with fs the stopband frequency and fp the passband frequency. My question is that the assignment is telling me to choose the more stringent ATT or TW if necessary. Does that mean the smaller of the two values? For the bandpass filter, I have TW_1 = 400 and TW_2 = 1000, so do I use TW = 400? I apologize since this is not clear and there are not a lot of details, but my confusion (for now) is jsut what is meant by "more stringent"?
Thank you.

#### Papabravo

Joined Feb 24, 2006
20,378
Just for starters:
1. When the gain is 1, as it would be in the passband, ATT = 0 dB
2. when the gain is 0, as it would be in the stopband, ATT = ∞ dB
We know that a gain 0 is not physically realizable, so we need to pick a gain for the stopband that is good enough. Usually when we write a filter specification we say the attenuation in the passband is less than 0.5 dB for example. So the end of the passband is at 0.5 dB. Then we say the attenuation in the stopband must be greater than 60 dB for example. So the beginning of the stopband is at 60 dB. This means the transition band is from 0.5 dB to 60 dB. What is the gain for 0.5 dB and 60 dB of attenuation?

$$60 = -20 \cdot log(\text gain)$$
$$-3=log(\text gain)$$
$$10^{-3}=\text gain=.001$$

similarly

$$.5 = -20 \cdot log(\text gain)$$
$$-0.025=log(\text gain)$$
$$10^{-0.025}=\text gain\approx 0.944$$

By reference to this example for a lowpass filter, more stringent would be
passband attenuation ≤ 0.1 dB
stopband attenuation ≥ 80 dB​
for example.

Does that clear things up for you?

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#### J_Rod

Joined Nov 4, 2014
109
The transition band begins then at some frequency where gain is 0.5 dB and ends at some other frequency where gain is 60 dB, right? So then you can determine the gain since ATT = -20 log_10(gain). But then how do you get the values 0.1 dB and 80 dB? Does this mean that the transition band would begin at the same frequency but have a gain of 0.1 dB < 0.5 dB, and end at the same other frequency but have a value of 80 dB > 60 dB? In relation to transition width, the more stringent would be the larger width, is that correct?

#### Papabravo

Joined Feb 24, 2006
20,378
The frequencies which define the transition band are the same, but in the more stringent case the passband characteristic is flatter because it can only drop 0.1 dB for the entire passband instead of 0.5 dB. The rolloff in the transition band is steeper because it must get all the way to -80 dB in the same transition bandwidth and the gain in the stopband must be another order of magnitude smaller than in the 60 dB case. I'd call that a more stringent filter specification.