# Series connection of digital PT1 filters

I was trying design $N^{th}$ order filter, I was suggested a series connection of low-pass PT1 filter. Basically, the filter coefficients are like this:

$$a_i = \dbinom{n}{i} T^{i}$$

This works for $N^{th}$ order, while $a_1, a_2, a_3....a_n$ being the filter coefficient.

I don't know how this works, in theory. I tried searching, but could not come up with anything. if anyone has any idea about this. I want to know how this works.

• What is $T$? Is $N=n$ in your equations?
– MBaz
Sep 15, 2016 at 16:40
• $T$ is filter time constant, $T=\frac{1}{2*\pi *fc}$ andand sorry $N=n$ Sep 16, 2016 at 8:02

You could have mentioned where you got that formula from. But anyway, I'll try to reconstruct it.

A first order discrete-time low pass filter has the following transfer function:

$$H(z)=\frac{1+c}{1+cz^{-1}},\qquad -1<c<0\tag{1}$$

where the constant $c$ determines the cut-off frequency. If you cascade $N$ of these filters, the total transfer function is

$$H_N(z)=\prod_{k=1}^NH(z)=\frac{(1+c)^N}{(1+cz^{-1})^N}\tag{2}$$

Using the Binomial Theorem this can be written as

$$H_N(z)=\frac{(1+c)^N}{\displaystyle\sum_{k=0}^N\dbinom{N}{k}c^kz^{-k}}=\frac{(1+c)^N}{\displaystyle\sum_{k=0}^Na_kz^{-k}}\tag{3}$$

with denominator coefficients

$$a_k=\dbinom{N}{k}c^k\tag{4}$$

Note that the numerator $(1+c)^N$ is just there to make sure that the scaling is correct, i.e., that the transfer function equals $1$ at $z=1$ (DC).

Following MBaz's suggestion, I add the relationship between the constant $c$ in $(1)$ and the filter's $3\,\text{dB}$ cut-off frequency $\omega_c$. For this we have to solve

$$|H(e^{j\omega_c})|^2=\frac12|H(1)|^2=\frac12\tag{5}$$

From $(1)$ we get

$$\frac{(1+c)^2}{1+2c\cdot \cos(\omega_c)+c^2}=\frac12\tag{6}$$

from which we obtain after some elementary algebra

$$c=\cos(\omega_c)-2+\sqrt{\cos^2(\omega_c)-4\cos(\omega_c)+3}\tag{7}$$

• Is there any paper or book i can cite this? Sep 16, 2016 at 8:04
• @Aashu10: I don't know, it's just a basic application of the binomial theorem, nothing special. Sep 16, 2016 at 8:16
• Thank you for your answer. I had actually derived in another domain, I wanted to read more about these, that's the reason i was asking for a source. If you know anything about where i can read more about them to understand more. That'd be helpful Sep 16, 2016 at 8:36
• @Aashu10: OK, but what I wrote in my answer is really everything there is to it. Is it clear? Sep 16, 2016 at 8:41
• @MattL. Nice analysis; the only thing I would add is how to calculate $c$ to obtain a specified cutoff frequency $f_c$.
– MBaz
Sep 16, 2016 at 15:00