# Cepstrum Calculation of Rational Function H(z)

I am trying to solve my first problems at cepstrum calculation. I want to calculate the complex cepstrum $$\hat{h}[n]$$ of a signal $$h[n]$$ with Z-Transform: $$H(z)=\frac{(1-0.5z^{-1})(1+4z^{-2})}{(1-0.64z^{-2})}$$ I know that $$\hat{H}[z]=\log(H(z))$$ and $$\hat{h}[n]=\frac{1}{2π}\int_{-π}^{π}\log[H(e^{jω})]e^{jωn}dω$$

Which is the easiest way to approach this?

I started by applying $$\log()$$ to H(z) which gave me: $$\log(H(z))=\log(1-0.5z^{-1})+\log(1+4z^{-2})-\log(1-0.64z^{-2})\Rightarrow$$ $$\log(H(z))=\log(1-0.5z^{-1})+\log(4(z^{-2}+\frac{1}{4}))-\log((1-0.8z^{-1})(1+0.8z^{-1}))\Rightarrow$$ $$\log(H(z))=\log(1-0.5z^{-1})+\log(4(z^{-1}+\frac{j}{2})(z^{-1}-\frac{j}{2}))-\log((1-0.8z^{-1})(1+0.8z^{-1}))\Rightarrow$$ $$\log(H(z))=\log(1-0.5z^{-1})+\log(4)+\log(z^{-1}+\frac{j}{2})+\log(z^{-1}-\frac{j}{2}))-\log(1-0.8z^{-1})-\log(1+0.8z^{-1})$$

Is this apporach right? If so, how do I go on? Any help is appreciated. Thanks in advance!

• Is this an exercise from a book? – Matt L. Apr 28 '19 at 11:03
• @MattL. No it's not from a book. Is my approach right? I also consider making use of the power series: $$\log(1-az^{-1})=-\sum_{n=1}^{\infty} \frac{a^{n}}{n}z^{-n}$$ but not every term is in such a form. – MJ13 Apr 28 '19 at 11:23
• I asked because I think that you have to remove a linear phase term in order to compute the complex cepstrum. If it had been from a book I would have been surprised at the necessity to discover that fact. I'll write up an answer as soon as I have the time to do so. – Matt L. Apr 28 '19 at 13:42
• I am not used to computing complex cepstrums. I am now trying to solve my first problems. So, I am not sure about my approach. I just tried to use the definition and make use of some properties of the log() function. However, I am finding difficulty calculating the inverse Z -transform of the expression I end up to so that I get the complex cepstrum $\hat{h}[n]$. Thanks a lot for the response. I am looking forward for your answer. – MJ13 Apr 28 '19 at 13:57

## 1 Answer

We're looking for a series representation

$$\log H(z)=C(z)=\sum_{n=-\infty}^{\infty}c[n]z^{-n}\tag{1}$$

which converges in an annular region $$r_1<|z| with $$0.

Note that the poles as well as the zeros of $$H(z)$$ lead to singularities of $$C(z)$$. All poles and zeros of $$H(z)$$ inside the unit circle contribute to the right-sided part of $$c[n]$$, whereas the poles and zeros of $$H(z)$$ outside the unit circle contribute to the left-sided part of $$c[n]$$. Consequently, a minimum-phase transfer function $$H(z)$$ has a right-sided cepstrum.

The given function $$H(z)$$ has all its poles inside the unit circle ($$z^{\infty}_{1,2}=\pm 0.8$$, $$z^{\infty}_{3}=0$$), one of its zeros is inside the unit circle ($$z^{0}_{1}=0.5$$), and two zeros are outside the unit circle ($$z^{0}_{2,3}=\pm 2j$$). Consequently, its complex cepstrum $$c[n]$$ is a two-sided sequence.

In order to find the sequence $$c[n]$$ we make use of the well-known Mercator series:

$$\log(1-z)=-\sum_{n=1}^{\infty}\frac{z^n}{n},\qquad |z|<1\tag{2}$$

From $$(2)$$ it is clear that terms of the form

$$\log(1-az^{-1})=-\sum_{n=1}^{\infty}\frac{a^n}{n}z^{-n},\qquad |z|>|a|\tag{3}$$

contribute to the right-sided part of $$c[n]$$, whereas terms of the form

$$\log(1-bz)=-\sum_{n=1}^{\infty}\frac{b^n}{n}z^{n}=\sum_{n=-\infty}^{-1}\frac{b^{-n}}{n}z^{-n},\qquad |z|<\frac{1}{|b|}\tag{4}$$

contribute to the left-sided part of $$c[n]$$. Note the argument can be easily generalized to higher powers of $$z$$ on the left-hand sides of $$(3)$$ and $$(4)$$.

The term $$(1+4z^{-2})$$ in the numerator of the given function $$H(z)$$ corresponds to two zeros outside the unit circle, and, consequently, it contributes to the left-sided part of $$c[n]$$. Hence, according to $$(4)$$, it must be rewritten in terms of positive powers of $$z$$:

$$(1+4z^{-2})=4z^{-2}(1+0.25z^2)\tag{5}$$

We have

$$H(z)=z^{-2}\frac{4(1-0.5z^{-1})(1+0.25z^2)}{(1-0.64z^{-2})}=z^{-2}\tilde{H}(z)\tag{6}$$

I think that due to the linear-phase term $$z^{-2}$$ in $$(6)$$ we actually cannot compute the complex cepstrum of $$H(z)$$, but we can compute the complex cepstrum of $$\tilde{H}(z)$$, which is just a shifted version of the original function $$H(z)$$.

From $$(6)$$ we obtain

$$\log \tilde{H}(z)=\log(4)+\log(1-0.5z^{-1})+\log(1+0.25z^{2})-\log(1-0.64z^{-2})\tag{7}$$

Using $$(3)$$ and $$(4)$$ we get

$$\log \tilde{H}(z)=\log(4)-\sum_{n=1}^{\infty}\frac{(0.5)^n}{n}z^{-n}+\sum_{n=-\infty}^{-1}\frac{(-0.25)^{-n}}{n}z^{-2n}+\sum_{n=1}^{\infty}\frac{(0.64)^n}{n}z^{-2n}\tag{8}$$

which is equivalent to

$$\log \tilde{H}(z)=\log(4)-\sum_{n=1}^{\infty}\frac{(0.5)^n}{n}z^{-n}+\sum_{n=-\infty\\n\textrm{ even}}^{-1}\frac{2(-0.5)^{-n}}{n}z^{-n}+\sum_{n=1\\n\textrm{ even}}^{\infty}\frac{2(0.8)^n}{n}z^{-n}\tag{9}$$

The region of convergence of the series representation $$(9)$$ is $$0.8<|z|<2$$, which includes the unit circle, as required by the definition of the complex cepstrum.

From $$(9)$$ we can directly write down the complex cepstrum of $$\tilde{H}(z)$$:

$$\tilde{c}[n]=\begin{cases}\log(4),&n=0\\ \frac{1}{n}\left[2(0.8)^n-(0.5)^n\right],&n>0 \land n\textrm{ even}\\-\frac{1}{n}(0.5)^n,&n>0 \land n\textrm{ odd}\\\frac{2}{n}(-0.5)^{-n},&n<0 \land n\textrm{ even}\\0,&n<0\land n \textrm{ odd}\end{cases}\tag{10}$$

• Great explanation!If I wanted to calculate the real cepstrum $c[n]$ as well, should I use the following property? : $$c[n]=\frac{\tilde{c}[n]+\tilde{c}[-n]}{2}$$ – MJ13 Apr 28 '19 at 15:50
• @MJ13: Yes, that should work. – Matt L. Apr 28 '19 at 15:58
• @MJ13: A right-sided sequence starts at some finite index (often $n=0$), and extends to $n\to\infty$. Same (but opposite) for a left-sided sequence. – Matt L. Apr 28 '19 at 16:25
• @MJ13: Yes, that's an option. $H_1(z)$ has all its poles and zeros inside the unit circle, so it's indeed minimum-phase. And the other filter is FIR but not (generalized) linear phase. – Matt L. May 3 '19 at 15:25
• @MJ13: Yes, thanks. Fixed. – Matt L. May 9 '19 at 9:22