Archive for April, 2009

Stability Of Discrete-Time LTI Systems

April 12, 2009 at 9:24 pm · Filed under Signal Processing

This time I’d like to show you an insight that I thought of a long time ago; it was during a semester in which I was taking both “Complex Analysis” and “Linear Systems” classes.

Let’s look at a discrete-time LTI system h[n], with a Z-Transform H(z)=\frac{N(z)}{D(z)}, where N(z) and D(z) are both polynomials.

Assuming that h[n] is causal, it is stable iff all poles of H(z) are inside the unit circle; i.e. all zeros of D(z) are inside the unit circle.

Since D(z) could be a polynomial of some degree, finding its zeros could be “difficult”. This gives rise to the need for systems stability criterion, which will not require finding them explicitly.

There is a direct Routh method for discrete-time systems, made up by Prof. Yuval Bistriz. However, what I would like to talk about is not a method, but rather a little “trick” which can be useful in certain cases.

Let D(z) be a polynomial of degree n:

D(z)=a_{n}z^{n}+a_{n-1}z^{n-1}+...+a_{1}z^{1}+a_{0}

Lemma:

Pick the coefficient a_{i} with the highest absolute value.

If |a_{i}|>\underset{j\neq i}{\sum}|a_{j}| then:

1. If i=n than the system is stable since all D(z) zeros are inside the UC.

2. If i\neq n than the system is unstable, since D(z) has zeros outside the UC.

Now comes the part in which Complex Analysis met Linear Systems in my mind -

Proof:

Let me define:

I(z)=a_{i}z^{i},

J(z)=\underset{\underset{j\neq i}{j=0}}{\overset{n}{\sum}}a_{j}z^{j}

Where

|a_{i}|>\underset{j\neq i}{\sum}|a_{j}|,

and obviously,

D(z)=I(z)+J(z).

What we are interested in is the number of zeros D(z) has inside the closed contour |z|=1, i.e the unit circle (alternatively: z=e^{jw}).

|I(z)|_{z=e^{jw}}=|a_{i}e^{jw}|=|a_{i}|\underset{given}{>}\underset{j\neq i}{\sum}|a_{j}|\underset{(1)}{\geq}|J(z)|_{z=e^{jw}}

While (1) is true since

|J(z)|_{z=e^{jw}}=|\underset{\underset{j\neq i}{j=0}}{\overset{n}{\sum}}a_{j}z^{j}|_{z=e^{jw}}\leq\underset{\underset{j\neq i}{j=0}}{\overset{n}{\sum}}|a_{j}e^{jw}|=\underset{\underset{j\neq i}{j=0}}{\overset{n}{\sum}}|a_{j}|.

What we have shown is that |I(z)|>|J(z)| anywhere on the unit circle. By Rouché’s theorem, we can conclude that I(z) and D(z)=I(z)+J(z) have the same number of zeros inside the unit circle.

Thus,

if i=n, than I(z) has n zeros inside the unit circle (all of them at z=0). It follows that D(z) also has n zeros inside the unit circle, i.e. all of its zeros are in it (as a polynomial of degree n, it has exactly n zeros).

However,

if i\neq n than I(z) has i zeros inside the unit circle (again, all of them at z=0), where i<n. It follows that D(z) also has only i zeros inside the unit circle, thus must have (n-i) zeros outside of it.

Q.E.D.

Note that this is Not a general method for checking whether a system is stable or not, but can be a nice trick for that purpose when one of the coefficients’ absolute value is “very large”.

Leave a Comment