Absolutely summable



Stability – Z Transforms

Absolutely summable

A discrete-time signal x is said to be absolutely summable if

[pic] exists and is finite. The “absolutely summable” refers to the use of absolute values in the summation.

BIBO stability

A system is said to be bounded input-bounded output stable (BIBO stable or just stable) if the output signal is bounded for all input signals that are bounded.

Consider a discrete-time system with input x and output y. The input is said to be bounded if there is a real number M < ∞ such that [pic]for all k.

An output is bounded if there is a real number N < ∞ such that [pic]for n.

The system is stable if for any input bounded by M, there is some bound N on the output.

Theorem:

A discrete –time LTI system is stable if and only if its impulse response is absolutely summable.

Proof of if:

Consider a discrete-time LTI system with impulse response h. The output y corresponding to the input x is given by the convolution sum,

[pic] (1)

Suppose that the input is bounded with bound M. Then, applying the triangle inequality, we see that

[pic]

Thus, if the impulse response is absolutely summable, then the output is bounded with bound

N=M [pic]

Proof of only if:

To show that the system is not stable, we need to find one bounded input for which the output either does not exist or is not bounded. Such an input is given by

[pic]

[pic]

The input is clearly bounded, with bound M=1. Plugging this input to the convolution sum (1) and evaluating at n=0, we get

[pic]

But since by assumption that the impulse response in not absolutely summable, y(0) does not exist or is not finite, so the system is not stable.

Z-transform

Consider a discrete-time signal x that is not absolutely summable. The scaled signal xr is given by

[pic][pic]

for some real number r≥0. Often, this signal is absolutely summable when r is chosen appropriately

The Z-transform of x is defined as [pic]

with region of convergence RoC(x) [pic]Complex is defined by

RoC(x) = {z=rejω [pic]Complex | x(n)r-n is absolutely summable }

Example:

h(n)=0, n [pic]0 [Bank Account]

= an-1, n > 0

a > 1 is not absolutely summable.

Z-transform of Impulse Response is

[pic]

Thus, hr(n)=h(n)r-n is absolutely summable if r > a i.e. ROC(h) = {z=rejω [pic]Complex | r > a }

If reiθ1 is in RoC(h), so is reiθ2 because of | reiθ1|=| reiθ2|=r.

There are three RoC structures.

[a] Anticausal RoC

[b] Two-sided RoC

[c] Causal-or right sided RoC

[pic]

RoC for u(n) = 0 when n 0

Delay

For any integer N (positive or negative) and signal x, let y=DN(x) be the signal given by

[pic]n [pic]Integers, y(n)=x(n-N)

Suppose x has a Z-transform X(z) with domain RoC(x). Then RoC(y) = RoC(x) and

[pic]z [pic]RoC(y),

Y(z) = [pic]= [pic]

Thus if a signal is delayed by N samples, it’s Z transform is multiplied by z-N.

Convolution

Suppose x and h have Z transforms X(z) and H(z). Let

y = x [pic]h

Then [pic]z [pic]RoC(y), Y(z)=X(z)H(z)

This follows from using the definition of convolution,

[pic]n [pic]Integers, y(n) = [pic]in the definition of the z-transform

[pic]

The Z-transform of y converges absolutely at least at the values of z where both X and H converge absolutely.

Thus, RoC(y) [pic]RoC(x) [pic]RoC(h)

We are interested in systems characterized by difference equations

[pic]

[pic]

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