2 Linear Diff. Eqns.

2.4 Transient ResponseAll Activities

The discrete exponential function is given by $e_2(k) = r^k, k ≥ 0$. The z-transform of this function is

$$\tag{a}E_2(z)=\sum^\infty_{k=0}{r^kz^{-k}}=\sum^\infty_{k=0}{(rz^{-1})^k}\optbreak{}=\frac{1}{1-rz^{-1}}=\frac{z}{z-r}$$

which has a single real pole at $z = r$, plus a zero at $z = 0$.

Figure 1 shows the pole and zero location of equation (a) in the z-plane, and the corresponding time domain response. Drag the pole (x) to change the pole location.

exponential response
Figure 1: Exponential pole/zero locations and response.

Drag the pole (x) on the z-plane in Figure 1 to see the corresponding response

  1. What pole locations result in an unstable (growing) solution?
  2. What pole location results in solution that doesn't grow or decay?
  3. How does the response decay if the pole is at -0.5 verses 0.5?

The discrete damped sinusoid is given by $e_3(k) = r^k/cos{(k\theta)}, k ≥ 0$. Using an Euler expansion of the cosine, we obtain

$$\tag{b}e_3(k)=r^k\Big(\frac{e^{jkθ}+e^{-jkθ}}{2}\Big)$$

The corresponding z-Transform is

$$\tag{c}E_3(z)=\frac{z(z-r\cos{θ})}{z^2-2r\cos{θ}z+r^2}$$

Figure 2 shows the pole and zero location of equation (c) in the z-plane, and the corresponding time domain response. Drag the poles (x) to change the pole location.

exponential response
Figure 2: Discrete damped sinusoidal response.

Drag the pole (x) on the z-plane in Figure 2 to see the corresponding response

  1. What value of $r$ results in an unstable (growing) solution?
  2. What is the effect on the response when you change only the imaginary part of the pole?
  3. What is the effect on the response when you change only the real part of the pole?