Question 2

For the 3 filters of assignment 1, find the impulse response h(n) in two ways:

1) Work out algebraically using the inverse z-transform, and evaluate the first few terms numerically.

2) Use the POI software to place the poles and zeros in the complex plane, and let the program calculate the impulse response. In POI, drag zeros on the z-plane from the top left, and drag poles from the top right. More details are in the POI on-line documentation under "ZPlane Design". Under the tap "analysis", choose "analysis graph", and then in the new window under the tab "graph" choose "impulse response". Under the tab "file" choose "save values" which will save the impulse response values in a file. For more documentation of POI, see the POI on-line documentation

Compare the results of algebra with POI. The sampling rate fs is not defined in Assignment 1, the POI software lets you choose the sampling rate. Choose fs = 44,100 Hz used for music CDs.

Solution.

1) Algebraic solution

(a)

$$\begin{eqnarray*}H \left (z \right) & = & \frac{1}{2} \frac{(z+1)^2}{z^2}\\ & = & 0.5 + z^{-1} + 0.5z^{-2} \end{eqnarray*}$$

Using the table of z-transform, ( $\delta[n] \Longrightarrow 1, \delta[n-k] \Longrightarrow z^{-k}$)

$$\begin{eqnarray*}h \left [ k \right ] & = & 0.5 \delta \left [ k \right ] + \delta \left [ k-1 \right ] + 0.5 \delta \left [ k-2 \right ] \end{eqnarray*}$$

Evaluate h[k] for different values of k

$$\begin{eqnarray*}k = 0: h[0] & = & 0.5\\ k = 1: h[1] & = & 1.0\\ k = 2: h[2] & = & 0.5\\ k = 3: h[3] & = & 0.0\\ k = 4: h[4] & = & 0.0 \end{eqnarray*}$$

(b)

$$\begin{eqnarray*}H \left (z \right) & = & \frac{z-1}{z}\\ & = & 1 - z^{-1} \end{eqnarray*}$$

From the table of z-transform,

$$\begin{eqnarray*}h \left [ k \right ] & = & \delta \left [ k \right ] - \delta \left [ k-1 \right ] \end{eqnarray*}$$

Evaluate h[k] for different values of k

$$\begin{eqnarray*}k = 0: h[0] & = & 1.0\\ k = 1: h[1] & = & -1.0\\ k = 2: h[2] & = & 0.0\\ k = 3: h[3] & = & 0.0 \end{eqnarray*}$$

(c)

$$\begin{eqnarray*}H \left ( z \right ) & = & \frac{1}{z-a} \end{eqnarray*}$$

Using the table of z-transform, ( $p^{k-1}u[k-1] \Longrightarrow \frac{1}{z-p}$)

$$\begin{eqnarray*}h \left[ k \right ] & = & a^{k-1}u[k-1]\\ & = & 0.9^{k-1}u[k-1] \end{eqnarray*}$$

Evaluate h[k] for different values of k

$$\begin{eqnarray*}k = 0: h[0] & = & 0.0\\ k = 1: h[1] & = & 1.0\\ k = 2: h[2] & = & 0.9\\ k = 3: h[3] & = & 0.9^2\\ k = 4: h[3] & = & 0.9^3 \end{eqnarray*}$$

2) Results from POI software

Impulse responses of the three filters are shown in Fig.1.

  

Figure 1: Impulse responses of the three filters

We can observe that the algebraic results are the same as POI's results.