Signal and System Spring 2023 Homework Requirements and Reference Answer Summary
Signals and Systems 2023 (Spring) Assignment Requirements - Eleventh Assignment

01 Basic exercises

1. Z-transform properties

1. Use the properties to find the z-transform of the sequence

（1）

Use the differential property in the transform domain of the z-transform:

therefore,

（2）

（3）

（4）

First there are:

Then, according to the displacement theorem of z transformation:

In exponentially weighted properties according to the z-transform:

Another solution method:

（5）

2. Initial value and final value

（1）

由于 $X\left( z \right)$ There is a pole z = 3 outside the unit circle $z = 3$ , so the final value of the sequence does not exist.

（2）

（3）

（4）

This is a false fraction, which is expanded into a polynomial and a proper fraction of z by the method of long division.

The constant item corresponds to the initial value of the sequence, namely: $x\left[ 0 \right] = 1.25$ 。

3. Find the convolution of the sequence

（1）

According to the convolution theorem of z transformation, $y\left[ n \right] = x\left[ n \right] * h\left[ n \right]$ ，那么 $y\left[ n \right]$ is:

（2）

序列 $x\left[ n \right],h\left[ n \right]$ are:

According to the convolution theorem of z transformation, we can know:

By factoring, we get:

so:

4. Find the z-transform of the sequence product

（1）

According to the z-transform time-domain product theorem,

According to $X\left( z \right)$ , we know that $\left| v \right| > {1 \over 3}$ , according to $H\left( z \right)$ , we know that $\left| {z/v} \right| < 1/3$ ，即 $\left| v \right| > 3\left| z \right|$ . So in the above perimeter integral, there are two poles are $1/3,\,\,3z$ . $_$ Use the residue theorem to calculate the above perimeter integral:

From this, we can know that $y\left[ n \right] = \delta \left[ n \right]$ 。

根据 $X\left( z \right),H\left( z \right)$ 的收敛域，可知： $\left| v \right| > e^{ - \beta } ,\left| v \right| < \left| z \right|$ . So the poles included in the above perimeter integral are $e^{ - \beta }$ . thus:

According to the z-transform table of typical signals, we can know: $x\left[ n \right] \cdot h\left[ n \right ] = e^{ - n\beta } \sin \omega _0 n \cdot u\left[ n \right]$ 。

5. Prove the z-transform accumulation theorem

The accumulation of sequences can be regarded as sequence and $u\left[ n \right]$ Convolution of $[$ $n$ $] , namely:$

According to the convolution theorem of z transformation, it can be known that the transformation of the sequence accumulation sum is equal to the z transformation of the sequence multiplied by $u\left[ n \right]$ z-transform. because:

so:

6. Z-transform scale characteristics

（1）

（2）

2. Solving Differential Equations and Difference Equations

1. Solve differential equations

（1）

Laplace transform is performed on both sides of the differential equation, and the initial conditions of the system can be substituted into the equation by using the Laplace transform differential theorem: