课程主页：https://www.coursera.org/learn/dsp2

这部分回顾第二周的习题。

1

(Difficulty: $\star$ ) Consider the following causal CCDE

Which of the following statements are correct?

  • If the input signal is $\delta[n]-\delta[n+1]$, then the z-transform of the output would be $(-3 z+1 / 2+$ $\left.5 / 2 z^{-1}\right) /\left(1+2 z^{-1}\right)$
  • It has two poles at $-$2 and $\frac{-5}{6}$
  • Its ROC contains the unit circle.
  • The system is stable.

使用$Z$变换可得

第一项：

如果输入为$\delta[n]-\delta[n+1]$，那么

所以

所以第一项正确。

第二项：

只有一个极点$-2$，所以第二项错误。

第三项：

极点为

所以ROC不包含单位圆。

第四项：

由上一项可得不稳定。

所以选第一项。

2

(Difficulty: $\star \star)$ Suppose that the ROC of the sequence $x[n]$ is $r_{L}<|z|<r_{U} .$ What is the ROC of $x^{*}[-n] ?$

  • $\frac{1}{r_{U}}<|z|<\frac{1}{r_{L}}$
  • $r_{L}<|z|<r_{U}$
  • $\frac{1}{r_{L}}<|z|<\frac{1}{r_{U}}$
  • $r_{U}<|z|<r_{L}$

对$x^*[-n]$做$z$变换可得

注意到$x[n]$的收敛域为$r_{L}<|z|<r_{U}$，不难得到$\sum_{n=0}^{\infty} x[n] z^{-n}$对应的收敛域为$r_L<|z|$，$\sum_{n=-\infty}^{0} x[n] z^{-n}$对应的收敛域为$|z|<r_U$，所以$x^*[-n]$对应的收敛域为

3

(Difficulty: $\star$ ) Consider an LTI system $h[n]$, whose transfer function’s ROC is $R_{h}$. Consider a second LTI system $g[n]$ with $\operatorname{ROC} R_{g} .$ Now consider the cascade of the two filters.
What is the ROC of the cascade?

  • It is only $R_{g}$
  • It contains $R_{g} \cup R_{h}$
  • It contains $R_{g} \cap R_{h}$
  • It is only $R_{h}$

显然为交集。

4

(Difficulty: $\star$ ) Consider the following CCDE

Let $H\left(e^{j \omega}\right)$ denote the transfer function of this system. What is $H\left(e^{j \pi}\right) ?$

取$z$变换可得

因为

所以

5

(Difficulty: $\star \star)$ Write some code in your preferred programming language that implements the following CCDE:

Use $y[n]=0$ for $n<0$ as initial conditions and run the algorithm for $x[n]=\delta[n]+\frac{1}{2} \delta[n-1]$

  • The output shows a diverging oscillation around zero: as $n$ grows, it assumes always larger values witha lternated signs.
  • The filter is mathematically unstable. Even in practice, you can see the output diverging $|y[50]|>10^{13}$
  • The filter is stable but the output $y[n]$ diverges because of the chosen input $x[n]$
  • $y[5]=1$
  • $y[4]=2$

计算后可得

所以

因此选则1,2,5

6

(Difficulty: $\star \star \star)$ A filter $H(z)$ has the following pole-zero plot:

Which of the following figures shows the magnitude response of the filter? Assume an implementation where the “internal clock” is $T_{s}=1 s$ so that the frequency axis is labeled in $\mathrm{Hz}$ and $1 \mathrm{Hz}$ corresponds to the digital frequency of $\omega=1$ radians.

极点附加取极大值，零点附加取$0$，所以选第二项。

7

(Difficulty: $\star \star)$ Let $h[n]$ represent the impulse response of the following system.

Select the correct statement about the poles and zeros of $H(z)$

  • It has one zero at $z_{1}=3 / 2$ and one pole at $z_{2}=5 / 6$
  • $H(z)$ has one zero at $z_{1}=1 / 4$ and two poles at $z_{3}=5 / 4$ and $z_{2}=3 / 8$
  • It has one pole at $z_{1}=3 / 2$
  • $H(z)$ has two zeros at $z_{1}=5 / 4$ and $z_{2}=3 / 8$ and one pole at $z_{3}=1 / 4$
  • It has one zero at $z_{1}=-3 / 4$

系统为

取$z$变换可得

所以零点为

极点为

8

(Difficulty: $\star \star)$ The following bit of Python code implements a discrete-time filter (assume $x[n]$ and $y[n]$ are suitably defined arrays):

f = 0;
g = 0;
for n in range(0, L-1):
    y[n] = x[n] + f;
    g = -f;
    f = -x[n] + 0.5 * y[n] + g;

What is the minimum number of delays necessary to implement this filter efficiently?

代码中只用了一次延迟，实际上系统的形式如下：

9

(Difficulty: $\star \star)$ Which of the following statements describes the system in this figure?

  • None of these statements describe this system.
  • The system is a resonator at $\omega_{0}=3 \pi / 4$
  • This is a resonator at $\omega_{0}=\pi / 2$
  • The system is a DC notch.
  • The system is a hum removal filter with $\omega_{0}=\pi / 2$
  • The system is a hum removal filter with $\omega_{0}=3 \pi / 4$

Hum removal的形式如下

对比可得

系统形式如下

取$z$变换可得