数字信号处理翻译

姓名班级学号成绩

考试内容：2010年第一学期数字信号处理

第一部分解答题

1.Please give the stages of digital processing of analog signals and the basic

components of DSP system.

2. Please describe Sampling Theorem Compute the z-transform of the following

sequences x(n)

x(n) = (-0.5)n u(n)

3. Try to test the linearity and time invariance of the discrete time systems defined

as follows:

x

=n

n

n

y

x

-

(

)1

(-

)

(

)

4. Given a causal IIR discrete-time system described by the difference equation

y[n]-0.4y[n-1]=x[n].

And it is known that the input sequence is x[n]= x[n]=(0.3)nμ[n]. .

(1)Determine the output sequence y[n] using the z-transform.

(2)Determine the expression of the frequency response H(e jω)in the form

|H(e jω)|e j?(ω)

第二部分文献翻译

参考文献：the scientist and engineer's guide to digital signal processing

具体内容：第123 页第二段---第127 页

原文：

Part 1. 解答下列习题

1. Please give the stages of digital processing of analog signals and the basic

components of DSP system. 答：

(1)信号的数字化需要进过采样、保持、量化和编码四个过程

(2)数字信号系统的基本组成为：前置预滤波器、A/D 变换器、数字信号处理器、D/A 变换器、模拟滤波器。

2. Please describe Sampling Theorem Compute the z-transform of the following

sequences x (n )，x (n ) = (-0.5)n u(n ) 解：(1)

采样定理：在进行模拟/数字信号的转换过程中，当采样频率大于信号中最高频率的2倍时，采样之后的数字信号完整地保留了原始信号中的信息，一般实际应用中保证采样频率为信号最高频率的5～10倍；采样定理又称奈奎斯特定理。

(2)

由题有：

()()()101

0.51

0.510.5n

n n

n X z x n z z z z

∞

-=-∞

∞

-=-=

??=-??=

>-+∑

∑

3. Try to test the linearity and time invariance of the discrete time systems defined

as follows:)1()()(--=n x n x n y 解：

验证线性：

()()()

()()()()()()()

12012121211x n ax n bx n y n ax n bx n ax n bx n ay n by n →+=+----=+

验证时不变性：

()()

()()()

()

0 n m 1n n m x n x n m y n x n m x n m y n m +→+∈→+=+-+-=+ 、

综上，此系统为线性时不变系统。

4. Given a causal IIR discrete-time system described by the difference equation y[n]-0.4y[n-1]=x[n].

And it is known that the input sequence is x[n]= x[n]=(0.3)n μ[n]. . (1)Determine the output sequence y[n] using the z-transform.

(2)Determine the expression of the frequency response H(e j ω) in the form |H(e j ω)|e j ?(ω)

解：

(1)

先对元方程做z 变换，有：

()()()10.4Y z z Y z X z --=

其中：

()1

1

0.310.3X z z z

-=

>- 我们得到：

()11

11

0.410.310.4Y z z z z --=

>--

对其做逆z 变换，即得所求：

()()()()()

()1

0.30.43140.4 n 0

314

n n

n n y n n n μμ+=*??- ???=≥-

(2)

由上可知：

()1

1

0.410.4H z z z -=

>-

代换变量后有：

()()10.4cos 0.4sin 1.160.8cos j j H e ωωω

ω

--=

-

其模为：

()1

1.160.8cos j H e ωω

=

-

相位为：

()0.4sin arcsin

1.160.8cos ω

?ωω

-=-

以上即所求解答。

第二部分：文献翻译

第七章：卷积的性质

一个线性系统特点可由系统的脉冲响应来彻底描述,在数学上表现为积分的性质。这是基本的多种信号处理技术。例如:数字滤波器是由设计合适的脉冲响应造成的。敌人的飞机被雷达通过分析测量脉冲响应探测到。回声抑制在长途电话中创建一个脉冲响应,抵消反射物对脉冲响应的回响。这种例子还有很多。本章就卷积的性质和使用从几个方面进行探讨。 首先, 对几种常见的脉冲响应进行了讨论。 第二,介绍了进行级联和并行组合的线性系统处理方法。第三, 简要介绍了相关的技术。第四,关于卷积有严重的问题被检测出来,利用常规算法和计算机有一个让人难以接受的响应时间。 常见的脉冲响应 δ函数

最简单的冲激响应,只不过是一个δ函数,如图7-1a 。也就是说,一个脉冲输入产生了相同的脉冲输出。这就意味着所有信号通过这个系统都没有变化。卷积任何信号的一个增量函数结果完全相同的信号。数学上,是这样写的: 方程7 - 1

[][][]x n n x n δ*=

函数是卷积的特性。 任何信号与卷积一个增量 函数功能保持不变。

此属性使 δ 函数具有卷积特性。这是类似于零正为补充的特性(a + 0 = a ),和一个乘法的特性 (a ×1 =a ). 乍一看，这种类型的系统

看似琐碎和无趣。其实不是这样的！这种系统是进行数据存储，通信和测量的理想选择。DSP 的大部分关注的是没有经过改变或退化系统的信息。

图7 - 1b 显示，以增量脉冲响应函数稍加修改..如果 δ 函数是在振幅较大或较小，由此产生的系统分别为一个放大器或衰减器。在方程的形式，放大的结果是k 大于1，衰减的结果是K 小于1： 方程7—2

[][][]x n k n kx n δ*=

一个放大或衰减系统有脉冲响应的δ函数，在这个方程式，K 确定放大或衰减。 在图7 - 1C 的的脉冲响应是一个具有转变功能的 δ函数。这导致在一个系统中引入了一个与输入和输出之间的信号相同的变化。让移参数来表示，s ，这可以写成方程式

方程7—3

[][][]x n n s x n s δ*+=+

一个输入与输出信号相对位移对应的脉冲响应对应的冲动响应是一个三角洲转

移的功能。变量，S,决定了这个方程的转变总数。

科学和工程是充满了一个信号的情况下是另一个版本的转移。例如，考虑一个无线电信号从太空探测器远程传输，以及相应的信号接收在地球上。所花费的时间的无线电波在传播之间的距离导致发送和接收信号的延迟。在生物学上，在邻近的神经细胞的电信号转变每个其他版本，如需要的时间一动作电位跨突触交界处，连接两个决定。

图7-1d显示了一维脉冲响应函数的加一δ函数，并调整δ函数组成。通过叠加，

这个系统的输出是输入信号加在输入信号，即延迟版本，一个回声装置是重要的。在许多DSP应用中回声是使录音听起来自然和舒适的重要组成部分。-雷达和声纳检测分析回声战机和潜艇。地质学家使用找寻石油回声。也是很重要的回声在电话网络,因为你想要避免他们。

a，特性

δ函数卷积的特性，一个信号与δ函数卷积得到的信号不变，这是目标系统传播与储存信号的原理。

b. 放大或衰减

增加或减少振幅δ函数形式的一种脉冲响应分别是放大或衰减,这个脉冲响应将

将此讯号放大1.6倍。

c. 移位

δ函数产生转移相应的之间变换的输入和输出信号。根据方向,这可以称为延迟或一个进步。这个脉冲响应拖延信号的4个采样点

d。回声

一个δ函数加上移位的规模δ函数结果导致了一个回声被加到原信号。在这个例子里,回声延迟到4分采样点，并且振幅只有原信号的60%

参看图7 - 1

简单的脉冲响应函数使用移位和规模δ函数。

类微积分操作

卷积也可用于处理离散信号，以类似积分和微分的操作。由于术语“积分”和“微分”是被用于对连续信号的处理，我们将使用其他的名字来称呼它们在离散情况下的对应操作。例如，离散情况下对应于一阶微分的操作被称为一阶差分。同样的，离散情况下对应于积分的操作被成为：

运行的数目。它也是普遍,听到这些操作称为离散导数和离散积分,虽然数学家皱眉的时候他们听见这些非正式的术语。

参看图7 - 2展示了脉冲响应,实现第一个不同点运行的数目。图7-3显示了一例使用这些操作。在7 -3a,原信号几部分组成的不同的斜坡上。卷积这个信号与第一个不同点脉冲响应产生如图7-3b所示信号。就如一阶导数,振幅值第一个不同点点信号等于相应边坡位置在原来的信号。)运行总数逆操作第一次的差异。也就是说,卷积信号(b),运行的总数脉冲响应,所生产的信号在(a)。

这些脉冲响应是简单地卷积程序，所以通常不需要实现他们。而是,考虑到他们选择的样本模式: 输入每一个样本从在输出信号中都是一笔加权样本。例如,第一个不同可以计算

方程7-4 [][][1]

=--

y n x n x n

计算第一个不同点。在这种关系中，[]

x n原信号,[]

y n是第一个不同点。

这就是说,每个样品在输出信号之间的区别等同于两种邻近的样品的输入信号之间的区别。例如,[40][40][39]

=-。它值得一提的是这种并不是唯一途径来y x x

定义一个离散的衍生工具。另一个普通的方法是定义斜坡信号在要点是检查,如: []([1][1])/2

=+--.

y n x n x n

a.第一差异

这是离散版本的第一衍生工具。在输出中每一个例子信号之间的区别等同于两种邻近的样品的输入信号之间的区别。换句话说,输出信号就是斜坡的输入信号。

b.运行和

运行总数是整数的离散版本。每个采样输出信号的总和等同于样品的输入信号翻转到左边。注意了脉冲响应延伸到无限大,一个相当严重的特征。

参看图7 - 2

脉冲响应模仿微积分操作。

图7-3

类微积分操作的例子。

类微积分操作的例子。

与（a ）图信号的第一个不同是（b ）图中的信号。相应地，（a ）图信号是（b ）图信号的运行总和。这些处理方法是使用离散信号正如连续信号中使用相同的分化和整合。

使用这个方法, 在运行和中的每个采样点可以通过总结原始信号中所有的点左边的采样点的位置计算出来。例如,如果[]y n 是[]x n 的运行和,然后在样品[40]y 等于通过[40]x 增加样品x[0]得到的值。同样地,样品[41]y 是通过[41]x 增加样品[0]x 得到的值。

当然, 运用这种方式来计算运行和效率是很低的。例如,如果[40]y 已经计算出来,则[41]y 只需要使用加法就可以算出：[41][41][40]y x y =+。在方程形式:

方程 7-5 计算运行的总数 在这个关系中, []x n 是原信号,

[]y n 是运行的总数。

[][][1]y n x n y n =+-

这种类型的关系被称为递归方程或差分方程。我们将在第19章详细介绍。现在，最重要的思想是理解这些关系与图 7-2用脉冲响应卷积得到的结果是相同的，表7-1提供了实现这些计算机程序演算式的操作。

Part2 翻译

A linear system's characteristics are completely specified by the system's impulse response, as

governed by the mathematics of convolution. This is the basis of many signal processing techniques. For example: Digital filters are created by designing an appropriate impulse response. Enemy aircraft are detected with radar by analyzing a measured impulse response.

Echo suppression in long distance telephone calls is accomplished by creating an impulse response that counteracts the impulse response of the reverberation. The list goes on and on.

This chapter expands on the properties and usage of convolution in several areas. First, several common impulse responses are discussed. Second, methods are presented for dealing with cascade and parallel combinations of linear systems. Third, the technique of correlation is introduced. Fourth, a nasty problem with convolution is examined, the computation time can be unacceptably long using conventional algorithms and computers.

Common Impulse Responses Delta Function

The simplest impulse response is nothing more that a delta function, as shown in Fig. 7-1a. That is, an impulse on the input produces an identical impulse on the output. This means that all signals are passed through the system without change . Convolving any signal with a delta function results in exactly the same signal. Mathematically, this is written: EQUATION 7-1

The delta function is the identity for [][][]x n n x n

δ*=

convolution. Any signal convolved with a delta function is left unchanged.

This property makes the delta function the identity for convolution. This is analogous to zero being the identity for addition (a + 0 = a ), and one being the identity for multiplication (a ×1 = a ). At first glance, this type of system may seem trivial and uninteresting. Not so! Such systems are the ideal for data storage, communication and measurement. Much of DSP is concerned with passing information through systems without change or degradation.

Figure 7-1b shows a slight modification to the delta function impulse response. If the delta function is made larger or smaller in amplitude, the resulting system is an amplifier or attenuator , respectively. In equation form, amplification results if k is greater than one , and attenuation results if k is less than one : EQUATION 7-2

A system that amplifies or attenuates has

[][][]x n k n kx n δ*=

a scaled delta function for an impulse response. In this equation, k determines the amplification or attenuation.

The impulse response in Fig. 7-1c is a delta function with a shift . This results in a system that introduces an identical shift between the input and output signals. This could be described as a signal delay , or a signal advance , depending on the direction of the shift. Letting the shift be represented by the parameter, s , this can be written as the equation:

EQUATION 7-3

A relative shift between the input and [][][]x n n s x n s δ*+=+ output signals corresponds to an impulse response that is a shifted delta function. The variable, s, determines the amount of

shift in this equation. Science and engineering are filled with cases where one signal is a shifted version of another. For example, consider a radio signal transmitted from a remote space probe, and the corresponding signal received on the earth.The time it takes the radio wave to propagate over the distance causes a delay between the transmitted and received signals. In biology, the electrical signals in adjacent nerve cells are shifted versions of each other,as determined by the time it takes an action potential to cross the synaptic junction that connects the two.

Figure 7-1d shows an impulse response composed of a delta function plus a shifted and scaled delta function. By superposition,

the output of this system is the input signal plus a delayed version of the input signal, i.e., an echo. Echoes are important in many DSP applications. The addition of echoes is a key part in making audio recordings sound natural and pleasant. Radar and sonar analyze echoes to detect aircraft and submarines. Geophysicists use echoes to find oil. Echoes are also very important in telephone networks, because you want to avoid them.

FIGURE 7-1

Simple impulse responses using shifted and scaled delta functions. Calculus-like Operations

Convolution can change discrete signals in ways that resemble integration and differentiation. Since the terms "derivative" and "integral" specifically refer to operations on continuous signals, other names are given to their discrete counterparts. The discrete operation that mimics the first derivative is called the first difference. Likewise, the discrete form of the integral is called the running sum. It is also common to hear these operations called the discrete derivative and the discrete integral, although mathematicians frown when they hear these informal terms used.

Figure 7-2 shows the impulse responses that implement the first difference and the running sum. Figure 7-3 shows an example using these operations. In 7-3a, the original signal is composed of several sections with varying slopes.

Convolving this signal with the first difference impulse response produces the signal in Fig. 7-3b. Just as with the first derivative, the amplitude of each point in the first difference signal is equal to the slope at the corresponding location in the original signal. The running sum is the inverse operation of the

first difference. That is, convolving the signal in (b), with the running sum's impulse response, produces the signal in (a).

These impulse responses are simple enough that a full convolution program is usually not needed to implement them. Rather, think of them in the alternative mode: each sample in the output signal is a sum of weighted samples from the input. For instance, the first difference can be calculated:

EQUATION 7-4

Calculation of the first difference. In [][][1

y n x n x n

=--this relation, x [n] is the original signal,

and y [n] is the first difference.

That is, each sample in the output signal is equal to the difference between two adjacent samples in the input signal. For instance, y[40] ’ x[40] & x[39] . It

should be mentioned that this is not the only way to define a discrete

derivative. Another common method is to define the slope symmetrically

around the point being examined, such as: y[n] ’ ( x[n%1]& x[n&1] )/2.

FIGURE 7-2

Impulse responses that mimic calculus operations.

Using this same approach, each sample in the running sum can be calculated by summing all points in the original signal to the left of the sample's location.

For instance, if y[n] is the running sum of x[n] , then sample y[40] is found by adding samples x[0] through x[40] . Likewise, sample y[41] is found by adding samples x[0] through x[41] . Of course, it would be very inefficient to calculate the running sum in this manner. For example, if y[40] has already been calculated, y[41] can be calculated wi th only a s ingle addi t ion: y[41] ’ x[41]% y[40] . In equation form:

EQUATION 7-5

Calculation of the running sum. In this

=+-

[][][1

y n x n y n

relation, x [n] is the original signal, and y [n]

is the running sum

Relations of this type are called recursion equations or difference

equations. We will revisit them in Chapter 19. For now, the important

idea to understand is that these relations are identical to convolution

using the impulse responses of Fig. 7-2. Table 7-1 provides computer

programs that

implement these calculus-like operations.