1、Introduction to Digital Signal Processing,6.1 Representation of Digital Signals 数字信号的表示,Messages sent by telegraph are digital signals. They are written using 26 alphabetical（字母的） characters, 10 numerals（数字）, and symbols（符号） such as commas（逗号） and periods（句号）. These symbols certainly can be represen
2、ted by different voltage levels, for example, A by 10V (volts), B by 9.5V, C by 9V, and so forth. 电报发送的消息是数字信号。这些消息由26个英文字母，10个数字，逗号句号等符号组成。它们可以用不同的电压来表示，例如用10伏表示字母A，9.5伏表示字母B，9伏表示字母C，等等。,This type of representation, however, is not used because it is susceptible to（易受影响的） noise, shifting of power s
3、upply, and any other disturbances. 然而并未采用这种表示法，因为它易受到噪声、电源起伏及其它干扰的影响。,In practice, these symbols are coded as sequences of dashes（划线） and dots（点） or, equivalently, ones and zeros. The symbol 0 can be represented by a voltage from 0 to 0.8V. The symbol 1 can be represented by a voltage from 2.0 to 5.
4、0V. 在实际应用中，这些符号被编成一串点、划，或者1和0。符号0可以用0到0.8伏之间的电压表示；符号1可以用2.0伏到5.0伏之间的电压表示。,The precise voltage of each symbol is not important, but it is critical that its value lies within one of the two allowable ranges. If the value lies outside the two allowable ranges, the telegraph must make a choice of either
5、 1 or 0, and an error may occur. 每个符号所表示的准确电压值并不重要，关键是它要落在上述任意一个允许范围之内，否则就必须在0，1之间作出选择而可能发生错误。,In this case, an error message can be sent, and the message be retransmitted(转发). On the compact disc（光盘）, 1 and 0 are represented by dimples（凹痕）. The leading and trailing edges of a dimple represent a 1（凹
6、痕的前端和后端表示1）; no change represents a 0（无变化则为0）. 在此情况下，可以发送错误信息，要求重传信号。在光盘上用凹痕来表示0和1，凹痕的前端和后端表示1，无变化则为0，,Each bit occupies approximately a distance of 1 micrometer (10-6 meter). This type of representation of 1 and 0 is much less susceptible to noise. Furthermore, it is easy to implement. Therefore, d
7、igital signals are always coded by ones and zeros in physical implementation. This is called binary coding.每位数据大约占据1微米的长度。这种表达方式受噪音的影响很小，而且，易于实现。因此，实现时数字信号总是用0，1编码，称做二进制编码。,There are many types of binary coding. In this section, we discuss the simplest one, called sign-and-magnitude coding（符号及值编码）.
8、Consider the following sequences of :,二进制编码的方式有很多种，本节讨论最简单的一种， 叫做符号及值编码。考虑如下序列,：,Each can assume（取值） only either 1 or 0 and is called a binary digit or bit（二进制位或比特）. There are six bits in the sequence. The left-most（最左边起） bit is called the sign bit（符号位）. The sequence represents a positive number if
9、= 0 and a negative number if = 1. 每一个 称做一个二进制位或比特，取值只能是1或者0。该序列中有6个位，最左边的是符号位， 0代表正数， 1代表负数。,The remaining five bits represent the magnitude of the number（剩下的5个比特代表该数的大小）. For example, we have,剩下的5个比特代表该数的大小。例如，,The left-most bit, excluding the sign bit, is the most significant bit (MSB) and the rig
10、ht-most bit is the least significant bit (LSB). If we use 10 bits to represent the decimal（小数的） part of a number, then the LSB represents = 1/1024 = 0.0009765, which yields(产生) the quantization step（这即是量化阶）. 除符号位外最左边的位是最高有效位，而最右边的位是最低有效位。如果用10个比特表示一个数的小数部分，最低位表示为1/1024 = 0.0009765, 这即是量化阶。exclude(v.
11、): 把.排除在外,Signals encountered in practice are mostly CT （continuous time）signals. To process an analog signal digitally, it must first be discretized in time to yield a discrete-time signal and then quantized in amplitude to yield a digital signal. Therefore, in actual digital signal processing, we
12、deal exclusively(排外地, 专有地) with digital signals. 在实际中常见的信号是连续时间信号，为了以数字的方式处理模拟信号，首先要将模拟信号进行时间离散化，产生一个离散时间信号，然后量化其幅度产生数字信号。因此，在实际信号处理中，我们只处理数字信号。,An analytical study of digital signals, however, is difficult, because quantization is not a linear process. To simplify the discussion, we use decimal num
13、bers to illustrate this point. Suppose every number is to be rounded to（四舍五入） the nearest integer (that is, the quantization step is 1); then we have ： 然而对于数字信号的分析研究却很困难，因为量化并非线性的过程。为了讨论简便，我们用十进制数来说明这一点。假设每个数值都量化为最近的整数（也就是量化阶为1），则有：,where Q stands for quantization. Because of these nonlinear phenome
14、na, analytical study of digital signals is complicated. There are, however, no such problems in studying discrete-time signals. For this reason, in analysis and design, all digital signals will be considered as discrete- time signals. In actual processing or implementation, all discrete-time signals
15、 must be converted into digital signals. 这里Q表示量化。由于非线性的存在，数字信号的分析研究变得很复杂。然而，离散时间信号的研究就没有这种问题。因此，在分析与设计中，所有的数字信号都被看作是离散时间信号。在实际处理和实现中，所有的离散时间信号都要转换为数字信号。,In quantization, if the amplitude（大小） of a discrete-time signal does not fall exactly on a quantization level, then the value must be approximated
16、by a quantization level either by truncation or rounding, in either case, errors will occur. Such errors are called quantization errors. 量化时，如果离散时间信号的大小与量化电平不一致，那么它就必须以截断或舍入的方式用相应的量化电平来近似。两种情况下都会产生误差。这种误差叫量化误差。,In general, quantization errors are studied separately in digital signal processing. Such
17、 a study is important in specialized（专门的） hardware that uses a small number of bits such as 4 or 8 bits. On digital computers and DSP processors that have 16 or 32 bits, quantization errors are very small and can often be simply ignored. 通常，在数字信号处理中常对量化误差进行专题研究。但是，在采用很少的位数，比如4位或者8位的硬件中，量化误差的研究是非常重要的
18、。在数字计算机和16位或者32位数字信号处理器中，量化误差很小，可以忽略不计。,For convenience, we use digital signals and discrete-time signals interchangeably （可交替地）with the understanding that （考虑到）all DT signals must be transformed into digital signals in implementation and all digital signals are considered as DT signals in analysis
19、and design. 考虑到实现时离散时间信号一定要转换为数字信号，而且在分析和设计中所有的数字信号被认为是离散时间信号，为了方便，我们不加区别地使用数字信号和离散时间信号。,6.2 Comparison of Digital and Analog Techniques 数字技术与模拟技术的比较,Digital techniques have become increasingly popular and have replaced, in many applications, analog techniques. We discuss some of the reasons in the
20、following. 数字技术越来越流行，而且在很多应用中取代了模拟技术。下面讨论一下其中的原因。,Digital techniques are less susceptible to noise and disturbance. In the transmission and processing of analog signals, any noise or disturbance will affect the signals. Digital signals are coded in 1s and 0s, which are represented by ranges of volta
21、ges; therefore, small noise, disturbance（干扰）, or perturbation（波动） in power supply may not affect the representation. Thus digital techniques are less susceptible to noise and disturbance. This reliability can further be improved by using error-detecting and error-correcting codes. 数字技术受噪音和干扰的影响小。在模拟
22、信号的传输和处理中任何噪音和干扰将会影响信号。数字信号由1和0编码，1和0代表不同范围的电压；因此小噪音、干扰和电压的波动不会影响信号的表示。因此数字技术受噪音和干扰的影响小。通过检错码和纠错码可以更进一步提高其可靠性。,On the compact disc, by using the so-called cross-interleaved Reed-Solomon code, it is possible to correct 2400 bit long errors (corresponding to 2-mm scratches). However, the number of bit
23、s must be increased considerably. For example, a portion（一部分） of the audio（音频的） signal represented originally by 192 bits now requires 588 bits on the disc. 在光盘上，通过采用所谓交叉纠错的Reed-Solomon 码可以纠正2400位长的错误（相当于2毫米的划痕）。然而其位数的增加也是相当可观的。例如，原来需要192位表示的音频信号现在在光盘上需要588位表示。 interleaved：交叉存取,Another example is th
24、e transmission of the pictures of the Mars, taken by a spacecraft, to the ground station on the earth. After traveling over kilometers, the received signal has a power level in the order of watts. If the signal is transmitted by analog techniques, the received signal will be severely corrupted by no
25、ise and it is not possible to reconstruct the pictures. However, the pictures of the Mars have been transmitted successfully to Earth by using digital techniques. 另一个例子就是由太空船采集的火星图片的传输到地面工作站。经过 2亿 千米的传输，收到的信号的功率为 瓦特量级。如果采用模拟信号传输，收到的信号会被噪声严重损坏而不可能重建图片。然而，火星图片已经成功采用数字信号传输到地球。,The precision in digital
26、techniques is much higher than that in analog techniques. In analog display, the accuracy of the reading is often limited. Generally, we can achieve only an accuracy of 1% of the full scale. In digital display, the accuracy can be increased simply by increasing the number of bits used. In analog sys
27、tems, it is difficult or very expensive to have a number of components with identical value. 数字技术的精度远高于模拟技术。采用模拟显示，读取的精度是有限的，大体上是满刻度的1%。而在数字显示中，仅仅提高采用的位数就可以提高其精度。在模拟系统中，很难做到使器件具有相同的数值。,For example, if we buy ten 1- resistors, probably the resistances of the ten resistors will be all different and no
28、ne exactly equals 1- . Even simple analog voltmeters require constant resetting in their use. Digital systems have no such problem; they can always be exactly reproduced. 例如，如果我们买10个1 电阻，很可能它们的阻值全都不同，而且没有一个是精确的1 。甚至是简单的电压表在使用中也需要经常重新设置。而数字系统没有这个问题，它们可以被精确的复制。,The storage of digital signals is easier
29、 than that of analog signals. Digital signals can be easily stored in shift registers, memory chips, floppy disks, or compact discs for as long as needed without loss of accuracy. These data can be retrieved for use in a few microseconds. This easy storage and rapid access is an important feature of
30、 digital techniques. The only convenient way to store analog signals is to tape or film them. Their retrieval is not as convenient and fast as in digital techniques. 数字信号的存储比模拟信号容易。在必要时数字信号可以无失真地存储在移位寄存器，内存，软盘和光盘中。检索这些数据只需要几微秒的时间。容易存储和易于读取是数字技术重要特征。存储模拟信号唯一方便的方式就是把它们存在磁带或者胶片上。它们的检索不象数字技术那么方便和快捷。,Dig
31、ital techniques are more flexible and versatile than analog techniques. In digital display, digital signals can easily be frozen, expanded, or manipulated. A digital system can be easily altered by resetting its parameters. To change an analog system, we must physically replace its components. Using
32、 time multiplexing（时分复用技术）, a digital system can be used to process a number of digital signals: for example, if the sampling period of a digital signal is 0.05s and if the processing of the signal requires only 0.005s, then the digital system will be free in the next 0.045s and can be used to proce
33、ss other signals. 数字技术比模拟技术更加灵活通用。采用数字显示，易于对数字信号进行定格、放大和处理。可以很容易地通过修改参数来改变一个数字系统。为了改变一个模拟系统，我们必须改变它的物理组建。利用时分复用技术，一个数字系统可以处理很多数字信号：例如，如果数字信号的采样周期是0.05秒，处理信号需要的时间是0.005秒，那么数字系统在剩下的0.045秒的时间里就是空闲的，在这段时间里就可以处理其它信号。,Another example of this type of arrangement is the digital transmission of human voices.
34、 On a telephone line, the voice is sampled 8000 times per second; each sample is coded by using eight bits. Thus the transmission of one channel of voices requires the transmission of 64,000 or 64K bits per second. The existing telephone line can transmit 1544K bits per second. Therefore, a single t
35、elephone line can be used to transmit 24 voice-hand channels. If we use fiber optics, then the number of channels is even larger. 这类安排的其它例子就是话音的数字传输。在电话线上，话音每秒钟采样8000次，每个采样用8个二进制位编码。因此一路声音信号的传输需要64k bps的带宽。现存的电话线路可以每秒传送1544k bps 的数据。因此一个电话线可以传输24路话音信号。如果采用光纤，传输的路数就会更多。,Personal computers are now wid
36、ely available, so is computer software. This fact speeds considerably the spreading of digital techniques. From the preceding discussion, it is not surprising to see the avalanche of digital techniques. In many applications digital techniques have almost completely replaced analog techniques such as
37、 in analog oscilloscopes and analog computers. 现在个人电脑得到广泛应用，电脑软件也是如此。这很大程度上加速了数字技术的推广。从前面的讨论中，不难看出数字技术越来越普及。在很多应用中，数字技术几乎完全取代了模拟技术，比如模拟示波器和模拟计算机。,The roots of DSP are in the 1960s and 1970s when digital computers first became available6. Computers were expensive during this era, and DSP was limited
38、to only a few critical applications. Pioneering efforts were made in four key areas: radar and medical imaging, where lives could be saved. DSP产生于二十世纪六七十年代，当时数字计算机刚问世。那时计算机很昂贵，DSP仅限于一些关键应用。在关系到国家安全的雷达和声纳等关键领域；在利润丰厚的石油勘探领域，在数据不能替代的太空探索中，在救死扶伤的医学影像领域，都做了开拓性应用。,The personal computer revolution of the 1
39、980s and 1990s caused DSP to explode with new applications. Rather than being motivated by military and government needs, DSP was suddenly driven by the commercial marketplace. Anyone who thought they could make money in the rapidly expanding field was suddenly a DSP vendor. DSP reached the public i
40、n such products as: mobile telephones, compact disc players, and electronic voice mail. Figure 6.1 illustrates a few of these varied（各式各样的） applications. 二十世纪八十年代到九十年代的个人计算机革命使DSP扩展到新的应用。DSP突然被商业市场所推动，而不是被军事和政府需要。那些认为自己能在快速扩张领域赚钱的人突然成为了DSP销售商。DSP通过如下产品进入公众视线：移动电话，光盘播放器，电子语音邮件。图1.3展示了其中的一些应用。,As you
41、go through each application, notice that DSP is very interdisciplinary, relying on the technical work in many adjacent fields. As Figure 6.2 suggests, the borders between DSP and other technical disciplines are not sharp（明显的） and well defined, but rather fuzzy and overlapping. If you want to specialize（专门研究） in DSP, these are the allied areas you will also need to study. 在看过每个应用后，你会发现DSP是一门跨学科的技术。它是基于许多相邻领域的技术工作。如图1.3所示，DSP和其它技术学科的界限并不清晰，也未被详细定义，而是模糊和交叉的。如果你想成为DSP领域的专家，相关领域的知识也需要学习。,
