同济大学数字信号处理教程

数字信号处理是电子信息类专业同学很重要的一门专业基础课。

数字信号处理，英文全称为digital signal processing，也简称DSP。主要是用数值表示和处理的方法来研究信号的一门学科，数字信号在时间和幅度上都是离散(量化)的，现在的一些芯片，单片机、微处理器、计算机内部都是以数值来存储和运算的，与之相对应的模拟信号处理，二者都属于信号处理的范畴。现在数字信号处理器具有体积小、功耗低、精度高、可靠性高、灵活性大、易于大规模集成、可进行二维与多维处理等优点，已经逐步取代模拟信号处理器了。

关于教程

教程系统地讨论了数字信号处理的基础理论、基本概念、基本分析方法、算法、设计。课程包括四部分内容。第一部分是离散时间信号（序列）与系统的基本概念和时域、频域（包括z变换域）的分析方法与算法，包括离散傅里叶变换及其快速算法，模拟信号用数字信号处理的原理方法；第二部分为各种IIR及FIR数字滤波器的基本概念、理论、结构与设计方法；第三部分为多抽样率数字信号处理的基础理论；第四部分为数字信号处理实现中的有限字长效应。

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回顾数字信号处理的基本原理

介绍

由于我们在本书中讨论的语音处理方案和技术本质上是离散时间信号处理系统，因此读者必须对数字信号处理的基本技术有很好的理解。在本章中，我们简要回顾了与语音信号数字处理最相关的数字信号处理概念。本文旨在为以后的章节提供方便的参考，并确定将在整本书中使用的符号。那些完全不了解离散时间信号和系统的表示和分析技术的读者可能会发现，当本章未提供足够详细的信息时，可以查阅有关数字信号处理的教科书[245、270、305]。

离散时间信号和系统

在涉及信息处理或通信的几乎每种情况下，自然都会从将信号表示为连续变化的模式或波形开始。人类语音中产生的声波无疑是这种性质的。在数学上方便的是，将这种连续变化的模式表示为代表时间的连续变量t的函数。我们使用xa（t）形式的标记来表示连续变化（或模拟）的时间波形。就像我们将看到的那样，还可以将语音信号表示为一系列（量化的）数字。实际上，这是本书的中心主题之一。通常，我们使用x [n]形式的表示法来表示在时间和幅度上均已量化的序列。如果像采样语音信号一样，一个序列可以看作是一个采样周期为T的周期采样的模拟信号的采样序列，那么我们通常发现使用x []来明确指出这一点很有用。 n] = xa（nT）对于通过采样和量化从模拟波形得出的任何数字序列，有两个变量决定离散表示的性质，即采样率Fs = 1 / T和量化级数，在图2B中，B是表示的每个样本的位数。尽管可以将采样率设置为满足Fs = 1 / T> 2Fv（其中Fv是连续时间信号中出现的最高频率）的关系的任何值，但是语音的各种“自然”采样率已经随着时间而发展。包括：对于电话带宽语音（FN = 3.2 kHz）为1Fs = 6.4 kHz;对于扩展电话带宽语音（FN = 4 kHz）为F = 8 kHz;对于过采样的电话带宽语音（Fy = 5 kHz）为Fs = 10 kHz：对于宽带（hi-f）带宽语音（FN = 8 kHz），F，= 16 kHz。语音信号的数字表示中的第二个变量是量化信号的每个样本的位数。在第11章中，我们将研究量化对数字化语音波形特性的详细影响。但是，在oresent一章中，我们假设采样值未量化。图2.1展示了一个语音信号的示例，该语音信号既以模拟信号形式，又以Fs = 16 kHz的采样率表示为一系列采样。在随后

图2.1语音波形图：（a）绘制为连续时间信号（使用MATLAB plot（）函数）;（b）绘制为采样信号（使用MATLAB stem（）函数）.1如第2.3节所述，我们使用大写字母表示模拟频率，如[270]中所示。

当然，即使考虑了离散表示，绘制的便利性也通常要求使用模拟表示（即连续函数）。在这种情况下，连续曲线是样本序列的包络，样本通过笔直连接。图2.1显示了MATLAB函数图（对于图顶部是模拟波形）和主干（对于图底部是离散样本集）的使用。它也说明了有关采样信号的重要一点。当绘制一个样本序列作为样本索引的函数时，时间尺度就会丢失。现在我们必须采样率为Fs = 1 / T = 16 kHz，以便通过采样周期T = 62.5 us将数字波形的持续时间（在这种情况下为320个采样）转换为20毫秒的模拟时间间隔在我们对数字语音处理系统的研究中，我们会发现许多重复出现的特殊序列。图2.2中描绘了其中的几个序列。图2.2a中所示的单位样本或单位冲量序列定义为

INTRODUCTION

Since the speech processing schemes and techniques that we discuss in this book are intrinsically discrete-time signal processing systems, it is essential that the reader have a good understanding of the basic techniques of digital signal processing. In this chapter we present a brief review of the concepts in digital signal processing that are most relevant to digital processing of speech signals. This review is intended to serve as a convenient reference for later chapters and to establish the notation that will be used throughout the book. Those readers who are completely unfamiliar with techniques fo representation and analvsis of discrete-time signals and systems may find it worthwhile to consult a textbook on digital signal processing [245, 270, 305] when this chapter does not provide sufficient detail.

DISCRETE-TIME SIGNALS AND SYSTEMS

In almost every situation involving information processing or communication, it is natural to begin with a representation of the signal as a continuously varying pattern or waveform. The acoustic wave produced in human speech is most certainly of this nature. It is mathematically convenient to represent such continuously varying patterns as functions of a continuous variable t, which represents time. We use notatior of the form xa(t) to denote continuously varying (or analog) time waveforms. As we will see, it is also possible to represent the speech signal as a sequence of (quantized) numbers; indeed, that is one of the central themes of this book. In general we use notation of the form, x[n], to denote sequences, quantized in both time and amplitude. If, as is the case for sampled speech signals, a sequence can be thought of as a sequence of samples of an analog signal taken periodically with sampling period, T, then we generally find it useful to explicitly indicate this by using the notation x[n]=xa(nT) For any digital sequence derived from an analog waveform via sampling and quantization, there are two variables that determine the nature of the discrete representation, namely the sampling rate Fs =1/T and the number of quantization levels, 2B where B is the number of bits per sample of the representation. Although the sampling rate can be set to any value that satisfies the relation that Fs= 1/T >2Fv (where Fv is the highest frequency present in the continuous-time signal), various "natural" sampling rates for speech have evolved over time, including:1Fs = 6.4 kHz for telephone bandwidth speech (FN = 3.2 kHz);F = 8 kHz for extended telephone bandwidth speech (FN =4 kHz);Fs = 10 kHz for oversampled telephone bandwidth speech (Fy =5 kHz): F, = 16 kHz for wideband (hi-f) bandwidth speech (FN =8 kHz).The second variable in the digital representation of speech signals is the number of bits per sample for the quantized signal. In Chapter 11, we will study the detailed effects of quantization on the properties of the digitized speech waveform; however, in the oresent chapter, we assume that the sample values are unquantized Figure 2.1 shows an example of a speech signal represented both as an analog signal and as a sequence of samples at a sampling rate of Fs = 16 kHz. In subsequent

FIGURE 2.1Plots of a speech waveform: (a) plotted as a continuous-time signal (with MATLAB plot ( ) function);(b) plotted as a sampled signal (with MATLAB stem( ) function).1As discussed in Section 2.3, we use capital letters to denote analog frequencies as in [270].

gures, convenience in plotting generally dictates the use of the analog representation (i.e., continuous functions) even when the discrete representation is being considered.In such cases, the continuous curve is the envelope of the sequence of samples, with the samples connected by straight lines for plotting (linear interpolation).2 Figure 2.1 illustrates the use of the MATLAB function plot (for the analog waveform at the top of the figure) and stem (for the set of discrete samples at the bottom of the figure). It also illustrates an important point about sampled signals in general; when a sequence of samples is plotted as a function of sample index, the time scale is lost. We must now the sampling rate, Fs = 1/T = 16 kHz, in order to convert the time duration of the digital waveform (320 samples in this case) to the analog time interval of 20 msec via the sampling period T = 62.5 us.In our study of digital speech processing systems, we will find a number of special sequences repeatedly arising. Several of these sequences are depicted in Figure 2.2.The unit sample or unit impulse sequence, shown in Figure 2.2a, is defined as

INTRODUCTION

介绍

Since the speech processing schemes and techniques that we discuss in this book are intrinsically discrete-time signal processing systems, it is essential that the reader have a good understanding of the basic techniques of digital signal processing.

由于我们在本书中讨论的语音处理方案和技术本质上是离散时间信号处理系统，因此读者必须对数字信号处理的基本技术有一个很好的理解。

In this chapter we present a brief review of the concepts in digital signal processing that are most relevant to digital processing of speech signals.

在这一章中，我们简要回顾了与语音信号数字处理最相关的数字信号处理的概念。

This review is intended to serve as a convenient reference for later chapters and to establish the notation that will be used throughout the book.

这篇评论的目的是为以后的章节提供一个方便的参考，并建立将在整本书中使用的记号。

Those readers who are completely unfamiliar with techniques fo representation and analvsis of discrete-time signals and systems may find it worthwhile to consult a textbook on digital signal processing [245, 270, 305] when this chapter does not provide sufficient detail.

那些完全不熟悉离散时间信号和系统的表示和分析技术的读者可能会发现，当本章没有提供足够的细节时，值得查阅关于数字信号处理的教科书[245,270,305]。

DISCRETE-TIME SIGNALS AND SYSTEMS

离散时间信号和系统

In almost every situation involving information processing or communication, it is natural to begin with a representation of the signal as a continuously varying pattern or waveform.

在几乎所有涉及信息处理或通信的情况下，很自然地，首先要把信号表示为连续变化的模式或波形。

The acoustic wave produced in human speech is most certainly of this nature.

人类语言中产生的声波肯定是这种性质的。

It is mathematically convenient to represent such continuously varying patterns as functions of a continuous variable t, which represents time.

用连续变量t的函数来表示这种连续变化的模式在数学上是很方便的，t代表时间。

We use notatior of the form xa(t) to denote continuously varying (or analog) time waveforms.

我们使用形式为xa(t)的符号来表示连续变化(或模拟)的时间波形。

As we will see, it is also possible to represent the speech signal as a sequence of (quantized) numbers;

正如我们将看到的，也可以将语音信号表示为(量化的)数字序列;

indeed, that is one of the central themes of this book.

事实上，这是本书的中心主题之一。

In general we use notation of the form, x[n], to denote sequences, quantized in both time and amplitude.

一般来说，我们使用x[n]这种形式的符号来表示序列，在时间和幅度上都量子化。

If, as is the case for sampled speech signals, a sequence can be thought of as a sequence of samples of an analog signal taken periodically with sampling period, T, then we generally find it useful to explicitly indicate this by using the notation x[n]=xa(nT) For any digital sequence derived from an analog waveform via sampling and quantization, there are two variables that determine the nature of the discrete representation,

如果是语音信号采样的情况下,一个序列可以看作是一系列的模拟信号采集样本定期与采样周期T,然后我们通常发现它有用的来显式地指示使用符号x [n] = xa (nT)对于任何一个数字序列来自一个模拟波形通过采样和量化,有两个变量,确定离散形式的性质,

namely the sampling rate Fs =1/T and the number of quantization levels, 2B where B is the number of bits per sample of the representation.

即采样率Fs =1/T和量化级别数，2B，其中B是表示的每个样本的位数。

Although the sampling rate can be set to any value that satisfies the relation that Fs= 1/T >2Fv (where Fv is the highest frequency present in the continuous-time signal), various "natural" sampling rates for speech have evolved over time, including:1Fs = 6.4 kHz for telephone bandwidth speech (FN = 3.2 kHz);F = 8 kHz for extended telephone bandwidth speech (FN =4 kHz);Fs = 10 kHz for oversampled telephone bandwidth speech (Fy =5 kHz):

虽然采样率可以设置为任何值满足Fs = 1 / T比的关系;2艘渔船(阵线是最高频率出现在连续时间信号),各种“自然”抽样率的演讲已经随着时间的推移,包括:电话1 F = 6.4千赫带宽演讲(FN = 3.2 kHz); F = 8 kHz长电话带宽演讲(FN = 4 kHz); Fs = 10 kHz的采样过量电话带宽演讲(= 5 kHz财政年度):

F, = 16 kHz for wideband (hi-f) bandwidth speech (FN =8 kHz).

F， = 16 kHz的宽带(hi-f)带宽语音(FN =8 kHz)。

The second variable in the digital representation of speech signals is the number of bits per sample for the quantized signal.

语音信号的数字表示的第二个变量是量化信号的每个样本的位数。

In Chapter 11, we will study the detailed effects of quantization on the properties of the digitized speech waveform;

在第11章中，我们将详细研究量化对数字化语音波形特性的影响;

however, in the oresent chapter, we assume that the sample values are unquantized Figure 2.1 shows an example of a speech signal represented both as an analog signal and as a sequence of samples at a sampling rate of Fs = 16 kHz.

然而，在本章中，我们假设样本值是未量化的。图2.1显示了一个语音信号的例子，该信号既表示为模拟信号，也表示为Fs = 16khz采样率下的一系列样本。

In subsequent

在随后的

FIGURE 2.1Plots of a speech waveform: (a) plotted as a continuous-time signal (with MATLAB plot ( ) function);(b) plotted as a sampled signal (with MATLAB stem( ) function).

图2.1语音波形图:(a)连续时间信号(用MATLAB plot()函数绘制);(b)采样信号(用MATLAB stem()函数绘制)

1As discussed in Section 2.3, we use capital letters to denote analog frequencies as in [270].

正如第2.3节所讨论的，我们用大写字母来表示[270]中的模拟频率。

gures, convenience in plotting generally dictates the use of the analog representation (i.e., continuous functions) even when the discrete representation is being considered.

图形，绘图的方便性通常要求使用模拟表示(即连续函数)，即使是在考虑离散表示时。

In such cases, the continuous curve is the envelope of the sequence of samples, with the samples connected by straight lines for plotting (linear interpolation).

在这种情况下，连续的曲线就是样本序列的包络，用直线连接样本进行绘图(线性插值)。

2 Figure 2.1 illustrates the use of the MATLAB function plot (for the analog waveform at the top of the figure) and stem (for the set of discrete samples at the bottom of the figure).

2图2.1说明了MATLAB函数图(用于图顶部的模拟波形)和stem(用于图底部的离散样本集)的使用。

It also illustrates an important point about sampled signals in general;

阐述了一般采样信号的一个重要问题;

when a sequence of samples is plotted as a function of sample index, the time scale is lost.

当一个样本序列被绘制成样本指数的函数时，时间尺度就丢失了。

We must now the sampling rate, Fs = 1/T = 16 kHz, in order to convert the time duration of the digital waveform (320 samples in this case) to the analog time interval of 20 msec via the sampling period T = 62.5 us.

我们必须现在的采样率，Fs = 1/T = 16 kHz，以便通过采样周期T = 62.5 us将数字波形(本例中为320个样本)的持续时间转换为20 msec的模拟时间间隔。

In our study of digital speech processing systems, we will find a number of special sequences repeatedly arising.

在我们对数字语音处理系统的研究中，我们会发现一些特殊序列反复出现。

Several of these sequences are depicted in Figure 2.2.

图2.2描述了其中几个序列。

The unit sample or unit impulse sequence, shown in Figure 2.2a, is defined as

图2.2a所示的单位样本或单位脉冲序列定义为

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