ADC与DAC：数字信号处理的关键

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"ADC and DAC，也称为模拟到数字转换（Analog-to-Digital Conversion）和数字到模拟转换（Digital-to-Analog Conversion），是工程和科学领域中将连续信号与数字系统交互的关键技术。这些过程使得计算机能够处理诸如光强度、电压变化、化学反应速率等日常信号。" 在数字信号处理领域，Steven W. Smith所著的《Digital Signal Processing: A Practical Guide for Engineers and Scientists》是一本非常实用的入门书籍，适合初学者和有一定经验的读者。书中以易于理解的方式阐述了物理概念，避免了复杂的公式推导，因此即便是英文版，阅读起来也不困难。 ### ADC (模拟到数字转换) ADC是将连续的模拟信号转换为离散的数字信号的过程。这个过程中包含两个关键步骤：**采样**和**量化**。 1. **采样**：根据奈奎斯特定理，采样频率至少需要是被采样信号最高频率的两倍，以确保能无失真地复现原始信号。如果采样频率不足，就会出现混叠现象，导致高频成分错误地表现为低频成分。 2. **量化**：量化是将采样后的连续值转化为有限精度的数字表示，通常通过比较采样值与量化级距来实现。这会导致信息损失，称为量化噪声。量化位数越多，量化误差越小，但所需的存储空间和计算资源也随之增加。 ### DAC (数字到模拟转换) 相反，DAC是将数字信号转换回模拟信号的过程。数字信号由二进制位序列表示，DAC将这些位转换为连续的电压或电流信号。选择合适的DAC分辨率（位数）对于还原数字信号的精度至关重要。 ### 信息管理在ADC和DAC的设计中，需要平衡保留的信息量和可接受的失真水平。这涉及到选择合适的采样频率、量化位数以及预/后滤波器设计。过高采样率会增加数据处理负担，而过低则可能导致信号失真。同样，更高的量化位数提供更好的分辨率，但成本和复杂性也会相应增加。 ### 概念澄清在数字信号处理中，数据被称为“digital data”而非“digit data”，因为数据是离散的，而不仅仅是单个数字。模拟到数字转换通常被称为“digitize”，因为它将连续信号数字化。《ADC and DAC》这一章讨论的是如何有效地在模拟世界和数字世界之间转换信息，同时考虑到信息保留和信息损失的权衡，这对任何涉及信号处理的工程应用都是至关重要的。

The Scientist and Engineer's Guide to Digital Signal Processing40

The sine wave shown in (b) has a frequency of 0.09 of the sampling rate. This

might represent, for example, a 90 cycle/second sine wave being sampled at

1000 samples/second. Expressed in another way, there are 11.1 samples taken

over each complete cycle of the sinusoid. This situation is more complicated

than the previous case, because the analog signal cannot be reconstructed by

simply drawing straight lines between the data points. Do these samples

properly represent the analog signal? The answer is yes, because no other

sinusoid, or combination of sinusoids, will produce this pattern of samples

(within the reasonable constraints listed below). These samples correspond to

only one analog signal, and therefore the analog signal can be exactly

reconstructed. Again, an instance of proper sampling.

In (c), the situation is made more difficult by increasing the sine wave's

frequency to 0.31 of the sampling rate. This results in only 3.2 samples per

sine wave cycle. Here the samples are so sparse that they don't even appear

to follow the general trend of the analog signal. Do these samples properly

represent the analog waveform? Again, the answer is yes, and for exactly the

same reason. The samples are a unique representation of the analog signal.

All of the information needed to reconstruct the continuous waveform is

contained in the digital data. How you go about doing this will be discussed

later in this chapter. Obviously, it must be more sophisticated than just

drawing straight lines between the data points. As strange as it seems, this is

proper sampling according to our definition.

In (d), the analog frequency is pushed even higher to 0.95 of the sampling rate,

with a mere 1.05 samples per sine wave cycle. Do these samples properly

represent the data? No, they don't! The samples represent a different sine wave

from the one contained in the analog signal. In particular, the original sine

wave of 0.95 frequency misrepresents itself as a sine wave of 0.05 frequency

in the digital signal. This phenomenon of sinusoids changing frequency during

sampling is called aliasing. Just as a criminal might take on an assumed name

or identity (an alias), the sinusoid assumes another frequency that is not its

own. Since the digital data is no longer uniquely related to a particular analog

signal, an unambiguous reconstruction is impossible. There is nothing in the

sampled data to suggest that the original analog signal had a frequency of 0.95

rather than 0.05. The sine wave has hidden its true identity completely; the

perfect crime has been committed! According to our definition, this is an

example of improper sampling.

This line of reasoning leads to a milestone in DSP, the sampling theorem.

Frequently this is called the Shannon sampling theorem, or the Nyquist

sampling theorem, after the authors of 1940s papers on the topic. The sampling

theorem indicates that a continuous signal can be properly sampled, only if it

does not contain frequency components above one-half of the sampling rate.

For instance, a sampling rate of 2,000 samples/second requires the analog

signal to be composed of frequencies below 1000 cycles/second. If frequencies

above this limit are present in the signal, they will be aliased to frequencies

between 0 and 1000 cycles/second, combining with whatever information that

was legitimately there.

Chapter 3- ADC and DAC 41

Time (or sample number)

-3

-2

-1

c. Analog frequency = 0.31 of sampling rate

Time (or sample number)

-3

-2

-1

d. Analog frequency = 0.95 of sampling rate

Time (or sample number)

-3

-2

-1

a. Analog frequency = 0.0 (i.e., DC)

Time (or sample number)

-3

-2

-1

b. Analog frequency = 0.09 of sampling rate

Amplitude Amplitude

Amplitude

FIGURE 3-3

Illustration of proper and improper sampling. A continuous signal is sampled properly if the samples contain all the

information needed to recreate the original waveform. Figures (a), (b), and (c) illustrate proper sampling of three

sinusoidal waves. This is certainly not obvious, since the samples in (c) do not even appear to capture the shape of the

waveform. Nevertheless, each of these continuous signals forms a unique one-to-one pair with its pattern of samples.

This guarantees that reconstruction can take place. In (d), the frequency of the analog sine wave is greater than the

Nyquist frequency (one-half of the sampling rate). This results in aliasing, where the frequency of the sampled data is

different from the frequency of the continuous signal. Since aliasing has corrupted the information, the original signal

cannot be reconstructed from the samples.

Amplitude

Two terms are widely used when discussing the sampling theorem: the

Nyquist frequency and the Nyquist rate. Unfortunately, their meaning is

not standardized. To understand this, consider an analog signal composed of

frequencies between DC and 3 kHz. To properly digitize this signal it must

be sampled at 6,000 samples/sec (6 kHz) or higher. Suppose we choose to

sample at 8,000 samples/sec (8 kHz), allowing frequencies between DC and 4

kHz to be properly represented. In this situation there are four important

frequencies: (1) the highest frequency in the signal, 3 kHz; (2) twice this

frequency, 6 kHz; (3) the sampling rate, 8 kHz; and (4) one-half the sampling

rate, 4 kHz. Which of these four is the Nyquist frequency and which is the

Nyquist rate? It depends who you ask! All of the possible combinations are

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