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# 物理代写|声学代写Acoustics代考|SIO190 Normal Error Function or the Gaussian Distribution

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## 物理代写|声学代写Acoustics代考|Normal Error Function or the Gaussian Distribution

“Everyone believes that the Gaussian distribution describes the distribution of random errors; mathematicians because they think physicists have verified it experimentally, and physicists because they think mathematicians have proved it theoretically.” [21]

When we think about random processes, a common example is the coin toss. If we flip a fair coin, there is a $50 \%$ probability that the coin will land “heads” up and an equal probability that it will land “tails” up. If we toss $N$ coins simultaneously, the probability of any particular outcome, say $h$ heads and $t=N-h$ tails, is given by a binomial distribution. The average of that distribution will still be $\bar{h}=$ $\bar{t}=N / 2$, but the likelihood of getting exactly $N / 2$ heads in any given toss is fairly small and grows smaller with increasing $N$.

The probability, $P_B(h, p, N)$, of obtaining $h$ heads and $t=N-h$ tails is given by a binomial distribution where the probability of obtaining a head is $p=0.5$. Of course, the probability of a tail is $q=0.5$.
$$P_B(h, p, \mathrm{~N})=\frac{N !}{h !(N-h) !} p^h q^{(N-h)}$$
For the binomial distribution, the average outcome with the largest probability is the mean, $\bar{h}=N p$, and the standard deviation about that mean is $\sigma=[N p(1-p)]^{1 / 2}$. It is worthwhile to recognize that $\bar{h}$ is proportional to the number of trials $N$, whereas $\sigma$ is proportional to $\sqrt{N}$. Therefore, the relative width of the distribution function, $\sigma / \bar{h}$, decreases in proportion to $\sqrt{N}$. This “sharpening” of the distribution with increasing $N$ is evident in Fig. 1.11.

## 物理代写|声学代写Acoustics代考|Systematic Errors (Bias)

Systematic error is not reduced by increasing the number of measurements. In the “target” example on the right-hand side of Fig. 1.10, taking more shots will not bring the average any closer to the bull’seye. On the other hand, an adjustment of the sighting mechanism could produce results that are far better than those shown in the left-hand side of Fig. $1.10$ by bringing the much tighter cluster of holes on the right-hand side toward the center of the target.

The right-hand side of the target example in Fig. $1.10$ represents a type of systematic error that I call a “calibration error.” These can enter a measurement in a number of ways. If a ruler is calibrated at room temperature but used at a much higher temperature, the thermal expansion will bias the readings of length. In acoustic and vibration experiments, often each component of the measurement system may be calibrated, but the calibration could be a function of ambient pressure and temperature. The “loading” of the output of one component of the system by a subsequent component can reduce the output or provide a gain that is load-dependent.

For example, a capacitive microphone capsule with a capacitance of $50 \mathrm{pF}$ has an electrical output impedance at $100 \mathrm{~Hz}$ of $Z_{e l}=(\omega C)^{-1}=32 \mathrm{M} \Omega$. If it is connected to a preamplifier with an input impedance of $100 \mathrm{M} \Omega$, then the signal amplified by that stage at $100 \mathrm{~Hz}$ is reduced by $(100 / 132) \cong 0.76$. Even though the capsule may be calibrated with a sensitivity of $1.00 \mathrm{mV} / \mathrm{Pa}$, it will present a sensitivity to the preamplifier of $0.76 \mathrm{mV} / \mathrm{Pa}$. At $1.0 \mathrm{kHz}$, the capsule’s output impedance drops to $3.2 \mathrm{M} \Omega$, so the effective sensitivity at that higher frequency will be $(100 / 103) \times 1.00 \mathrm{mV} /$ $\mathrm{Pa}=0.97 \mathrm{mV} / \mathrm{Pa}$. A typical acoustic measurement system may concatenate many stages from the sensor to its preamplifier, through the cabling, to the input stage of the data acquisition system, through some digital signal processing, and finally out to the display or some recording (storage) device.
As important as it is to know the calibration or gain (i.e., transfer function) of each stage in the measurement system, it is also imperative that the entire system’s overall behavior be tested by an endto-end calibration that can confirm the calculation of the overall system’s sensitivity. This is usually accomplished by providing a calibrated test signal to the sensor and reading the output at the end of the signal acquisition chain. Commercial calibrators are available for microphones, hydrophones, and accelerometers. If a commercial calibrator is not available for a particular sensor, some end-to-end calibration system should be designed as part of the test plan for every experiment.

## 物理代写|声学代写声学代考|正态误差函数或高斯分布

“大家都认为高斯分布描述了随机误差的分布;数学家是因为他们认为物理学家已经通过实验验证了它，物理学家是因为他们认为数学家已经从理论上证明了它。”[21]

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。