Question

Derivative in the Presence of Noise. We will now explore the effects of input noise on the quality of a numerical derivative. First, generate an input vector containing 100 values of the function \(\sin x\) starting at \(x=0\), using a step size \(\Delta x\) of \(0.05\), just as you did in the previous problem. Next, use function randomo to generate a small amount of random noise with a maximum amplitude of \(\pm 0.02\), and add that random noise to the samples in your input vector (see Figure 5.8). Note that the peak amplitude of the noise is only \(2 \%\) of the peak amplitude of your signal, since the maximum value of \(\sin x\) is 1. Now take the derivative of the function using the derivative function that you developed in the last problem. How close to the theoretical value of the derivative did you come?

Short answer

The short answer to this problem is as follows: After generating the noisy \(\sin(x)\) vector and taking its numerical derivative, you can compare the result with the theoretical \(\cos(x)\) values. Calculate the difference between the numerical approximation and the theoretical values and compute the average, root mean square difference, or maximum difference to evaluate the accuracy and impact of noise on the numerical derivative calculation.

Step-by-step solution

Generating the \(\sin(x)\) vector

Start by creating a sequence of 100 x-values with a step size of 0.05. To do so, initialize an empty list or array for x-values, and start from 0, each time incrementing the value by 0.05 until you have 100 values. Next, compute the \(\sin(x)\) value for each x in the sequence, and store these values in another array. Remember that the x-values must be in radians when calculating sine of x.

Generating random noise

Following the first step, use any random number generator function which can generate 100 noise values ranging from -0.02 to 0.02. This function should take the count of random numbers needed (100 in this case) and the range within which these values should fall (from -0.02 to 0.02). Store these noise values.

Adding noise to the \(\sin(x)\) vector

The third step is to add the noise to the \(\sin(x)\) vector. For each \(\sin(x)\) value, add the corresponding noise value (index-wise). This should give you a new "noisy" sine wave vector.

Taking the derivative of the function

Now, apply the numerical derivative function that was developed in the previous problem. However, the basic idea is that for a function \(f\) defined at \(x[i]\), the derivative \(\frac{df}{dx}\) at \(x[i]\) can be approximated as \(\frac{f(x[i+1])-f(x[i])}{x[i+1]-x[i]}\). Do this computation for all but the last value of \(x[i]\) (because the formula uses \(x[i+1]\)).

Compare with the theoretical derivative

Remember, the theoretical derivative of \(\sin(x)\) is \(\cos(x)\). Therefore, compare the calculated numerical derivative from step 4 with the theoretical values achieved by calculating \(\cos(x)\) for the same x-values. You can calculate the difference for each point and form average, root mean square difference, or maximum difference and gain insights about the accuracy of the numerical derivative calculation in the presence of noise.

Key concepts

MATLAB Programming

MATLAB is a powerful computing environment that allows users to perform mathematical computations, algorithm development, data analysis, and visualization. For students tackling numerical analysis, MATLAB provides an array of built-in functions which make the implementation of mathematical concepts like derivatives simple. To generate a vector containing values of a sine function, one can use a combination of MATLAB's loop structures and built-in trigonometric functions. For instance, using for loops to iterate over values from 0 with an increment of 0.05 to create x-values and applying the sin function to compute the sine of each value.

Adding noise in MATLAB involves generating random numbers, which can be done using functions like rand or randn. These functions allow for the creation of uniformly distributed random numbers in a specified range. When adding random noise, the + operator in MATLAB can be used to combine the sine function values with generated noise, simulating real-world data which often contains some level of inaccuracy or 'noise'.

Sin Function Analysis

The sine function, represented mathematically as \(\sin(x)\), is a periodic function commonly used in trigonometry and it describes the y-coordinate of a point on the unit circle as it moves around the circumference. When analyzing \(\sin(x)\) in a numerical setting such as MATLAB programming, one often samples this function at various points to study its behavior. An important tool for analyzing the dynamics of \(\sin(x)\) is its derivative. In theory, the derivative of \(\sin(x)\) is \(\cos(x)\), which implies that the rate of change of the sine function at any point x is given by the cosine value at that same point.

Understanding this relationship is essential when analyzing the effects of noise on numerical derivatives. With noisy data, we expect the calculated numerical derivative to deviate from the smooth curve of \(\cos(x)\), but the overall shape and periodic nature should be preserved. In exploratory analysis, we can graph both the noisy and theoretical derivatives to visually assess the impact of added noise.

Random Noise Generation

Random noise generation is an essential aspect of simulating real-world data collection in a controlled environment. Noise can represent measurement errors, data transmission errors, or environmental fluctuations. In MATLAB, generating random noise that simulates such imperfections is done using random number generators such as rand, which generates uniformly distributed values, or randn, which generates normally distributed numbers.

The amplitude of generated noise is crucial; it must be realistic in its influence on the signal of interest. In our exercise, the noise has a maximum amplitude of \(\pm 0.02\), a significant but not overshadowing level compared to the signal's peak amplitude. This scenario is reflective of many scientific measurements where the signal-to-noise ratio is a key indicator of data quality. The goal is to generate a series of random numbers within the desired amplitude range and apply them to the signal to replicate the presence of 'noise' in empirical data.

Theoretical vs Numerical Derivative Comparison

Comparing theoretical and numerical derivatives offers insight into the accuracy of numerical methods in approximating derivatives, especially amidst noisy data. Theoretical derivatives are calculated based on the known mathematics of a function, while numerical derivatives are practical approximations that use discrete data points and difference quotients. The numerical derivative, especially for functions like \(\sin(x)\), is determined by calculating the slope of the tangent to the points on the wave.

In the context of the exercise, the derivative is approximated using a finite difference method, which evaluates the slope between successive points of the noisy sine function. The exercise encourages comparing the numerical derivative—affected by noise—to the smooth curve of the theoretical derivative, \(\cos(x)\). One expects the numerical derivative to closely follow the \(\cos(x)\) curve, but discrepancies will arise, reflecting the noise's effect. Measuring the differences using various methods (average difference, RMSE, etc.) reveals the robustness of the numerical method against noise and informs improvements for more precise calculations.