fourier transform homework solutions

fourier transform homework solutions

Understanding Fourier Transform and Its Applications

1. Introduction to Fourier Transform

Fourier transform was named after Jean-Baptiste Joseph Fourier, a French mathematician. It is a mathematical method that transforms a function of time space into a function of frequency space. Many engineering and physics problems require a study of how a quantity varies with time. The concepts of time- and frequency-domain representations are used to analyze and represent different types of time functions. The main reason that Fourier transforms are used is that they can help simplify the analysis of complex waveforms by turning them into a combination of individual sine and cosine waves. The simplest type of Fourier transform is called the continuous transform, and it is used to transform functions of time space into functions of continuous frequency space. Hopefully now you have a good, general understanding of what Fourier transforms do. Later, I will go more into depth for each of the steps that I have just outlined and will describe how Fourier transforms can be used in real life situations.

2. Properties and Mathematics of Fourier Transform

Fourier Transform is a complex operation, but it has very simple and elegant properties. These properties are so fundamental that the Parseval’s theorem, for example, which is one of the most important theorems about Fourier Transform, can use any of a number of different properties to generate a proof. That is, these properties are so important that they can be used to prove other theorems of Fourier Transform, thus making the mathematics easier on the one hand, when trying to understand different ways of approaching Fourier Transform, and the same mathematics that we develop for just studying Fourier Transform can be quite useful for studying many other things in mathematics and in practical applications. Speaking of the mathematics of it, let’s show the formula of continuous-time and a periodic-time Fourier transform first. We use the formula of continuous-time for continuous-time signals and we use the formula of periodic-time for discrete-time signals. Most of the time, the variance of a discrete-time stochastic process is defined by the power spectral density (per hertz) of the process. And probably the most important property is the linearity. This means if we have a linear system, as long as the input can be broken down into pieces, the output will be the sum of the Fourier series corresponding to each of these pieces. This is what makes Fourier Transform so useful – we can take apart a function that we care about, piece by piece and examine each of its unique frequency content. Also, linearity also suggests that if you have a sum of functions in the time domain – each of them, let’s say, are representing an individual force loading or individual motion excitation you could transform each of these to the frequency domain separately and then sum up the results in the frequency domain to find out the total frequency content. The duality is also one of the important properties of Fourier Transform. This means if we have the function f(x) and its Fourier Transform F(w), as f(x) transforms into F(w), F(w) should transform back into f(-x) and that is the meaning of duality. The duality says that we can do Fourier Transform in both ways. The first equation below shows the formula of Fourier Transform in terms of x and w. The second equation below shows the formula of reverse Fourier Transform in terms of w and x. And it’s the most historical way of popularity of driving the formulas of Fourier Transform. Any quantity that is can be defined in terms of sums of sinusoids can be called a spectrum of it and that’s why the continually main of application of Fourier Transform is to either study the steady state response of certain linear time invariance systems or to break down a given input function into pieces such that we can examine each of its unique frequency contents. And so, a periodic-time Fourier transform (as we have talked about in the previous page) is used to study a discrete-time and periodic-time signal in frequency domain and anytime we talk about power spectral density, such power for example, the turbine blades of a jet engine – we use the continuous-time Fourier Transform. These help a lot in the study of a wide variety of things such as engineering, physics, chemistry, biology and etc. It is not hard to see why the knowledge of the properties and the ability to apply such properties come in very helpful when dealing with Fourier Transform in order to solve different problems. And the wonderful thing is that over the last decade and current decade, Fourier Transform has been used and will be used to derive and solve problems in fascinating research fields like signal processing, magnetic resonance image processing in medical field and the study of quantum mechanics in substances. The power of it can lead to breakthrough discoveries that may change the world. It provides many different tools by combining what we know and what we have studied so far to generate new technologies and solve many practical problems. We should be excited to – and we should anticipate – learn new things in the future along with many ever-evolving digital technologies.

3. Solving Fourier Transform Problems

Every time a new parameter is discovered in the method, it should allow a better understanding of why the Fourier transform must have that variable input. The variable output from the new method could be transformed through a continuous Fourier series, which is not restricted to only time, but to any function of a spatial variable. The interactive time-moving app of a bouncing spring-mass system with damping and parameter changes. Visualize the motion of a mass-spring system in both real position-time space and Fourier frequency space. The length of the animation can be customized, as the entire motion shows the entire discrete sum of a Fourier series. Mass/spring and damping constant can be changed, as well as the period, magnitude, and phase of the driving force. The initial position or initial velocity of the mass can be customized with the checkboxes, but the simulate button creates the motion. By using these functions in the class code of each object (mass, spring, and spring), a Fourier series can be computed for each important dynamic output of the animation, like velocity or kinetic energy. Also, the initial conditions of the signals in motor movement could be solved for and graphical illustrations of the transform rules can be shown. Specifically, the graphs of functions in the time domain and their transform in the frequency domain could be compared with their datasets. This would provide more evidence to students as to why the continuous Fourier series is a powerful and useful method to solve problems.

public void service(HttpServletRequest request, HttpServletResponse response) throws IOException { double t = 0; double sumreal = 0, sumimag = 0; double[] rp = new double[N]; double[] ip = new double[N]; double[] cmagn = new double[N]; double[] ph = new double[N]; PrintWriter out = response.getWriter(); for (int j = 0; j < N; j++) { out.println(” j = ” + j + “<br>”); for (t = 0; t < T; t = t + dT) { sumreal = sumreal + x(t) * Math.cos(2 * 3.14159 * j * t / T) * dT; sumimag = sumimag – x(t) * Math.sin(2 * 3.14159 * j * t / T) * dT; } rp[j] = sumreal * 1 / T; ip[j] = sumimag * 1 / T; cmagn[j] = Math.sqrt(rp[j] * rp[j] + ip[j] * ip[j]); ph[j] = Math.atan2(ip[j], rp[j]); out.println(” rp = ” + rp[j] + ” ip = ” + ip[j] + “<br>”); out.println(” cmagn = ” + cmagn[j] + ” ph = ” + ph[j] + “<br>”); sumreal = 0; sumimag = 0; } out.close(); }

When solving a problem with Fourier transform, the function f(t) is known and the function F(ω) is sought. We must now perform the integration. If the integral of f(t)e^(-iωt) from -∞ to ∞ can be evaluated, then that space of f(t) is fully explored, as that integral gives us F(ω), the frequency domain expression of the function. Without using the ‘solve’ function, you must input the range over which you would like to solve the Fourier transform and the function that says what the integrand is, show the variable that is changing. By using the int function in MATLAB and placing the right parameters in the brackets, the computer program can solve for the integral of the function multiplied by the complex exponential, which comes directly from the definition of the Fourier transform. The parameters of the int function come directly from properties of the Fourier transform. For example, one of the parameters would be the range for t, which if not specified otherwise would be from 0 to 2*pi. This is because the parameters for the range of t are in terms of the angular velocity ω, which goes from 0 to 2*pi by definition of the continuous time Fourier series. To show the graph of a Fourier transform in MATLAB, input the correct parameters for the function that you would like to solve the transform for and use the ‘fplot’ function for Fourier transforming plotting three dimensions.

4. Applications of Fourier Transform

Also, in weather radars, they constantly emit radio wave signals. These signals can hit raindrops and get reflected back to the radar. The radar picks up the returned signals and the intensities are digitized. Then the wind speed and direction can be calculated. This is done by using the Fourier Transform to analyze how rain intensity varies with distance from the radar. However, the wind is the thing moving the raindrops and so when the radar picks up a returned signal of a certain intensity, the Fourier Transform is showing all the possible speeds and directions the rain could be falling based upon that one signal. So, in fact, the speed and direction of the rain can be worked out by finding the maximum amplitude of the Fourier Transform of the signal.

The MRI scanner uses the spectroscopy of hydrogen atoms. When a patient is placed in a magnetic field, all the hydrogen atoms line up in a way that creates a net magnetization vector. This causes the atoms to spin. When the patient is slightly disturbed by a radio frequency pulse which provides energy, the spinning atoms move back into alignment. Once the radio frequency pulse is turned off, this energy is released and it gets picked up by a receiving coil in the scanner. Then the Fourier Transform can be used to give a tiny electrical signal which can be processed to make an image.

Next to the time domain, the Fourier Transform can show how many amplitudes of frequencies compose a wave. This is used in electronics to show all the frequencies that make up a time signal. The instrument to do this is called a signal spectrum. Some examples of how it is used in electronics include noise reduction, AM and FM radio, and video and audio signal processing.

5. Conclusion and Further Resources

Now that you have understood the Fourier Transform, which is at the heart of understanding frequency domain analysis and finding and isolating certain frequency components of a continuous time signal, you may further try to understand and see how the Fourier Series becomes a special case of the Fourier Transform and how the frequency domain analysis becomes more reflective over different time domains in the series. The continuous-time Fourier Transform as demonstrated in this article is pivotal in claiming the power of representing a signal in time and that in frequency. Also, by tuning the delta function’s frequency and phase, accurate frequency digital signal can be successfully presented digitally in the frequency domain. Aren’t you amazed about how a tiny delta signal at a given frequency can be employed to digitally present an entirely different frequency signal and hence, is applied in that particular frequency? This is a spectacular phenomenon that we can only understand and unveil the mystery of it by mastering the knowledge of Fourier Transform. For those who are interested in this topic, below list further resources that you may gather and enhance your knowledge in understanding the frequency domain analysis through Fourier Transform. First, mathematical theories and advanced knowledge on how the modern technology represent and analyze the digital signal in both time and frequency domain are introduced in “Signals and Systems” by Dr. Daniel W. Easley and Dr. Bonnie S. Heck. Second, a deeper understanding on how the Fourier Transform and its application in digital signal processing system help in decoding the complexity of real world analog signal processing can be obtained from “Principles of Digital Analysis and Synthesis” by Andrew Bateman and Ian Stobie. Third, another advanced reference for both theoretical and practical aspects of representing a signal in frequency domain that is useful for research in areas of medical imaging and diagnostic, communication and most importantly, the emerging field in technology, digital signal processing, is “Digital Signal Processing Using Matlab and Wavelets” by Dr. Michael Weeks.