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Questions and Answers
What is the key feature of the Discrete Fourier Transform (DFT)?
What is the key feature of the Discrete Fourier Transform (DFT)?
What role does stock addition play in signal analysis?
What role does stock addition play in signal analysis?
In the Fourier series representation, what does Bk represent?
In the Fourier series representation, what does Bk represent?
What is the significance of the parameter T in the square wave function y(t)?
What is the significance of the parameter T in the square wave function y(t)?
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What does x(t) repeating itself every M periods imply?
What does x(t) repeating itself every M periods imply?
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Why is stock addition necessary in analyzing signals with more than one period?
Why is stock addition necessary in analyzing signals with more than one period?
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What does it mean when the spectrum of x(mt) repeats M times per cycle of X(e^{j ext{ω}T})?
What does it mean when the spectrum of x(mt) repeats M times per cycle of X(e^{j ext{ω}T})?
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How can we express the DFT of x(t) if we wish to analyze a 6-periodic function instead of the original 4-periodic one?
How can we express the DFT of x(t) if we wish to analyze a 6-periodic function instead of the original 4-periodic one?
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What modification should be made to the MATLAB code to handle multiple periods within each call to the dft() function?
What modification should be made to the MATLAB code to handle multiple periods within each call to the dft() function?
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What is the consequence of performing an eight-period DFT instead of a four-period one?
What is the consequence of performing an eight-period DFT instead of a four-period one?
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Why is stock addition important in expanding the range of applications for the DFT?
Why is stock addition important in expanding the range of applications for the DFT?
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Study Notes
Stock Addition to the Discrete Fourier Transform (DFT)
The Discrete Fourier Transform (DFT) is a fundamental tool used in signal analysis and processing. One of its key features is its ability to break down a time domain sequence into frequency components using complex coefficients. However, what if you want to analyze signals with more than one period? This is where stock addition comes in.
Stock addition allows us to extend our analysis beyond single periods by representing periodic functions using their corresponding Fourier series representation. These representations consist of an infinite summation of sines and cosines. To understand how this works, let's consider a simple example of the square wave function y(t), defined over two periods on the interval ( -T \le t \le T):
[y(t) = \begin{cases} A_n & -\frac{T}{2} \leq t < 0 \ -A_n & 0 \leq t < \frac{T}{2} \end{cases}]
where (A_n) is the amplitude of the nth component. We can represent this function using the following expression:
[y(t) = \sum_{k=-\infty}^{\infty} B_k \cos\left(\frac{2\pi k}{T}t + \phi_k \right)]
Here, (B_k) represents the magnitude of each coefficient, while (\phi_k) indicates the phase shift between them.
Now, let's say we have another periodic function (x(t)) which has the property that it repeats itself every (M) periods. In order to analyze this function using the DFT, we need to first express it as follows:
[X(e^{jwT}) = X(e^{jwMT})]
This means that the spectrum of (x(mt)) will repeat M times per cycle of (X(e^{j\omega T})). By multiplying these spectra together, we obtain the final expression for the DFT of (x(t)):
[X(e^{j\omega T}) = F(e^{j(\omega / M)T}), \quad m = 0, 1,\ldots, M]
To implement this concept in practice, we simply modify our code to handle multiple periods within each call to the dft() function in MATLAB. For instance, if you wish to perform an eight-period DFT instead of a four-period one, you would call dft(8*input).
In summary, stock addition expands the range of applications for the DFT by allowing us to analyze multi-periodic functions without having to adjust our existing code structure.
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Description
Learn how to extend the analysis of multi-periodic signals beyond single periods using stock addition in the Discrete Fourier Transform (DFT). Explore how periodic functions are represented using Fourier series and understand the concept of spectrum repetition for functions with multiple periods. Discover the practical implementation of stock addition in MATLAB for analyzing multi-periodic functions.
