Stock Addition in Discrete Fourier Transform (DFT)

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.