Theorem of Discrete Fourier Transform

Last edited time

Jan 20, 2024 03:23 AM

Representation of discrete Fourier transform

The Discrete Fourier Transform (DFT) is a fundamental theorem in the field of signal processing and harmonic analysis. It allows us to take a discrete signal, which is a finite sequence of data points in time or space, and transform it into its frequency components. The theorem can be formally stated as follows:

Theorem: Discrete Fourier Transform (DFT) Given a finite sequence of \( N \) complex numbers \( x_0, x_1, x_2, ..., x_{N-1} \), the DFT produces another sequence of \( N \) complex numbers \( X_0, X_1, X_2, ..., X_{N-1} \), which is defined by the formula:

\[ X_k = \sum_{n=0}^{N-1} x_n \cdot e^{-\frac{2\pi i}{N} nk} \]

for \( k = 0, 1, 2, ..., N-1 \), where:

\( x_n \) are the input sequence samples,

\( X_k \) are the DFT coefficients or the output sequence,

\( N \) is the number of samples,

\( n \) is the current sample index,

\( k \) corresponds to the index of the DFT coefficient and can be thought of as the frequency index,

\( i \) is the imaginary unit, and

\( e \) is the base of the natural logarithm.

The exponential term \( e^{-\frac{2\pi i}{N} nk} \) is a complex exponential that, when graphed, traces out a circle in the complex plane as \( n \) varies from 0 to \( N-1 \). It represents a discrete set of points on the unit circle, and these points correspond to discrete frequencies.

The DFT maps the time domain signal, which is a sequence of values in time, into the frequency domain. Each \( X_k \) represents the amplitude and phase of a frequency component of the original signal.

The inverse DFT (IDFT) allows you to reconstruct the original time domain signal from its DFT coefficients and is given by:

\[ x_n = \frac{1}{N} \sum_{k=0}^{N-1} X_k \cdot e^{\frac{2\pi i}{N} nk} \]

for \( n = 0, 1, 2, ..., N-1 \).

Key Properties of the DFT:

Periodicity: The DFT is a periodic transform with a period of \( N \), meaning that \( X_{k+N} = X_k \).

Linearity: The DFT is a linear transform; the DFT of a sum of signals is the sum of their DFTs.

Symmetry: For real-valued signals, the DFT exhibits symmetry in its coefficients; \( X_{N-k} \) is the complex conjugate of \( X_k \) for \( k = 1, 2, ..., N-1 \).

Efficiency: While the direct computation of the DFT from its definition requires \( O(N^2) \) operations, fast algorithms like the Fast Fourier Transform (FFT) can compute it in \( O(N \log N) \) operations.

The DFT is widely used in many applications such as digital signal processing, image processing, solving partial differential equations, and in various fields of engineering and science. It is particularly useful because it enables the analysis and manipulation of signals in the frequency domain, where many signal properties are more intuitive and easier to handle.

Use the Euler function to transform the formation

To establish a connection between Euler's formula and the given functions, we'll look at how a sinusoidal function can be represented using complex exponentials, which can then be used to express the Fourier series coefficients.

Starting with the Euler's formula:

\[ e^{ix} = \cos(x) + i\sin(x) \] \[ e^{-ix} = \cos(x) - i\sin(x) \]

From these identities, you can solve for the sine and cosine functions:

\[ \sin(x) = \frac{e^{ix} - e^{-ix}}{2i} \] \[ \cos(x) = \frac{e^{ix} + e^{-ix}}{2} \]

Given the input function \( v_I = E \sin(\omega t) \), we can express it using Euler's formula as:

\[ v_I = E \left( \frac{e^{i\omega t} - e^{-i\omega t}}{2i} \right) \]

This is the complex representation of the original sinusoidal input signal. Now, for the output function \( v_0 \), which is a response of a system with potential harmonics due to nonlinearity, we use the complex form of the Fourier series. The Fourier series of a periodic function can be expressed in terms of complex exponentials as:

\[ v_0 = \sum_{n=-\infty}^{\infty} C_n e^{in\omega t} \]

Where \( C_n \) are the complex Fourier coefficients. By considering both positive and negative \( n \), this series includes both the sine and cosine terms, since \( e^{in\omega t} \) and \( e^{-in\omega t} \) will yield \( \sin(n\omega t) \) and \( \cos(n\omega t) \) when decomposed using Euler's formula.

To find the Fourier coefficients \( C_n \) for the non-negative values of \( n \), we would use:

\[ C_n = \frac{1}{T} \int_{0}^{T} v_0 e^{-in\omega t} \, dt \]

And for the non-positive values of \( n \), \( C_{-n} \) would be the complex conjugate of \( C_n \), since the original function is real.

Now, if we look back at the provided function:

\[ v_0 = A_1(E, \omega) \cos(\omega t) + B_1(E, \omega) \sin(\omega t) + A_2(E, \omega) \cos(2\omega t) + B_2(E, \omega) \sin(2\omega t) + \cdots \]

We can see that this is essentially the real part of the Fourier series represented by complex exponentials, where \( A_n(E, \omega) \) and \( B_n(E, \omega) \) correspond to the real and imaginary parts (up to a constant factor) of the complex Fourier coefficients \( C_n \) for each harmonic \( n \).

In summary, Euler's formula allows us to express sinusoidal functions as complex exponentials, which in turn allows us to represent the output of a nonlinear system as a sum of complex exponentials. The real and imaginary parts of these exponentials then correspond to the cosine and sine terms in the Fourier series representation of the system's output.