Name: from fourier series to fourier transform
Uploaded: 2024-09-26T13:17:03.000Z
Duration: 1 min 11 s
Channel: JoVE

The Fourier series is an effective tool for representing periodic functions like a train of square waves.
Consider a pulse-train waveform consisting of a sequence of rectangular pulses. If the period of these pulses is finite, the waveforms can be represented by a Fourier series.
Now, if the period of the pulse-train increases, the frequency of the obtained line spectra decreases.
But what if the period goes to infinity, resulting in a single pulse? In this case, the summation in the Fourier series evolves into a continuous integral, known as the Fourier transform.
According to Dirichlet conditions, a periodic function can be expanded in terms of
sinusoids if it has a finite number of discontinuities, maxima, and minima and is
integrable. If a function fails these conditions, it cannot be represented by a Fourier series.
Fourier transform is commonly used in image processing, where it helps enhance images and filter out noise, making the details clearer and sharper.

continuous -time fourier transform

Continuous -time Fourier Transform

The Fourier series is instrumental in representing periodic functions, offering a powerful method to decompose such functions into a sum of sinusoids. This technique, however, necessitates modification when applied to nonperiodic functions. Consider a pulse-train waveform consisting of a series of rectangular pulses. When these pulses have a finite period, they can be accurately represented by a Fourier series. Yet, as the period approaches infinity, resulting in a single, isolated pulse, the discrete summation of the Fourier series transforms into a continuous integral known as the Fourier transform.
The transition from the Fourier series to the Fourier transform is pivotal for analyzing nonperiodic functions. The Fourier series decomposes a periodic function x(t) into a sum of sines and cosines, expressed as:
<img alt="Fourier series equation, x(t) sum, formula for harmonic signal analysis." src="/files/ftp_upload/16066/16066_PT%20EQUATION_1.svg" style="text-align: center; height: 40px;" data-altupdatedat="1762173717000" data-alt="Equation1" />
Where xn are the Fourier coefficients and &#969;0&#8203; is the fundamental angular frequency. As the period of the function extends to infinity, the fundamental frequency &#969;0 tends to zero, and the summation over discrete frequencies n&#969;0 evolves into an integral over a continuous frequency variable &#969;:
<img alt="Fourier Transform equation, mathematical formula, continuous signal transformation analysis." src="/files/ftp_upload/16066/16066_PT%20EQUATION_2.svg" style="text-align: center; height: 60px;" data-altupdatedat="1762173717000" data-alt="Equation2" />
This integral defines the Fourier transform X(&#969;), representing the original function x(t) in the frequency domain.
The initial skepticism about representing any periodic function with sinusoids led to the establishment of the Dirichlet conditions. These conditions provide criteria under which a periodic function can be expanded in terms of sinusoids. Specifically, a function x(t) can be represented by a Fourier series if the function has finite discontinuities, finite number of maxima and minima, and it is absolutely integrable over the period.
In practical applications, particularly in image processing, the Fourier transform plays a crucial role. It aids in enhancing images and filtering out noise, thereby making details more distinct and sharper. By transforming an image into the frequency domain, various filtering techniques can be applied to emphasize certain features or reduce noise, and then the inverse Fourier transform is used to convert the processed image back to the spatial domain. This approach is foundational in modern image analysis, enabling advanced techniques in medical imaging, remote sensing, and digital photography.

The Fourier series is an effective tool for representing periodic functions like a train of square waves.
Consider a pulse-train waveform consisting of a ...

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Electrical Engineering

Unit 1

The Fourier Transform

KEY TERMS AND CONCEPTS

from fourier series to fourier transform

The Fourier transform extends Fourier series ideas to nonperiodic functions. A Fourier series breaks a periodic signal into a sum of sines and cosines. When the period becomes infinite, that discrete sum changes into a continuous integral.

A pulse-train waveform shows this change clearly. With a finite period, the repeating rectangular pulses can be represented by a Fourier series. As the period grows without bound, the waveform becomes a single isolated pulse, and the description shifts to the Fourier transform.

In the series form, a periodic function x(t) is written using Fourier coefficients xn and a fundamental angular frequency ω0. As the period increases, ω0 approaches zero. The frequencies nω0 no longer appear as separate terms, and they become a continuous frequency variable ω in an integral that defines X(ω), the frequency-domain form of x(t).

Not every periodic function can be expanded in sinusoids without limits. The Dirichlet conditions give the rules for this expansion. A function can be represented by a Fourier series if it has a finite number of discontinuities, a finite number of maxima and minima, and is absolutely integrable over one period.

The Fourier transform is also useful in image processing. It can help sharpen images and remove noise by moving image data into the frequency domain. After filtering, the inverse Fourier transform brings the image back to the spatial domain. This method is used in medical imaging, remote sensing, and digital photography.

From Fourier Series to Fourier Transform

The Fourier transform extends Fourier series ideas to nonperiodic functions. A Fourier series breaks a periodic signal into a sum of sines and cosines. When the period becomes infinite, that discrete sum changes into a continuous integral. A pulse-train waveform shows this change clearly. With a finite period, the repeating rectangular pulses can be represented by a Fourier series. As the period grows without bound, the waveform becomes a single isolated pulse, and the description shifts to...

basic signals of fourier transform

Fourier Transform Signals in Time and Frequency

The Fourier Transform links time-domain signals to frequency-domain signals. It is a key tool in signal processing. It helps show how a waveform can be described by frequency parts instead of only by time changes. The sinc function is one of the basic signals in this topic. It is written as sinc(x) = sin(πx)/(πx). It equals 1 when x is 0, and it has even symmetry about the y-axis. In the frequency domain, it appears as the Fourier transform of a rectangular pulse. A rectangular pulse has a...

properties of fourier transform i

Fourier Transform in Radio Broadcasting

Fourier Transform properties help radio broadcasting manage signals in the time and frequency domains. These properties are useful for sending multiple audio channels at once, changing clip speed, adding broadcast delays, adjusting audio frequencies, and demodulating signals. They make it easier to transmit and receive audio clearly. When broadcasters combine several audio signals, the Fourier Transform simplifies the process. If f(t) and g(t) have Fourier Transforms F(ω) and G(ω), then the...

properties of fourier transform ii

Fourier Transform Rules in Signal Analysis

The Fourier Transform helps analyze signals by linking the time domain and the frequency domain. Its properties show how a signal changes when it is shifted, differentiated, or combined with another signal. These rules are important in signal processing, radio broadcasting, and audio work. The frequency shifting property shows that moving a signal in frequency causes a phase change in time. If x(t) has a Fourier Transform X(f), then multiplying x(t) by e^j2πf0t gives a transform of X(f−f0).

parseval's theorem for fourier transform

Signal Energy in Time and Frequency

Parseval's theorem links signal energy in the time domain and the frequency domain. In signal processing, it shows that the same energy can be calculated in either view. This makes it an important idea for comparing time-based and frequency-based analysis. To find signal energy, the standard approach uses a resistor value of 1 ohm. In that case, power equals the square of the signal's voltage or current. This gives a simple way to relate power to energy before moving into Fourier analysis. <br...

discrete-time fourier transform

DTFT and Frequency Analysis of Digital Signals

The Discrete-Time Fourier Transform (DTFT) shows how a discrete-time signal can be analyzed in the frequency domain. It takes a signal from the time domain and reveals its frequency components and spectral features. In the DTFT, the integral used for continuous-time signals is replaced by a summation to match the discrete form of the signal. The DTFT spectrum X(Ω) is periodic with a period of 2π. This periodicity means that X(Ω) can be written as a Fourier series, which helps with analysis and...

properties of dtft i

DTFT Signal Properties and Frequency Effects

Discrete-Time Fourier Transforms (DTFTs) help describe discrete-time signals in the frequency domain. The main DTFT properties include linearity, time shifting, frequency shifting, time reversal, conjugation, and time scaling. These rules make it easier to analyze and change signals in signal processing. Linearity is the most basic property. If two discrete-time signals are multiplied by constants a and b and then added together, the DTFT of the result is the weighted sum of the two individual...

properties of dtft ii

DTFT Rules for Differentiation and Energy

The DTFT has useful rules for frequency differentiation, convolution, accumulation, and energy checks. These rules help students see how a discrete-time signal changes in the frequency domain. For frequency differentiation, start with a DTFT pair and differentiate both sides with respect to ω. After multiplying by j, the result gives the DTFT of n x[n]. This property is useful for studying how the slope of a spectrum relates to the signal in time. Convolution brings out another key DTFT rule.

discrete fourier transform

DFT for Finding Signal Frequencies

The Discrete Fourier Transform (DFT) turns a finite set of time-domain samples into frequency components. It is a key tool in signal processing because it evaluates a discrete signal at evenly spaced frequency intervals. The result shows the magnitude and phase of each complex sinusoid in the signal. The DFT also follows several important properties. It is linear, so the DFT of a sum of sequences equals the sum of their individual DFTs. When a sequence shifts in time, its DFT changes by a...

fast fourier transform

Fast Fourier Transform for Signal Analysis

The Fast Fourier Transform (FFT) is a computational algorithm that makes the Discrete Fourier Transform (DFT) faster to calculate. It works by splitting the calculation into smaller parts. This reduces the amount of work needed to move data between the time domain and the frequency domain. A direct N-point DFT needs N² complex multiplications. The FFT lowers that cost to about (N/2)log₂N multiplications. That difference becomes more important as N gets larger, because the FFT cuts the number...

key terms and concepts

See how a periodic Fourier series becomes a continuous Fourier transform, and how the frequency-domain idea helps with image filtering and noise reduction.

From Fourier Series to Fourier Transform | JoVE K12

JoVE Core Electrical Engineering offers easy-to-understand animated videos that cover fundamental concepts of electric circuits along with circuit analysis techniques. It incorporates first-order and second-order circuits using operational amplifiers and circuit transformation concepts. The inclusion of practical design examples and real-world applications through scientist-in-action videos adds significant educational value, making complex concepts accessible.

electrical engineering

From Fourier Series to Fourier Transform

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