# How is Fourier transform related to Fourier series?

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Fourier series is an expansion of periodic signal as a linear combination of sines and cosines while Fourier transform is the process or function used to convert signals from time domain in to frequency domain.

### Is Fourier series and Fourier transform same?

5 Answers. The Fourier series is used to represent a periodic function by a discrete sum of complex exponentials, while the Fourier transform is then used to represent a general, nonperiodic function by a continuous superposition or integral of complex exponentials.

**Why do we need Fourier transform even we have Fourier series?**

The Fourier Transform is an important image processing tool which is used to decompose an image into its sine and cosine components. The Fourier Transform is used in a wide range of applications, such as image analysis, image filtering, image reconstruction and image compression.

**Can Fourier Transform be used for periodic signals?**

The Fourier series and the Fourier transform can both be used for periodic and aperiodic signals. A periodic signal can be expressed in the time domain as a Fourier series, which is nothing but a series of exponentials.

## How do you find the Fourier Transform of a periodic signal?

So, X(ω)=δ(ω-ω0) and x(t)=ejω0t/2π x ( t ) = e j ω 0 t / 2 π form a Fourier Transform pair. This result will be used below to find the Fourier Transform of Sines, Cosines, and any periodic function that can be represented by a Fourier Series.

### What is Fourier transform of signals?

The Fourier transform is a mathematical formula that relates a signal sampled in time or space to the same signal sampled in frequency. In signal processing, the Fourier transform can reveal important characteristics of a signal, namely, its frequency components. y k + 1 = ∑ j = 0 n – 1 ω j k x j + 1 .

**How do Fourier series represent periodic signals?**

Fourier Series Representation of Continuous Time Periodic Signals

- x(t)=cosω0t (sinusoidal) &
- x(t)=ejω0t (complex exponential)
- These two signals are periodic with period T=2π/ω0.
- A set of harmonically related complex exponentials can be represented as {ϕk(t)}
- Where ak= Fourier coefficient = coefficient of approximation.

**What does Fourier transform do to a signal?**

At a high level the Fourier transform is a mathematical function which transforms a signal from the time domain to the frequency domain. This is a very powerful transformation which gives us the ability to understand the frequencies inside a signal.

## What is a Fourier transform and how is it used?

The Fourier transform is a mathematical function that can be used to show the different parts of a continuous signal. It is most used to convert from time domain to frequency domain. Fourier transforms are often used to calculate the frequency spectrum of a signal that changes over time.

### Why is the Fourier transform so important?

Fourier transforms is an extremely powerful mathematical tool that allows you to view your signals in a different domain, inside which several difficult problems become very simple to analyze.

**What are the different types of the Fourier transform?**

Types of Fourier Transforms Fourier Series. – If the function f ( x) is periodic, then the expression of f ( x) as a series of frequency terms with varying terms can be performed Fourier Integral Discrete Fourier Transform. Note that k is simply an integer counter, k = 0, 1, 2. Fast Fourier Transform. Send Mail:

**What are the properties of Fourier transform?**

The Fourier transform is a major cornerstone in the analysis and representa- tion of signals and linear, time-invariant systems, and its elegance and impor- tance cannot be overemphasized. Much of its usefulness stems directly from the properties of the Fourier transform, which we discuss for the continuous- time case in this lecture.