What are the differences between Fourier series and Fourier transform?

What are the differences between Fourier series and Fourier transform?

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.

Is the Fourier series unique?

for any n . So the Fourier series of f is unique.

How does Fourier series make it easier to represent periodic?

Explanation: Fourier series makes it easier to represent periodic signals as it is a mathematical tool that allows the representation of any periodic signals as the sum of harmonically related sinusoids.

Can two functions have the same Fourier series?

In particular, taking differences, if two functions have the same Fourier coefficients, then they are the same (except on a set of measure zero).

What does Fourier series represent?

A Fourier series is a way of representing a periodic function as a (possibly infinite) sum of sine and cosine functions. It is analogous to a Taylor series, which represents functions as possibly infinite sums of monomial terms. A sawtooth wave represented by a successively larger sum of trigonometric terms.

How to describe the Fourier transform of a periodic function?

Fourier Transform of a Periodic Signal Described by a Fourier Series Given a periodic function xT(t) and its Fourier Series representation (period= T, ω0=2π/T): xT (t) = +∞ ∑ n=−∞cnejnω0t x T (t) = ∑ n = − ∞ + ∞ c n e j n ω 0 t we can use the fact that we know the Fourier Transform of the complex exponential

Is the Fourier series a function of ω?

Recall that the Fourier series is defined by discrete coefficients with index n (and amplitude cn ), not a function of ω. The relationshop between Fourier Series and Transform is discussed in more detail later. In the next section several properties of the Fourier Transform are derived.

How to find Fourier series of periodic pulse?

Find the Fourier Series representation of the periodic pulse train xT(t)=ΠT (t/Tp). Since xT(t) is the periodic extension of x (t)=Π (t/Tp), and we know from a Fourier Transform table (or from previous work) Find the Fourier Series representation of the periodic triangular pulse xT(t)=ΛT(t/Tp).

Why do we use the discrete time Fourier transform?

For periodic signals this representation be- comes the discrete-time Fourier series, and for aperiodic signals it becomes the discrete-time Fourier transform. The motivation for representing discrete-time signals as a linear combi- nation of complex exponentials is identical in both continuous time and dis- crete time.