Frequencies of complex exponentials in Discrete Fourier Transform

1. What is the Fourier transform of complex exponential function?
2. What are the frequencies in DFT?
3. What is complex exponential signal?
4. How do you find the complex exponential Fourier series?

What is the Fourier transform of complex exponential function?

Hence, the Fourier transform of the complex exponential function is given by, [ejω0t]=2πδ(ω−ω0) Or, it can also be represented as, ejω0tFT↔2πδ(ω−ω0)

What are the frequencies in DFT?

The set of frequency samples which define the spectrum X(k), are given on a frequency axis whose discrete frequency locations are given by Equation 2.63 where k = 0, 1,…., N−1. The frequency resolution of the DFT is equal to the frequency increment F/N and is referred to as the bin spacing of the DFT outputs.

What is complex exponential signal?

A complex exponential is a signal of the form. (1.15) where A = ∣A∣e^{j θ} and a = r + j Ω 0 are complex numbers. Using Euler's identity, and the definitions of A and a, we have that x(t) = A e^at equals. We will see later that complex exponentials are fundamental in the Fourier representation of signals.

How do you find the complex exponential Fourier series?

In order to derive the exponential Fourier series, we replace the trigonometric functions with exponential functions and collect like exponential terms. This gives f(x)∼a02+∞∑n=1[an(einx+e−inx2)+bn(einx−e−inx2i)]=a02+∞∑n=1(an−ibn2)einx+∞∑n=1(an+ibn2)e−inx.