## What is discrete Fourier transform (DFT)?

Discrete Fourier Transform (DFT) Recall the DTFT: X(ω) = X∞ n=−∞ x(n)e−jωn. DTFT is not suitable for DSP applications because •In DSP, we are able to compute the spectrum only at speciﬁc discrete values of ω, •Any signal in any DSP application can be measured only in a ﬁnite number of points. A ﬁnite signal measured at N

## How do you write the Fourier transform of a function?

Fourier Transform Properties Rather than write “the Fourier transform of an X function is a Y function”, we write the shorthand: X $Y.Ifz is a complex number and z Dx Ciy where x and y are its real and imaginary parts, then the complex conjugate of z is z Dx −iy.

**How do you calculate the periodicity of a DFT spectrum?**

= X(k)e−j2πn= X(k) =⇒ the DFT spectrum is periodic with period N (which is expected, since the DTFT spectrum is periodic as well, but with period 2π). Example: DFT of a rectangular pulse: x(n) = ˆ 1, 0 ≤n ≤(N −1), 0, otherwise. X(k) = NX−1 n=0

Discrete Fourier transforms are often used to solve partial differential equations, where again the DFT is used as an approximation for the Fourier series (which is recovered in the limit of infinite N ). The advantage of this approach is that it expands the signal in complex exponentials

### What is convolution in discrete time Fourier transform?

by a linear phase. Mathematically, if The convolution theorem for the discrete-time Fourier transform (DTFT) indicates that a convolution of two sequences can be obtained as the inverse transform of the product of the individual transforms. An important simplification occurs when one of sequences is N-periodic, denoted here by

### Are there Hermite-Gaussian eigenvectors of the discrete Fourier transform?

Magdy Tawfik Hanna, Nabila Philip Attalla Seif, and Waleed Abd El Maguid Ahmed (2004). “Hermite-Gaussian-like eigenvectors of the discrete Fourier transform matrix based on the singular-value decomposition of its orthogonal projection matrices”.

**What is the Fourier transform of sinusoids?**

Fourier transform (bottom) is zero except at discrete points. The inverse transform is a sum of sinusoids called Fourier series. Center-right column: Original function is discretized (multiplied by a Dirac comb) (top).