What are the conditions for existence of Fourier transform?

The conditions are: f must be absolutely integrable over a period. f must be of bounded variation in any given bounded interval. f must have a finite number of discontinuities in any given bounded interval, and the discontinuities cannot be infinite.

Why do we compute short time Fourier transform?

7.2 Short-Time Fourier Transform (STFT) STFT provides the time-localized frequency information for situations in which frequency components of a signal vary over time, whereas the standard Fourier transform provides the frequency information averaged over the entire signal time interval.

How do you find the continuous time of a Fourier transform?

Continuous time Fourier transform of x(t) is defined as X ( ω ) = ∫ − ∞ + ∞ x ( t ) e − j ω t d t and discrete time Fourier transform of x(n) is defined as X(ω)=Σ∀nx(n)e−ωn.

What is the sufficient condition for the existence of discrete time Fourier transform?

Sufficient Condition for Existence of the DTFT A sequence x[n] satisfying (7.7) is said to be absolutely summable, and when (7.7) holds, the infinite sum defining the DTFT X(ej ˆω) in (7.2) is said to converge to a finite result for all ˆω.

Why DFT is preferred over Fourier Transform?

The Fast Fourier Transform (FFT) is an implementation of the DFT which produces almost the same results as the DFT, but it is incredibly more efficient and much faster which often reduces the computation time significantly. It is just a computational algorithm used for fast and efficient computation of the DFT.

How does short time Fourier transform work?

In practice, the procedure for computing STFTs is to divide a longer time signal into shorter segments of equal length and then compute the Fourier transform separately on each shorter segment. This reveals the Fourier spectrum on each shorter segment.

What does the Fourier transform compute?

The Fourier Transform is a mathematical technique that transforms a function of time, x(t), to a function of frequency, X(ω). As before, we write ω=nω0 and X(ω)=Tcn. A little work (and replacing the sum by an integral) yields the synthesis equation of the Fourier Transform.