Discrete-time Fourier transform
The term discrete-time refers to the fact that the transform operates on discrete data, often samples whose interval has units of time.
In simpler terms, when you take the DTFT of regularly-spaced samples of a continuous signal, you get repeating (and possibly overlapping) copies of the signal's frequency spectrum, spaced at intervals corresponding to the sampling frequency.
The DTFT itself is a continuous function of frequency, but discrete samples of it can be readily calculated via the discrete Fourier transform (DFT) (see § Sampling the DTFT), which is by far the most common method of modern Fourier analysis.
The inverse DTFT reconstructs the original sampled data sequence, while the inverse DFT produces a periodic summation of the original sequence.
represents time in seconds: We can reduce the integral into a summation by sampling
seconds (see Fourier transform § Numerical integration of a series of ordered pairs).
Thus, we obtain one formulation for the discrete-time Fourier transform (DTFT): This Fourier series (in frequency) is a continuous periodic function, whose periodicity is the sampling frequency
The utility of the DTFT is rooted in the Poisson summation formula, which tells us that the periodic function represented by the Fourier series is a periodic summation of the continuous Fourier transform:[b]The components of the periodic summation are centered at integer values (denoted by
Therefore, an alternative definition of DTFT is:[A] The modulated Dirac comb function is a mathematical abstraction sometimes referred to as impulse sampling.
For instance, the inverse continuous Fourier transform of both sides of Eq.3 produces the sequence in the form of a modulated Dirac comb function: However, noting that
The standard formulas for the Fourier coefficients are also the inverse transforms: When the input data sequence
can be expressed in terms of the inverse transform, which is sometimes referred to as a Discrete Fourier series (DFS):[1]: p 542 With these definitions, we can demonstrate the relationship between the DTFT and the DFT: Due to the
this can be simplified to: which satisfies the inverse transform requirement: When the DTFT is continuous, a common practice is to compute an arbitrary number of samples
For instance, a long sequence might be truncated by a window function of length
The DFT then goes by various names, such as: Recall that decimation of sampled data in one domain (time or frequency) produces overlap (sometimes known as aliasing) in the other, and vice versa.
summation/overlap causes decimation in frequency,[1]: p.558 leaving only DTFT samples least affected by spectral leakage.
So multi-block windows are created using FIR filter design tools.
[14][15] Their frequency profile is flat at the highest point and falls off quickly at the midpoint between the remaining DTFT samples.
That is a common practice, but the truncation affects the DTFT (spectral leakage) by a small amount.
-length DFT of the truncated window produces frequency samples at intervals of
The samples are real-valued,[16]: p.52 but their values do not exactly match the DTFT of the symmetric window.
In this case, the DFT simplifies to a more familiar form: In order to take advantage of a fast Fourier transform algorithm for computing the DFT, the summation is usually performed over all
decreases, is detrimental to certain important performance metrics, such as resolution of multiple frequency components and the amount of noise measured by each DTFT sample.
sequence is a noiseless sinusoid (or a constant), shaped by a window function.
Then it is a common practice to use zero-padding to graphically display and compare the detailed leakage patterns of window functions.
To illustrate that for a rectangular window, consider the sequence: Figures 2 and 3 are plots of the magnitude of two different sized DFTs, as indicated in their labels.
Contributing factors to the illusion are the use of a rectangular window, and the choice of a frequency (1/8 = 8/64) with exactly 8 (an integer) cycles per 64 samples.
is also discrete, which results in considerable simplification of the inverse transform: For s and y sequences whose non-zero duration is less than or equal to N, a final simplification is: The significance of this result is explained at Circular convolution and Fast convolution algorithms.
is a Fourier series that can also be expressed in terms of the bilateral Z-transform.
The terms of S1/T(f) remain a constant width and their separation 1/T scales up or down.
