The discrete-time Fourier transform

For a continuous signal, we can move from the time domain to the frequency domain via the Fourier transform, but as we are now dealing with digital signals, this formula cannot be used directly. We will now derive an equivalent transform for digital signals. Starting with the previously defined Fourier transform (Equation 10.21) 

Ts is independent of the integral, we move it outside, and use x[n in place oix nTs) giving  

We can get rid of the Ts by making use of normalised frequency, d) = (o/r. The result is the discrete-time Fourier transform or DTFT  

It should be clear that the DTFT is a transform in which time is discrete but frequency is continuous. As we shall see in later sections, the DTFT is a very important transform in its own right but for computational purposes, it has two drawbacks. Firstly, the infinite sum is impractical, and secondly we wish to obtain a representation for frequency that is useful computationally, and such a representation must be discrete. 

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The discrete-time Fourier transform converts a digital waveform into a continuous spectrum ... 

The discrete-time Fourier transform (DTFT) for aperiodic, finite energy discrete-time waveforms x n) is given by... 

Suppression rules. Let X(p,Qk) denote the short-time Fourier transform of x[ri, where p is the time index, and Qk the normalized frequency index (0t lies between 0 and 1 and takes N discrete values for k = 1,N, Wbeing the number of sub-bands). Note that the time index p usually refers to a sampling rate lower than the initial signal sampling rate (for the STFT, the down-sampling factor is equal to hop-size between to consecutive short-time frames) [Crochiere and Rabiner, 1983]. 

Notice, that the power series expansion of (f) does not involve the usual time-Fourier transform, but it does correspond to the discrete half-Fourier transform of the... 

Although the discrete cosine Fourier transform yields N calculated frequency-domain values for N measured time-domain data points, the second half of the cosine F.T. data is a mirror image of the first half, and thus gives no new information. The other half of the time-domain information is contained in the first N/2 frequency-domain values of the sine Fourier transform. [It is possible to put all the information into the cosine transform by first adding N... 

The transformation given by Eq. (20.87) can also be applied to some periodic functions as well as functions that are not absolutely summable. Examples of discrete-time Fourier transform pairs are shown in Table 20.4. 

Discrete-time waveform A waveform, herein represented by x( ), that takes on values at a countable, discrete set of sample times or sample numbers , the assumed independent variable. The discrete Fourier transform, the discrete-time Fourier series, and discrete-time Fourier transform apply to discrete-time waveforms. Compare with continuous-time waveform. 

Many DSP concepts can be demonstrated by examples which involve a great deal of computation. A list of some of the concepts is as follows convolution, filtering, quantization effects, etc. The curriculum begins with discrete Fourier transform (DFT). DFT is derived from discrete-time Fourier transform expression. The continuous and discrete Fourier transform are covered in Signals and Systans. The flow of the topics is as follows DFT, properties of DFT, Fast Fourier Transform, Infinite Impulse Response filter and Finite Impulse Response fillers and filter structures. If the topics are linked to a project with each block of the project demonstrating the various topics of the curriculum, it is easier for the student to comprehend what is being taught. 

It is important to bear in mind that there are at least four other terms that are closely related to STFT, which are prone to cause confusion Fourier transform (FT), discrete-time Fourier transform (DTFT), discrete Fourier transform (DFT) and fast Fourier transform (FFT). An in-depth discussion of the differences between these terms is beyond the scope of this book. On the whole, they differ in the way that they consider time and frequency, as follows ... 

In the simulation, the value of the F(t) function is calculated at discrete points in a finite time interval then discrete complex Fourier transform is performed on this array to obtain the simulated spectrum.101... 

The continuous, infinite Fourier transform defined in Equation 10.9, unfortunately, is not convenient for signal detection and estimation. Most physically significant data are recorded only at a fixed set of evenly spaced intervals in time and not between these times. In such situations, the continuous sequence h(t) is approximated by the discrete sequence hn... 

The calculation of the Fourier space contribution is the most time consuming part of the Ewald sum. The essential idea of P M is to replace the simple continuous Fourier transformations in (3) by discrete Fast Fourier Transformations, that are numerical faster to calculate. The charges are interpolated onto a regular mesh. Since this introduces additional errors, the simple Coulomb Green function as used in the second term in (3), is cleverly adjusted in... 

Figure 17. Demonstrate of foldover aliasing, (a) Hypothetical spectrum, with peaks located at their true frequencies, (b) Discrete cosine Fourier transform of the time-domain signal corresponding to (a), with sampling and Nyquist frequencies as shown. The peaks in (b) have correct relative intensities, but are folded-back to lower apparent displayed frequencies.

The Fourier spectrum is obtained from the final time spectrum by application of a standard discrete fast Fourier transform algorithm ... 

Using FFT, find the Fourier transform X(,a>) of the discrete time signal A-point signal x(n). 

The discrete-time waveform counterparts of the Fourier series and Fourier transform provide viable alternatives for estimating the frequency content of signals if discrete measurements of the desired waveforms are available. This avenue of analysis, which is popular in benchtop instrumentation, is considered in further detail in Sec. 20.6.7. 

The reciprocal of the time interval A is called the sampling rate. If A is measured in seconds, for example, then the sampling rate is the number of samples recorded per second [73]. A sequence of sampled values in a relevant time domain, starting from a given time origin, are considered, and the one-sided Fourier Transform is taken into account. When a discrete set of experimental points is available, the usual procedure is to approximate the integral, appearing in the analytical expression, by a discrete sum ... 

In practical applications, x(t) is not a continuous function, and the data to be transformed are usually discrete values obtained by sampling at intervals. Under such circumstances, I hi discrete Fourier transform (DFT) is used to obtain the frequency function. Let us. suppose that the time-dependent data values are obtained by sampling at regular intervals separated by [Pg.43]

From the time function F t) and the calculation of [IT], the values of G may be found. One way to calculate the G matrix is by a fast Fourier technique called the Cooley-Tukey method. It is based on an expression of the matrix as a product of q square matrices, where q is again related to N by = 2 . For large N, the number of matrix operations is greatly reduced by this procedure. In recent years, more advanced high-speed processors have been developed to carry out the fast Fourier transform. The calculation method is basically the same for both the discrete Fourier transform and the fast Fourier transform. The difference in the two methods lies in the use of certain relationships to minimize calculation time prior to performing a discrete Fourier transform. 

Frequency-domain data are obtained by converting time-domain data using a mathematical technique referred to as Fast Fourier Transform (FFT). FFT allows each vibration component of a complex machine-train spectrum to be shown as a discrete frequency peak. The frequency-domain amplitude can be the displacement per unit time related to a particular frequency, which is plotted as the Y-axis against frequency as the X-axis. This is opposed to time-domain spectrums that sum the velocities of all frequencies and plot the sum as the Y-axis against time... 

The frequency-domain format eliminates the manual effort required to isolate the components that make up a time trace. Frequency-domain techniques convert time-domain data into discrete frequency components using a mathematical process called Fast Fourier Transform (FFT). Simply stated, FFT mathematically converts a time-based trace into a series of discrete frequency components (see Figure 43.19). In a frequency-domain plot, the X-axis is frequency and the Y-axis is the amplitude of displacement, velocity, or acceleration. 

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