Discrete-time Fourier transform

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

We can get rid of the Ts by making use of normalised frequency, 6) = ojfTs. The result is the discrete-time Fourier transform or DTFT ... 

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

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. 

TABLE 20.4 Example Discrete-Time Fourier Transform Pairs... 

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 ... 

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