Discrete Fourier Series | Discrete Fourier Transform | Discrete Time Fourier Transform
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The document discusses discrete Fourier series, discrete Fourier transform, and discrete time Fourier transform. It provides definitions and explanations of each topic. Discrete Fourier series represents periodic discrete-time signals using a summation of sines and cosines. The discrete Fourier transform analyzes a finite-duration discrete signal by treating it as an excerpt from an infinite periodic signal. The discrete time Fourier transform provides a frequency-domain representation of discrete-time signals and is useful for analyzing samples of continuous functions. Examples of applications are also given such as signal processing, image analysis, and wireless communications.
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• The DTFT is a frequency-domain representation for a wide range of both finite
and infinite-length discrete-time signals x[n].
• The DTFT is denoted as X(ejωˆ), which shows that the frequency dependence
always includes the complex exponential function ejωˆ .
• The operation of taking the Fourier transform of a signal will become a
common tool for analyzing signals and systems in the frequency domain.
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Discrete Fourier Series | Discrete Fourier Transform | Discrete Time Fourier Transform
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- 2. DISCRETE FOURIER SERIES DISCRETE FOURIER TRANSFORM DISCRETE TIME FOURIER TRANSFORM Meer Zafarullah Noohani K-F16ES-33 Kaleem Ullah Magsi K-F16ES-57 Ms. Sumera Hashim K-F16ES-53
- 3. Fair Use Notice The material used in this presentation i.e., pictures/graphs/text, etc. is solely intended for educational/teaching purpose, offered free of cost to the reader for and may include copyrighted material - the use of which may not have been specifically authorized by Copyright Owners. It’s application constitutes Fair Use of any such copyrighted material as provided in globally accepted law of many countries. The contents of presentations are intended only for the learning purpose. 8/2/2020 2
- 5. DISCRETE FOURIER SERIES • The discrete Fourier series is a periodic function • Fourier series represent either: (a) A periodic function, or (b) A function that is defined only over a finite-length interval; the values produced by the Fourier series outside the finite interval are irrelevant. 4
- 6. • When the function being represented is discrete, that set is called a discrete Fourier transform (DFT). • The term discrete Fourier series (DFS) is intended for use instead of DFT when the original function is periodic, defined over an infinite interval. • DFS is a frequency analysis tool for periodic infinite-duration discrete-time signals. 5
- 7. 6 The discrete Fourier series (DFS) is called “the Fourier transform for periodic sequences,” in that it plays the same role for them that the Fourier transform plays for nonperiodic (ordinary) sequences. These are discrete signals that repeat themselves in a periodic fashion from negative to positive infinity. This class of Fourier transform is sometimes called the discrete Fourier series, but is most often called the discrete Fourier transform.
- 8. Periodicity in Discrete Fourier Series 7
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- 13. Find Discrete Fourier Series
- 14. APPLICATIONS OF DFS Signal Processing. The best application of Fourier analysis. Approximation Theory. Used to write a function as a trigonometric polynomial.
- 15. Control Theory. Useful to find out the dynamics of the solution. Partial Differential equation. We use it to solve higher order partial differential equations by the method of separation of variables. The Fourier series of functions in the differential equation often gives some prediction about the behavior of the solution of differential equation. 14
- 16. Computation of power spectra. Cross spectra. Coherence.
- 17. One dimensional heat equation and wave equation Advanced noise cancellation Cell phone network technology
- 18. • Original time function can be uniquely recovered from it. • Fourier methods give us a set of powerful tools for representing any periodic function as a sum of sines and cosines. • The main advantage of Fourier analysis is that very little information is lost from the signal during the transformation. 17 ADVANTAGES
- 20. • All periodic functions can't be expressed in terms of Fourier series, but only those which follow Dirichlet conditions. • It can be used only for periodic inputs and thus not applicable for aperiodic one. (Fourier Transform) • It cannot be used for unstable or even marginally stable systems. (Laplace transform) 19 DISADVANTAGES
- 22. INTRODUCTION • The DFT is one of the most powerful tools in digital signal processing which enables us to find the spectrum of a finite-duration signal. • There are many circumstances in which we need to determine the frequency content of a time-domain signal. • For example, we may have to analyze the spectrum of the output of an LC oscillator to see how much noise is present in the produced sine wave. This can be achieved by the discrete Fourier transform (DFT).
- 23. • Basically, computing the DFT is equivalent to solving a set of linear equations. • The DFT provides a representation of the finite-duration sequence using a periodic sequence, where one period of this periodic sequence is the same as the finite-duration sequence. • As a result, we can use the discrete-time Fourier series to derive the DFT equations. • The algorithms for the efficient computation of the DFT are collectively called fast Fourier transforms (FFTs). 22
- 25. WHAT IS DFT? ▪ Assume that x(t)x(t), shown in Figure 1, is the continuous-time signal that we need to analyze. 24 Figure 1
- 26. ▪ Assume that the finite-duration sequence that we need to analyze is as shown in Figure 5 (a). To calculate the N-point DFT, we need to make a periodic signal, p(n)p(n), from x(n)x(n) with period NN, as shown in Figure 5(b). ▪ Considering the fact that p(n)=x(n)p(n)=x(n) for n=0,1,…,N−1n=0,1,…,N−1, we obtain the discrete-time Fourier series of this periodic signal ▪ Equation 1 25
- 27. • Where N denotes the period of the signal. The time-domain signal can be obtained as follows: • Equation 2 26
- 28. ▪ Multiplying the coefficients given by Equation 6 by NN, we obtain the DFT coefficients, X(k)X(k), as follows: ▪ Equation 3 → ▪ The inverse DFT will be ▪ Equation 4 → 27
- 29. EXAMPLE OF DFT
- 33. • To what degree does the DFT approximate the Fourier transform of the function underlying the data? Clearly the DFT is only an approximation since it provides only for a finite set of frequencies. But how correct are these discrete values themselves? There are two main types of DFT errors: • 1. Aliasing • 2. Leakage 32 ERRORS IN DFT
- 34. ALIASING • If the initial samples are not sufficiently closely spaced to represent high-frequency components present in the underlying function, then the DFT values will be corrupted by aliasing 33
- 35. LEAKAGE • The continuous Fourier Transform requires the integration to be performed over the interval – P to FP or over an integer number of cycles of the waveform. If we attempt to complete the DFT over a non-integer number of cycles of the input signal, then we might expect the transform to be corrupted in some way 34
- 36. Advantages The main advantage of discrete Fourier Transform analysis is that very little information is lost from the signal during the transformation. The DFT can find a system's frequency response from the system's impulse response, and vice versa. The DFT is especially useful for efficiently representing signals that are comprised of a few frequency components. Like ECG recorded signal etc. 35 ADVATAGES
- 37. Applications Image filtering Seismograms- Frequency content Eigenmodes of the Earth Seismo-weather with FFTS
- 38. IMAGE APPLICATIONS, WHICH INCLUDES IMAGE ANALYSIS, IMAGE FILTERING, IMAGE RECONSTRUCTION AND IMAGE COMPRESSION. INFRA-RED SPECTROSCOPY FT-RAMAN SPECTROSCOPY MEDICAL MRI IMAGING
- 39. DISCRETE TIME FOURIER TRANSFORM 38
- 40. DEFINITION OF DTFT • In mathematics, the discrete-time Fourier transform is a form of Fourier analysis that is applicable to a sequence of values. The DTFT is often used to analyze samples of a continuous function. 39
- 41. OR • The DTFT is a frequency-domain representation for a wide range of both finite and infinite-length discrete-time signals x[n]. • The DTFT is denoted as X(ejωˆ), which shows that the frequency dependence always includes the complex exponential function ejωˆ . • The operation of taking the Fourier transform of a signal will become a common tool for analyzing signals and systems in the frequency domain. 40
- 42. • The application of the DTFT is usually called Fourier analysis, or spectrum analysis or “going into the Fourier domain or frequency domain.” • Thus, the words spectrum, Fourier, and frequency-domain representation become equivalent, even though each one retains its own distinct character. 41
- 43. DISCRETE-TIME SIGNALS IN FREQUENCY DOMAIN • For continuous-time signals, we can use Fourier series and Fourier transform to study them in frequency domain. • With the use of sampled version of a continuous-time signal we can obtain the discrete-time Fourier transform (DTFT) or Fourier transform of discrete-time signals as follows. 42
- 44. We start with studying the sampled signal produced by multiplying by the impulse train 43
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- 50. APPLICATIONS Orthogonal Frequency Division Multiplex (OFDM) Image compression to MPEG format Digital filters
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