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Chapter 2 D iscrete Fourier Transform (DFT)

Chapter 2 D iscrete Fourier Transform (DFT). Topics: Discrete Fourier Transform. Using the DFT to Compute the Continuous Fourier Transform. Comparing DFT and CFT Using the DFT to Compute the Fourier Series. Huseyin Bilgekul Eeng360 Communication Systems I

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Chapter 2 D iscrete Fourier Transform (DFT)

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  1. Chapter 2 Discrete Fourier Transform (DFT) Topics: • Discrete Fourier Transform. • Using the DFT to Compute the Continuous Fourier Transform. • Comparing DFT and CFT • Using the DFT to Compute the Fourier Series Huseyin Bilgekul Eeng360 Communication Systems I Department of Electrical and Electronic Engineering Eastern Mediterranean University

  2. Where n = 0, 1, 2, …., N-1 where k = 0, 1, 2, …., N-1. Discrete Fourier Transform (DFT) • Definition: The Discrete Fourier Transform (DFT)is defined by: The Inverse Discrete Fourier Transform (IDFT) is defined by: The Fast Fourier Transform (FFT) is a fast algorithm for evaluating the DFT.

  3. Using the DFT to Compute the Continuous Fourier Transform • Suppose the CFT of a waveform w(t) is to be evaluated using DFT. • The time waveform is first windowed (truncated) over the interval (0, T) so that only a finite number of samples, N, are needed. The windowed waveform ww(t) is • The Fourier transform of the windowed waveform is • Now we approximate the CFT by using a finite series to represent the integral, • t = k∆t, f = n/T, dt = ∆t, and ∆t = T/N

  4. f = n/T and ∆t = T/N Computing CFT Using DFT • We obtain the relation between the CFT and DFT; that is, • The sample values used in the DFT computation are x(k) = w(k∆t), • If the spectrum is desired for negative frequencies • – the computer returns X(n) for the positive n values of 0,1, …, N-1 • – It must be modified to give spectral values over the entire • fundamental range of -fs/2 < f <fs/2. • For positive frequencies we use For Negative Frequencies

  5. Comparisonof DFT and the Continuous Fourier Transform (CFT) • Relationship between the DFT and the CFT involves three concepts: • Windowing, • Sampling, • Periodic sample generation

  6. Comparisonof DFT and the Continuous Fourier Transform (CFT) • Relationship between the DFT and the CFT involves three concepts: • Windowing, • Sampling, • Periodic sample generation

  7. The Fast Fourier Transform (FFT) is a fast algorithm for evaluating DFT. Fast Fourier Transform Block diagrams depicting the decomposition of an inverse DTFS as a combination of lower order inverse DTFS’s. (a) Eight-point inverse DTFS represented in terms of two four-point inverse DTFS’s. (b) four-point inverse DTFS represented in terms of two-point inverse DTFS’s. (c) Two-point inverse DTFS.

  8. The Discrete Fourier Transform (DFT) may also be used to compute the complex Fourier series. • Fourier series coefficients are related to DFT by, Using the DFT to Compute the Fourier Series • Block diagram depicting the sequence of operations involved in approximating the FT with the DTFS.

  9. Ex. 2.17 Use DFT to compute the spectrum of a Sinusoid

  10. Ex. 2.17 Use DFT to compute the spectrum of a Sinusoid Spectrum of a sinusoid obtained by using the MATLAB DFT.

  11. Using the DFT to Compute the Fourier Series The DTFT and length-N DTFS of a 32-point cosine. The dashed line denotes the CFT. While the stems represent N|X[k]|. (a) N = 32 (b) N = 60 (c) N = 120.

  12. Using the DFT to Compute the Fourier Series The DTFS approximation to the FT of x(t) = cos(2(0.4)t) + cos(2(0.45)t). The stems denote |Y[k]|, while the solid lines denote CFT. (a) M = 40. (b) M = 2000. (c) Behavior in the vicinity of the sinusoidal frequencies for M = 2000. (d) Behavior in the vicinity of the sinusoidal frequencies for M = 2010

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