Eeng 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

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

Eeng  Suppose the CFT of a waveform w(t) is to be evaluated using DFT. 1.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 w w (t) is 2.The Fourier transform of the windowed waveform is 3.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 Using the DFT to Compute the Continuous Fourier Transform

Eeng 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 f = n/T and ∆t = T/N

Eeng Relationship between the DFT and the CFT involves three concepts: Windowing, Sampling, Periodic sample generation Comparison of DFT and the Continuous Fourier Transform (CFT)

Eeng Relationship between the DFT and the CFT involves three concepts: Windowing, Sampling, Periodic sample generation Comparison of DFT and the Continuous Fourier Transform (CFT)

Eeng Fast Fourier Transform  The Fast Fourier Transform (FFT) is a fast algorithm for evaluating DFT. 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.

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

Eeng Ex Use DFT to compute the spectrum of a Sinusoid

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

Eeng 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.

Eeng 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 = (c) Behavior in the vicinity of the sinusoidal frequencies for M = (d) Behavior in the vicinity of the sinusoidal frequencies for M = 2010