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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 BilgekulEeng360 Communication Systems I

Department of Electrical and Electronic Engineering Eastern Mediterranean University

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Discrete Fourier Transform Discrete Fourier Transform (DFT)(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.

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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

ww(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 TransformUsing the DFT to Compute the Continuous Fourier Transform

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Computing CFT Using DFTComputing 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

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Relationship between the DFT and the CFT involves three concepts:• Windowing, • Sampling, • Periodic sample generation

ComparisonComparison of of DFTDFT and and the Continuous the Continuous Fourier TransformFourier Transform

(CFT)(CFT)

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Fast Fourier TransformFast 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.

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Using the DFT to Compute the Fourier Using the DFT to Compute the Fourier SeriesSeries

The Discrete Fourier Transform (DFT) may also be used to compute the complex Fourier series. Fourier series coefficients are related to DFT by,

1( )

1( ), 0

21

( ), - 02

n

n

n

c X nN

Nc X n n

NN

c X N n nN

Block diagram depicting the sequence of operations involved in approximating the FT

with the DTFS.

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Ex. 2.17 Use DFT to compute the spectrum of a Ex. 2.17 Use DFT to compute the spectrum of a SinusoidSinusoid

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Ex. 2.17 Use DFT to compute the spectrum of a Ex. 2.17 Use DFT to compute the spectrum of a SinusoidSinusoid

Spectrum of a sinusoid obtained by using the Spectrum of a sinusoid obtained by using the MATLAB DFT.MATLAB DFT.

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Using the DFT to Using the DFT to Compute the Compute the Fourier SeriesFourier 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.

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Using the DFT Using the DFT to Compute to Compute the Fourier the Fourier

SeriesSeries

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