Bài giảng xử lý tín hiệu số fourier transform ngô quốc cường

Fourier series for periodic signals • Example 1: Determine the spectra of the signal... Fourier series for periodic signals • Solution of example 1: 6... Fourier transform of aperiodic s

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Xử lý tín hiệu số Fourier Transform

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• Frequency analysis of discrete time signal

• Properties of Fourier transform

• Frequency domain characteristics of LTI systems

• Discrete Fourier Transform

• Fast Fourier Transform

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1.Frequency analysis of discrete time signal

1.1 Fourier series for periodic signals

– Given a periodic signal x(n) with period N

(DTFS)

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• The spectrum of a signal x(n) which is periodic with period N,

is a periodic sequence with period N

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• Example 1: Determine the spectra of the signal

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• Solution of example 1:

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1.2 Fourier transform of aperiodic signals

• The Fourier transform of a finite energy signal x(n) is defined

as

• X(w) is periodic with period 2𝜋:

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• In summary, the Fourier transform pair of a discrete time is as follows

• Uniform convergence is guaranteed if x(n) is absolutely

summable

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• The spectrum X(w) is, in general, a complex valued function

of frequency

• The energy density spectrum of x(n) is

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• Example 2:

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• Solution of example 2:

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• Solution of example 2 (cont’d):

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2 Properties of Fourier transform

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• Example 3:

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• Solution of Example 3:

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• Solution of Example 3 (cont’d):

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a=0.8

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• Example 4:

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3 Frequency domain characteristics of LTI

systems

• The response of any relaxed-system to arbitrary input signal is:

• Excite the system with the complex exponential

• Obtain the response

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• The Fourier transform of the unit sample response h(k) of the system

• The function H(𝜔) exists if the system is BIBO stable

• The response of the system to the complex exponential is

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systems

• Example

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systems

• In general, H(𝜔) is a complex function of 𝜔, hence

• Expressed in terms of real and image components

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• Example

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systems

• The impulse response

• It follows that

• Hence

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systems

• Example

• Determine the response of the system (with given impulse

response h(n)) to the input signal x(n)

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• Solution

• The frequency response

• With each term in the input signal

• The response to the input signal

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4 Discrete Fourier Transform

4.1 Frequency domain sampling

• Aperiodic finite- energy signals have continuous spectra

• Sample X(𝜔) periodically at a spacing of 𝛿𝜔 radians, take N equidistant samples in the interval 0 ≤ 𝜔 ≤ 2𝜋 with spacing

𝜔 = 2𝜋/𝑁

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4.1 Frequency domain sampling

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4.2 Discrete Fourier Transform

• A finite-duration sequence x(n) of length L has the DFT

where N ≥ L

• Recover x(n) from its DFT (inverse DFT)

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• 4.2 Discrete Fourier Transform

• Exercise

– Compute the 8-point DFT of

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• 4.2 Discrete Fourier Transform

• Exercise

– Compute the 6-point DFT of

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4.2 Properties of the DFT

Denote

• Periodicity, Linearity, Circular symmetries

• Time reversal

• Circular time shift

• Circular frequency shift

• Complex conjugate properties

• Circular correlation

• Parseval ’s theorem

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4.2 Properties of the DFT

• Circular symmetries

– x’(n) is related to x(n) by a circular shift

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4.3 Multiplication of two DFTs and Circular convolution

– Suppose we have two finite-duration sequences of length

N, x1(n) and x2(n) Their respective N-point DFTS are:

– Multiply two sequences, the result is a DFT, say X3(k), of a sequence x3(n)

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– The inverse of X3(k) is

– Circular convolution

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• Example

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• Solution

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• The circular of the two sequences yields the sequence

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• Example

• By means of DFT and IDFT, determine the sequence x3(n)

corresponding to the circular convolution of x1(n) and x2(n)

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• Solution

• Compute the DFTs

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• Solution

• Thus,

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• Exercise

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5 Fast Fourier Transform

• The DFT can be written in the formula

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• Direct computation of DFT does not exploit the symmetry and periodicity properties

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5.1 Radix-2 FFT algorithms

• Split N-point data sequence x(n) into two N/2-point sequence

• The N-point DFT of x(n) can be expressed as

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• We can repeat the process for each of the sequences f1(n)

and f2(n)

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• Thus

• The total number of multiplication is reduced again

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• Comparison of computational complexity

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• Perform 8-point DFT in 3 stages

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• Basic butterfly computation

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• Compute 8-point DFT (using FFT algorithm) of

x(n)={1, 2, 3, 0, 0, 0, 0, 0}

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