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## Chapter 15. The Fast Fourier Transform

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**Wireless & Emerging Networking System Laboratory**Chapter 15.The Fast Fourier Transform 09 December 2013 Oka DanilSaputra (20136135) IT Convergence Kumoh National Institute of Technology**Fourier Analysis**• Represent continuous function by sinusoidal (sine and cosine) functions. • Discrete fouriertransform as a sequence function in time domain to another sequence frequency domain .**Fourier Analysis (Cont…)**Figure 15.1 (a) A set of 16 data points representing sample of signal strength in the time interval 0 to 2 • Example of the discrete fouriertransform.**Fourier Analysis (Cont…)**f3 f4 f2 f1 To calculate the coefficient , for each frequency divide the amplitude by 8 (half of 16, the number of sample point) • The frequency 1 component is 16. • The frequency 2 component is -8. • The frequency 3 component is -16. • The frequency 4 component is 4. Figure 15.1 (b) The discrete fouriertransform yields the amplitude and Frequencies of the constituent sine and cosine functions • The function generating the signal is of the form:**Fourier Analysis (Cont…)**Figure 15.1 (c) A plot of the four constituent functions and their sum a continuous function. (d) A plot of the continuous function and the original 16 sample • The generating signal are:**Fourier Analysis (Cont…)**Figure 15.2 Discrete fouriertransform for human speech • This plot can be used as inputs to speech recognition system with identify spoken through pattern recognition.**The Discrete Fourier Transform**• Given an element vector , the DFT is the matrix-vector product , where is the primitive th root of unity. • Example, compute DFT of the vector (2,3) where the primitive square root of unity is -1. • Compute the DFT of the vector (1,2,4,3) using the primitive fourth root of unity, which is .**The Discrete Fourier Transform (Cont…)**• Let’s put the DFT for previous section where we have a vector of 16 complex. • The DFT of this vector is: • To determine the coefficients of the sine and cosine, we examine the nonzero element in the first half. • Thus the combination of sine and cosine functions making up the curve is:**Inverse Discrete Fourier Transform**• Given an n element vector x, the inverse DFT is:**Sample Application: Polynomial Multiplication**• For example, to multiply the two polynomials. • Yielding: • Convolute the coefficient vectors: • The result:**Sample Application: Polynomial Multiplication (Cont…)**Another way to multiply two polynomials of degree n-1 is: To evaluate at the n complex throots of unity. Perform an element-wise multiplication of the polynomials value at these points. Interpolate the results to produce the coefficients of the product polynomial.**Sample Application: Polynomial Multiplication (Cont…)**We perform the DFT on the coefficients of p(x). Perform the DFT on the coefficients of q(x).**Sample Application: Polynomial Multiplication (Cont…)**3. We perform an element-wise multiplication. Last step, perform the inverse DFT on the product vector. The vector produced by the inverse DFT contains the coefficients.**The Fast Fourier Transform**• The FFT uses a divide-and-conguer strategy to evaluate a polynomial of degree n at the n complex nth roots of unity. • Having Lemma: If is an even positive number, then the squares of the complex th roots of units are identical to the /2 complex (/2)th root of unity.**The Fast Fourier Transform**The time complexity of this algorithm is easy to determine. Lets T(n) denote the time needed to perform the FFT on a polynomial of degree n. • The most natural way to express the FFT algorithm is using recursion.**The Fast Fourier Transform (Cont…)**Figure 15.4 (a) Recursive implementation of FFT • Figure 15.4 illustrates the derivation of an iterative algorithm from recursive algorithm. • Performing the FFT on input vector (1,2,4,3) produces the result vector (10,-3-,0,-3+).**The Fast Fourier Transform (Cont…)**Figure 15.4 (b) Determining which computations are performed for each function invocation • In figure 15.4b we look inside the functions and determine exactly which operations are performed for each invocation. • The expressions of form a+b(c) and a-b(c) correspond the pseudocode statements.**The Fast Fourier Transform (Cont…)**Figure 15.4 (c) Tracking the flow of data values • Iterative algorithm: • After an initial permutation step, the algorithm will iterate log n time. • Each iteration corresponds to a horizontal layer in Figure 15.4c. • Within an iteration the algorithm updates value for each of the indices.**The Fast Fourier Transform (Cont…)**Iterative algorithm has the same time complexity as the recursive algorithm :