Chapter 15. The Fast Fourier Transform

09 December 2013

Oka Danil Saputra (20136135) IT Convergence

Kumoh National Institute of Technology

Wireless & Emerging Networking System Laboratory

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• Represent continuous function by sinusoidal (sine and cosine)

functions.

• Discrete fourier transform 𝑓 𝑘 as a sequence function in

time domain to another sequence frequency domain 𝑓 𝑗 .

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• Example of the discrete fourier transform.

Figure 15.1 (a) A set of 16 data points representing sample of signal strength in the time interval 0 to 2𝜋.

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• The function generating the signal is of the form:

Figure 15.1 (b) The discrete fourier transform yields the amplitude and Frequencies of the constituent sine and cosine functions

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 16𝑖. • The frequency 2 component is -8. • The frequency 3 component is -16𝑖. • The frequency 4 component is 4.

f1 f2 f3 f4

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• The generating signal are:

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

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• This plot can be used as inputs to speech recognition system

with identify spoken through pattern recognition.

Figure 15.2 Discrete fourier transform for human speech

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• Given an 𝑛 element vector 𝑥, the DFT is the matrix-vector

product , where is the

primitive 𝑛th 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 𝑖.

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• 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:

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• Given an n element vector x, the inverse DFT is:

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• For example, to multiply the two polynomials.

• Yielding:

• Convolute the coefficient vectors:

• The result:

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Another way to multiply two polynomials of degree n-1 is:

1. To evaluate at the n complex 𝑛th roots of unity.

2. Perform an element-wise multiplication of the polynomials

value at these points.

3. Interpolate the results to produce the coefficients of the

product polynomial.

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1. We perform the DFT on the coefficients of p(x).

2. Perform the DFT on the coefficients of q(x).

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3. We perform an element-wise multiplication.

4. Last step, perform the inverse DFT on the product vector.

5. The vector produced by the inverse DFT contains the

coefficients.

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• 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 𝑛 is an even positive number, then the

squares of the 𝑛 complex 𝑛th roots of units are identical to the

𝑛/2 complex (𝑛/2)th root of unity.

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• The most natural way to express the FFT algorithm is using recursion.

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.

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• 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-𝑖,0,-3+ 𝑖).

Figure 15.4 (a) Recursive implementation of FFT

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• 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.

Figure 15.4 (b) Determining which computations are performed for each function invocation

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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 𝑛 indices.

Figure 15.4 (c) Tracking the flow of data values

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Iterative algorithm has the same time complexity as the recursive algorithm :

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