Question

This exercise develops a fast algorithm, called the fast Fourier transform or FFT, for computing the function \(\mathcal{E}_{n, \omega} .\) This is a recursive algorithm 17.6 Faster polynomial arithmetic (*) FFT \(\left(n, \omega ; a_{0}, \ldots, a_{2^{n}-1}\right)\) that takes as input an integer \(n \geq 0,\) a primitive \(2^{n}\) th root of unity \(\omega \in R,\) and elements \(a_{0}, \ldots, a_{2^{n}-1} \in R,\) and runs as follows: if \(n=0\) then return \(a_{0}\) else $$ \begin{array}{l} \left(\alpha_{0}, \ldots, \alpha_{2^{n-1}-1}\right) \leftarrow \mathrm{FFT}\left(n-1, \omega^{2} ; a_{0}, a_{2}, \ldots, a_{2^{n}-2}\right) \\\ \left(\beta_{0}, \ldots, \beta_{2^{n-1}-1}\right) \leftarrow \mathrm{FFT}\left(n-1, \omega^{2} ; a_{1}, a_{3}, \ldots, a_{2^{n}-1}\right) \\\ \text { for } i \leftarrow 0 \text { to } 2^{n-1}-1 \mathrm{do} \\ \quad \gamma_{i} \leftarrow \alpha_{i}+\beta_{i} \omega^{i}, \gamma_{i+2^{n-1}} \leftarrow \alpha_{i}-\beta_{i} \omega^{i} \\ \text { return }\left(\gamma_{0}, \ldots, \gamma_{2^{n}-1}\right) \end{array} $$

Short answer

Question: Explain the Fast Fourier Transform (FFT) algorithm in a few steps. Answer: The Fast Fourier Transform (FFT) algorithm is a recursive algorithm that computes the discrete Fourier transform of a sequence. Here are the main steps: 1. Handle the base case: When the input integer n=0, return a single-element result, a_0. 2. Make recursive calls: Call the FFT algorithm for even-indexed elements (a_0, a_2, ..., a_(2^n-2)) and odd-indexed elements (a_1, a_3, ..., a_(2^n-1)) of the input sequence, with n-1 and omega^2 as arguments. 3. Calculate gamma values: Compute gamma_i and gamma_(i+2^(n-1)) using the results from the previous step, i.e., alpha_i and beta_i, for all i from 0 to 2^(n-1)-1. 4. Return the result: Combine the calculated gamma values as a tuple and return it. The FFT algorithm can efficiently compute the discrete Fourier transform using these steps, enabling various applications in signal and image processing, among others.

Step-by-step solution

Base case

When \(n=0\), the algorithm should return \(a_0\). This is the base case of the recursion, and it helps to terminate the algorithm when it reaches the end point.

Recursive calls

Next, we need to make two recursive calls to the FFT algorithm with \(n-1\) and \(\omega^2\). For the first call, use even-indexed elements of the input: $$\left(\alpha_{0}, \ldots, \alpha_{2^{n-1}-1}\right) \leftarrow \mathrm{FFT}\left(n-1, \omega^2 ; a_0, a_2, \ldots, a_{2^{n}-2}\right)$$ For the second call, use odd-indexed elements of the input: $$\left(\beta_{0}, \ldots, \beta_{2^{n-1}-1}\right) \leftarrow \mathrm{FFT}\left(n-1, \omega^2 ; a_1, a_3, \ldots, a_{2^{n}-1}\right)$$

Calculate gamma values

Now, we need to compute the gamma values \(\gamma_i\) and \(\gamma_{i+2^{n-1}}\) using the alpha and beta values from the previous step. For this, iterate through all \(i\) values from \(0\) to \(2^{n-1}-1\) and calculate: $$\gamma_{i} \leftarrow \alpha_{i}+\beta_{i} \omega^{i}$$ $$\gamma_{i+2^{n-1}} \leftarrow \alpha_{i}-\beta_{i} \omega^{i}$$

Return the result

Finally, return the calculated gamma values as a tuple: $$\left(\gamma_0, \ldots, \gamma_{2^{n}-1}\right)$$ The Fast Fourier Transform algorithm is now complete and can be used to compute the function \(\mathcal{E}_{n, \omega}\).

Key concepts

Recursive Algorithm

When we talk about the Fast Fourier Transform (FFT), one of its core methods involves using a recursive algorithm. Recursion is a technique where a function calls itself to solve a problem, typically breaking it down into smaller, more manageable sub-problems.
In the case of the FFT, the recursion is used to split a problem into halves until it reaches a base case. This is particularly efficient with problems that naturally divide, like the FFT, which separates input data into odd and even indexed components.

• Base Case: The recursive process must define a stopping point, known as the base case. In this FFT algorithm, if the input size parameter \( n = 0 \), the function returns the single input term \( a_0 \), indicating that the recursion has processed down to the most straightforward scenario.
• Recursive Calls: The function makes two recursive calls to process the even and odd parts of the data separately. This allows the algorithm to process each subset independently before combining the results.

By breaking down the FFT problem using recursion, we can simplify the computations and achieve quicker processing, which is the backbone of the algorithm's efficiency.

Polynomial Arithmetic

Understanding polynomial arithmetic is crucial for grasping how the FFT works since it deals with polynomials evaluated at several points. FFT is often used to speed up polynomial multiplication.
Let's consider a polynomial with coefficients \( a_0, a_1, \ldots, a_{n-1} \). To multiply two polynomials, you can convert them into point-value form, which is what FFT helps to achieve efficiently.

• Coefficient Representation: Each polynomial is initially expressed in terms of its coefficients \( (a_0, a_1, \ldots) \). Polynomial arithmetic begins by evaluating these at several points.
• Point-Value Representation: The FFT transforms the coefficient representation of a polynomial into its point-value form, using the given roots of unity. This conversion allows for element-wise multiplication of corresponding points, simplifying the multiplication process.
• Efficient Computation: By transforming into the point-value form via the FFT, the overall polynomial multiplication process becomes more efficient because it reduces complexity from \( O(n^2) \) to \( O(n \log n) \).

The FFT doesn't solve polynomial multiplication directly but instead facilitates a transformation that makes it far more efficient in terms of computational resources.

Primitive Root of Unity

In the FFT, the concept of the primitive root of unity is pivotal. A complex number \( \omega \) is defined as a primitive \( n \)-th root of unity if \( \omega^n = 1 \) and cannot be equal to any lower power of 1. This root plays a significant role in the FFT by allowing polynomials to be evaluated efficiently at different points.

• Properties: These roots of unity have specific properties: They're evenly spaced on the complex unit circle, which allows the transformation to maintain symmetry and balance.
• Use in FFT: Within the FFT, the primitive root \( \omega \) helps to form the basis for transforming data into the frequency domain and back, enabling the FFT to split data into its constituent periodic components with efficiency.
• Calculation: In the FFT process, \( \omega \) is raised to increasing powers as part of the recursive call to ensure that all necessary evaluations are performed. This results in an evaluation at points \( 1, \omega, \omega^2, \ldots, \omega^{n-1} \).

The ingenious use of primitive roots of unity is what underpins the rapid computation abilities of the FFT, turning a laborious polynomial evaluation into a swift operation.