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Each Fourier coefficient on its own is the average of $2^n$ independent uniformly random $\pm1$ variables. Its distribution is roughly normal with mean $0$ and variance $2^{-n}$.

Let us denote the input signal by $x(0),\ldots,x(n-1)$ and the output signal by $y(0),\ldots,y(n-1)$. Your program is using the following formula: $$ y(k) = \sum_{t=0}^{n-1} e^{-2\pi i tk/n} x(t). $$ …

Here is an attempt. The Fourier transform decomposes a signal into superimposed sine waves of various frequencies (this can be demonstrated visually). It turns out that we can find out the strength of …

Suppose we use fourth roots of unity, which corresponds to substituting $1,i,-1,-i$ for $x$. We also use decimation-in-time rather than decimation-in-frequency in the FFT algorithm. (We also apply a b …

Suppose that $A$ and $B$ are vectors of length $n$, where $n$ is a power of 2. We index $A$ and $B$ using binary vectors of length $\log_2 n$, and define their convolution $C$ as $$ C_i = \sum_{j \opl …

The index $\mathbf{k}$ goes over all vectors in $\{0,1\}^n$ (which can be identified with $\mathbb{Z}_2^n$). The index $\mathbf{x}$ also has the same range, and $\mathbf{k} \cdot \mathbf{x} = \sum_{i= …

Suppose that the charges are $q_1,\ldots,q_n$. You need to calculate, for all $i$, $$ F_i = q_i \sum_{j>i} \frac{q_j}{(i-j)^2} - q_i \sum_{j<i} \frac{q_j}{(i-j)^2}. $$ In matrix form, we get $$ \begin …

There are two variants of FFT: Decimation in Time (DIT) and Decimation in Frequency (DIF). What you describe is DIT, whereas the diagram shows DIF. The only difference is the order in which the bits a …

You haven't explained what you mean by the DFT of a polynomial, so I'm assuming you mean the polynomial evaluated at all $n$th roots of unity, that is, $$ v_i = P(\omega^i), $$ where $\omega = e^{2\pi …

An efficient change of basis doesn't change the hardness of a problem. Consider some decision problem $Q_1$, which consists of all objects $x$ of some type that satisfy some property. Now let $f$ be a …

Morgenstern first defines the notion of a linear algorithm. A linear algorithm gets as input $x_1,\ldots,x_n$ and its goal is to compute some $y_1,\ldots,y_m$, each of which is a (specific) linear com …

Hint: Taking the derivative has a certain, simple diagonal effect on the Fourier transform, that hopefully was covered in class. Hence if you compute the Fourier transform of the original function, yo …

Let's answer your questions one by one. You don't compute $W^2$ by squaring $W$ algorithmically. Rather, you compute $W^2$ "on paper", and you find out a formula for $W^2$. Given this formula, you c …

Here is the idea of the proof. Let $a_i,b_i$ be the distance thrown by the $i$th male/female. Using FFT, we calculate $$ \left(\sum_i x^{a_i}\right) \left(\sum_j x^{b_j}\right) = \sum_{i,j} x^{a_i+b_j …

The trivial algorithm that multiplies every monomial in $A$ by every monomial in $B$ takes time $O(|A| \cdot |B|)$ (where $|A|$ is the number of monomials in $A$ or $\deg A + 1$, depending on the repr …