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Mathematics of the DFT
    Fourier Theorems for the DFT
       Signal Operators
          Convolution
             Polynomial Multiplication

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Polynomial Multiplication

Note that when you multiply two polynomials together, their coefficients are convolved. To see this, let $ p(x)$ denote the $ m$th-order polynomial

$\displaystyle p(x) = p_0 + p_1 x + p_2 x^2 + \cdots + p_m x^m $

with coefficients $ p_i$, and let $ q(x)$ denote the $ n$th-order polynomial

$\displaystyle q(x) = q_0 + q_1 x + q_2 x^2 + \cdots + q_n x^n $

with coefficients $ q_i$. Then we have [1]

\begin{eqnarray*} p(x) q(x) &=& p_0 q_0 + (p_0 q_1 + p_1 q_0) x + (p_0 q_2 + p_1... ...\qquad\qquad\; \mathop{+} p_{n+m-1} q_1 + p_{n+m} q_0) x^{n+m}. \end{eqnarray*}

Denoting $ p(x) q(x)$ by

$\displaystyle r(x) \isdef p(x) q(x) = r_0 + r_1 x + r_2 x^2 + \cdots + r_{m+n} x^{m+n}, $

we have that the $ i$th coefficient can be expressed as

\begin{eqnarray*} r_i &=& p_0 q_i + p_1 q_{i-1} + p_2 q_{i-2} + \cdots + p_{i-1}... ...=-\infty}^\infty p_j q_{i-j}\\ &\isdef & (p \circledast q)(i), \end{eqnarray*}

where $ p_i$ and $ q_i$ are doubly infinite sequences, defined as zero for $ i<0$ and $ i>m,n$, respectively.

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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