Convolution Theorem - Functions of A Discrete Variable... Sequences

Functions of A Discrete Variable... Sequences

By similar arguments, it can be shown that the discrete convolution of sequences and is given by:



where DTFT represents the discrete-time Fourier transform.

An important special case is the circular convolution of and defined by where is a periodic summation:

It can then be shown that:

\begin{align} x_N * y\ &=\ \scriptstyle{DTFT}^{-1} \displaystyle \big\\ &=\ \scriptstyle{DFT}^{-1} \displaystyle \big, \end{align}

where DFT represents the discrete Fourier transform.

The proof follows from DTFT#Periodic_data, which indicates that can be written as:

The product with is thereby reduced to a discrete-frequency function:

(also using Sampling the DTFT).

The inverse DTFT is:

\begin{align} (x_N * y)\ &=\ \int_{0}^{1} \frac{1}{N} \sum_{k=-\infty}^{\infty} \scriptstyle{DFT}\displaystyle\{x_N\}\cdot \scriptstyle{DFT}\displaystyle\{y_N\}\cdot \delta\left(f-k/N\right)\cdot e^{i 2 \pi f n} df\\ &=\ \frac{1}{N} \sum_{k=-\infty}^{\infty} \scriptstyle{DFT}\displaystyle\{x_N\}\cdot \scriptstyle{DFT}\displaystyle\{y_N\}\cdot \int_{0}^{1} \delta\left(f-k/N\right)\cdot e^{i 2 \pi f n} df\\ &=\ \frac{1}{N} \sum_{k=0}^{N-1} \scriptstyle{DFT}\displaystyle\{x_N\}\cdot \scriptstyle{DFT}\displaystyle\{y_N\}\cdot e^{i 2 \pi \frac{n}{N} k}\\ &=\ \scriptstyle{DFT}^{-1} \displaystyle \big, \end{align}

QED.

Read more about this topic:  Convolution Theorem

Famous quotes containing the words functions of, functions, discrete and/or variable:

    “One of the most highly valued functions of used parents these days is to be the villains of their children’s lives, the people the child blames for any shortcomings or disappointments. But if your identity comes from your parents’ failings, then you remain forever a member of the child generation, stuck and unable to move on to an adulthood in which you identify yourself in terms of what you do, not what has been done to you.”
    —Frank Pittman (20th century)

    “The English masses are lovable: they are kind, decent, tolerant, practical and not stupid. The tragedy is that there are too many of them, and that they are aimless, having outgrown the servile functions for which they were encouraged to multiply. One day these huge crowds will have to seize power because there will be nothing else for them to do, and yet they neither demand power nor are ready to make use of it; they will learn only to be bored in a new way.”
    —Cyril Connolly (1903–1974)

    “The mastery of one’s phonemes may be compared to the violinist’s mastery of fingering. The violin string lends itself to a continuous gradation of tones, but the musician learns the discrete intervals at which to stop the string in order to play the conventional notes. We sound our phonemes like poor violinists, approximating each time to a fancied norm, and we receive our neighbor’s renderings indulgently, mentally rectifying the more glaring inaccuracies.”
    —W.V. Quine (b. 1908)

    “There is not so variable a thing in nature as a lady’s head-dress.”
    —Joseph Addison (1672–1719)