What is continuous convolution?

For suitable functions, $(f*g)(t)=\int_{-\infty}^{\infty} f(\tau)g(t-\tau)\,d\tau$; it produces a function of the shift $t$.

What is discrete convolution?

For integer-indexed sequences, $(f*g)[n]=\sum_{m=-\infty}^{\infty} f[m]g[n-m]$.

What does convolution with the delta distribution do?

It leaves a suitable function unchanged: $f*\delta=f$.

How does convolution behave under translations?

Shifting either input shifts the convolution output by the same amount.

How is convolution related to integration over the full domain?

For integrable functions, the integral of their convolution equals the product of their integrals.

Signal processingLesson 2 of 2 in the course Fourier analysis

Convolution: Definitions, Properties, Transforms, and Applications

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Summary

Convolution combines two functions to produce a third. It is useful for describing how one shape, signal, or response modifies another.

For suitable functions $f$ and $g$ , their convolution is written $f*g$ and defined by

$(f*g)(t)=\int_{-\infty}^{\infty} f(\tau)g(t-\tau)\,d\tau.$

The variable $\tau$ is integrated over, while $t$ specifies the output location. Evaluating the integral for each value of $t$ produces the function $f*g$ .

All flashcards

What is continuous convolution?: For suitable functions, $(f*g)(t)=\int_{-\infty}^{\infty} f(\tau)g(t-\tau)\,d\tau$ ; it produces a function of the shift $t$ .
What is discrete convolution?: For integer-indexed sequences, $(f*g)[n]=\sum_{m=-\infty}^{\infty} f[m]g[n-m]$ .
What does convolution with the delta distribution do?: It leaves a suitable function unchanged: $f*\delta=f$ .
How does convolution behave under translations?: Shifting either input shifts the convolution output by the same amount.
How is convolution related to integration over the full domain?: For integrable functions, the integral of their convolution equals the product of their integrals.

Quiz questions

In circular convolution of length $N$ , how are indices beyond the sequence length handled?Answer: They wrap around by interpreting indices modulo $N$ .
With a consistent Fourier-transform convention, what is the transform of a convolution?Answer: The pointwise product of the transforms of the two functions.
When the required regularity conditions hold, how can the derivative of a convolution be expressed?Answer: As either factor differentiated and then convolved with the other.
For a linear time-invariant system with impulse response $h$ , how is the output $y$ related to input $x$ ?Answer: The output is the convolution of the input with the impulse response.
How is convolution defined for functions on $\mathbb{R}^d$ ?Answer: By integrating $f(y)g(x-y)$ over the vector domain.

Based on the Wikipedia article Convolution, licensed CC BY-SA 4.0. Adapted into a summary, flashcards and quizzes.

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