How does the convolution relate to the most popular transforms in signal processing?

The *convolution property* appears in at least in three very important transforms: the Fourier transform, the Laplace transform, and the

### The Convolution Series

- Definition of convolution and intuition behind it
- Mathematical properties of convolution
**Convolution property of Fourier, Laplace, and z-transforms**- Identity element of the convolution
- Star notation of the convolution
- Circular vs. linear convolution
- Fast convolution
- Convolution vs. correlation
- Convolution in MATLAB, NumPy, and SciPy
- Deconvolution: Inverse convolution
- Convolution in probability: Sum of independent random variables

## Recap

Let us briefly recap the definition of the discrete convolution

To understand these properties more easily, we can think of

## In Short

The main takeaway from this article is that convolution in the time domain changes to multiplication (possibly with some additional constraints) in the transform domain. In particular,

- For the Fourier transform,
$x(t) \ast h(t) \stackrel{\mathcal{F}}{\longleftrightarrow} X(j\omega)H(j\omega)$ . - For the Laplace transform,
$x(t) \ast h(t) \stackrel{\mathcal{L}}{\longleftrightarrow} X(s)H(s)$ with the region of convergence (ROC) containing the intersection of$X(s)$ 's ROC and$H(s)$ 's ROC. - For the
$z$ -transform,$x[n] \ast h[n] \stackrel{\mathcal{Z}}{\longleftrightarrow} X(z)H(z)$ with the ROC containing the intersection of$X(z)$ 's ROC and$H(z)$ 's ROC.

Analogously, convolution in the transform domain changes to multiplication in the time domain. This symmetry should be clear from the derivations, so I only mention it once.

The article explains these relations in detail and gives proofs of the corresponding convolution property versions.

## Fourier Transform

The Fourier transform is without a doubt the most important transform in signal processing. For a continuous signal

**Fourier transform pair**. We can denote this by

While **angular frequency** with the unit of rad/s.

### The Convolution Property

The behavior of convolution under any of the three discussed transforms bears the name of the **convolution property**. When the following transforms exist

the transform of their convolution is the multiplication of their transforms [1, Eq. 4.56]

*Note: The same holds for the Fourier transform of discrete signals (not to be confused with the discrete Fourier transform). For details, see [2, p. 60].*

#### Proof

We can prove the convolution property by definining

and deriving its Fourier transform

#### Application

The convolution property of the Fourier transform has a number of practical applications, namely, it enables

- fast convolution algorithms,
- efficient implementations of various signal processing algorithms via frequency-domain filtering,
- deconvolution in the frequency domain,
- frequency-based filter design,
- cascaded systems analysis,
- further transform properties derivations.

Additionally, the convolution property makes the commutativity property from the previous article immediately obvious, as the multiplication operands

## Laplace transform

The Laplace transform is another frequently used transform even outside the field of engineering. For example, it plays an important role in solving differential equations.

The Laplace transform of

where **region of convergence (ROC)** of the Laplace transform. In this article, I denote ROC of

The relationship between

### Convolution Property

When the following transforms exist

the Laplace transform of their convolution is the multiplication of their transforms [1, Eq. 9.95]

where

#### Proof

By substituting

#### Application

Without going into details, let me just mention that the convolution property of the Laplace transform plays an important role in the analysis of linear time-invariant (LTI) systems.

## Z-transform

What the Laplace transform does for continuous-time systems, the

The

where

Again, the relation between

### Convolution Property

When the following transforms exist

the

where

#### Proof

The proof of the convolution property is rather straightforward

## Summary

In this article the convolution property of the Fourier, Laplace, and

## Bibliography

[1] Alan V. Oppenheim, Alan S. Willsky, with S. Hamid *Signals and Systems*, 2nd Edition, Pearson 1997.

[2] Alan V. Oppenheim, Ronald W. Schafer *Discrete-Time Signal Processing*, 3rd Edition, Pearson 2010.

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