What is convolution integral?

Among all the electrical engineering students, this topic of convolution integral is very confusing. It is a mathematical operation of two functions f and g that produce another third type of function (f * g) , and this expresses how the shape of one is modified with the help of the other one. The process of computing it and the result function is known as convolution. After one is reversed and shifted, it is defined as the integral of the product of two functions. After producing the convolution function, the integral is evaluated for all the values of shift. The convolution integral has some similar features with the cross-correlation. The continuous or discrete variables for real-valued functions differ from cross-correlation (f * g) only by either of the two f(x) or g(x) is reflected about the y-axis or not. Therefore, it is a cross-correlation of f(x) and g(-x) or f(-x) and g(x), the cross-correlation operator is the adjoint of the operator of the convolution for complex-valued piecewise functions.

Definition

Denoting the operator with the symbol "*", the convolution of the two f and g is written as f*g. After one is reversed and shifted, it is known as the integral of the product of two functions. It is given by the equation,

$(f*g)\left(t\right) = {\int }_{-\infty }^{\infty } f\left(\tau \right)g(t-\tau )d\tau$

Another definition equivalently is given by the equation,

$(f*g)\left(t\right) = {\int }_{-\infty }^{\infty } f(t-\tau )g\left(\tau \right)d\tau$

The convolution process formula can be defined as the area under the function $f\left(\tau \right)$ weighted by the function $g(-\tau )$ which is shifted by t amount. The function $g(t-\tau )$ emphasizes different parts of the input function $f\left(\tau \right)$ as t changes.

For interval [0, $\infty$], the functions f and g are supported and the integration limits can be truncated which might result in the equation,

$(f*g)\left(t\right) = {\int }_0^t f\left(\tau \right)g(t-\tau )d\tau$

Derivations

The linear time-invariant (LTI) systems output signal is described with the help of convolution integral equation operations. No frequency components are created in terms of the Fourier transforms of the input and output of a linear time-invariant or LTI system operation. The modification is done only on the existing ones. The output transform is the product of the input transform with another transform known as the transfer function. As the inverse Fourier transform of the product of two Fourier transforms, the convolution can be derived.

Circular convolution

With period T, when a function $g_T$ is periodic, then for functions f , f*$g_T$ exists and the convolution is also periodic and identical to,

$(f*g_T)\left(t\right)\equiv {\int }_{t_o}^{t_o+T}\left[\sum _{k=-\infty }^{\infty }f\right(\tau + kT\left)\right] g_T(t-\tau )d\tau$

where $t_o$= arbitrary choice. This summation is known as the periodic summation of the function f. The f*$g_T$ is known as the circular or cyclic convolution of f and g when $g_T$ is a periodic summation of another function g. 

Discrete convolution

The discrete convolution of f and g, for complex-valued functions f ,g defined on the set of integers is given by-

$$\begin{gathered} (f*g)\left[n\right] = \sum _{m=-\infty }^{\infty } f\left[m\right]g[n-m] or given by \\ (f*g)\left[n\right] = \sum _{m=-\infty }^{\infty }f[n-m]g\left[m\right] \end{gathered}$$

On the set of integers by extending the sequence to finitely supported functions, the convolution of the two finite sequences is defined. The coefficients of the product of the two types of polynomials are the convolution of the original sequences when the sequences are the coefficients of two polynomials. This is called the Cauchy product. A finite summation may be used as listed below when g has finite support in the set {-M, -M+1,......, M-1, M},

$(f*g) \left[n\right] = \sum _{m=-M}^{M}f[n-m]g\left[m\right]$

Properties

Algebraic properties

A product is defined on the linear space of integrable functions by the convolution. Without identity, the space of integrable functions with the product which is given by convolution operation is a commutative associative algebra. The space of continuous functions of compact support is a type of linear space of function which is closed under the convolution and also forms commutative associative algebras.

Commutativity

f*g = g*f

Proof: We will now prove the commutativity using the variables x, y, and z and consider the generalized equation,  xy=yx.

Fix y first and let R be the set of all x. 

We have y . 1 = y and also, 1 . y = y . 1 so that 1 belongs to R.

If x belongs to R then xy = yx

Hence   xy + y = yx + y = y$x^{\text{'}}$

so now we have $x^{\text{'}}y = xy + y$

hence, $x^{\text{'}}y = y{x}^{\text{'}}$ so that $x^{\text{'}}$ belongs to R. The assertion therefore holds for all x.

Similarly using the above methodology, the equation f*g = g*f can be proved.

Associativity

f*(g*h) = (f*g)*h

Proof: We will prove the associativity using variables x, y and z, and considering the generalized equation, which is  f(f(x,y),z) = f(x,f(y,z))

A binary operation $*$ on a set of S is known as associative if it satisfies the law of associativity.

(x * y) * z = x * (y * z) for all x, y, z in S

Here the operator * is used in order to replace the symbol of the operation.

(xy)z = x(yz) = xyz for all x, y, z in S

Thus the associative law can also be expressed in functional notation thus :

f(f(x,y),z) = f(x,f(y,z)).

Similarly using the above approach, the equation f*(g*h) = (f*g)*h can be proved.

Distributivity

f*(g+h) = (f*g)+(f*h) 

Proof: We will prove the distributivity using the variable x, y and z, and using the generalized equation,  (y+z) * x = (y*x) + (z*x)

Given a set of S and two types of operators which are binary operators * and + on S

The operation * is left distributed over the operator + if given any elements x, y and z of S,

x*(y+z) = (x*y) + (x*z)

the operation * is right distributed over + if given any elements x, y and z of S,

(y + z) * x = (y * x) + (z * x) and the operation * is distributed over + if it is left and right distributed.

Hence, by the above approach the equation f*(g+h) = (f*g)+(f*h) can be proved.

Multiplicative identity

An identity for the convolution is possessed by any algebra of functions. Under the convolution, the linear space of compactly supported distributions however admits an identity.  Especially, f * $\delta$ = f where $\delta$ is the delta function distribution.

Integration

The integral of the convolution of the integrable functions f and g on the whole space is obtained as the product of their integrals given as - 

${\int }_{R^d}(f*g)\left(x\right)dx = \left({\int }_{R^d}f\right(x\left)dx\right) \left({\int }_{R^d}g\right(x\left)dx\right)$

This is given by Fubini's theorem.  Fubini's theorem states that if ${\{ a_{m,n}\}}_{m=1, n=1}^{\infty }$ is a doubly indexed sequence of real numbers and if $\underset{ }{{\sum }_{(m,n)\in NxN} a_{m,n}}$ is absolutely convergent then $\sum _{(m,n)\in NxN } a_{m,n} = \sum _{m=1}^{\infty }\sum _{n=1}^{\infty } a_{m,n} = \sum _{n=1}^{\infty }\sum _{m=1}^{\infty } a_{m,n}$

Differentiation

In the case of one variable -

$\frac{d }{dx}(f*g) = \frac{df}{dx}*g = f * \frac{dg}{dx}$ where d/dx is the derivative. An analogus formula holds with the partial derivative in the case of functions of several variables,

$\frac{\partial }{\partial x_i}(f*g) = \frac{\partial f}{\partial x_i}*g = f * \frac{\partial g}{\partial x_i}$

The convolution can be viewed as a "smoothing operator". As many times as f and g are in total, the convolution of f and g is differentiable.

An analogous relationship is satisfied by the equation f(n) = f (n+1) - f (n) in the discrete case by the difference operator D which is given by,

D(f*g) = (Df)*g = f*(Dg)

Context and Applications

This topic is significant in the professional  exam for Undergraduate, Graduate, and Postgraduate courses:

• Bachelors of Science in Electrical Engineering
• Masters of Science in Electrical Engineering

Practice problems

Question 1: Find the convolution of the given continuous time signals.

$x\left(t\right) = e^{-t}u\left(t\right) and h \left(t\right) = e^{-2t}u\left(t\right)$

1. $y\left(t\right) = [e^{-t}- e^{-2t}] u\left(t\right)$
2. $y\left(t\right) = [e^{-2t}- e^{-2t}] u\left(t\right)$
3. $y\left(t\right) = [e^{-t}+e^{-2t}] u\left(t\right)$
4. $y\left(t\right) = [e^t+e^{2t}] u\left(t\right)$

Answer: Correct option 1

Explanation:

 $$\begin{gathered} y\left(t\right) = {\int }_{-\infty }^{\infty }x\left(\tau \right)h(t-\tau )d\tau \\ = {\int }_{-\infty }^{\infty }{e}^{-\tau }u\left(\tau \right).e^{-2(t-\tau )}u(t-\tau )d\tau \\ = f_0^t{e}^{-\tau }{e}^{-2(t-\tau )}.d\tau ; t\ge 0 = e^{-2t}{\int }_0^t{e}^{\tau }d\tau \\ = e^{-2t}(e^t-1) ; t\ge o \\ \sin ce, y\left(t\right) = 0 for t<0 \\ ⇨ y\left(t\right) = [e^{-t}-e^{-2t}]u\left(t\right) \end{gathered}$$

Question 2: Convolution is done between how many types of functions?

1. 1
2. 2
3. 3
4. 4

Answer: Correct option 2

Explanation: Convolution integral is a mathematical operation done on two functions that are f and g which produces another third type of function (f * g) and this expresses how the shape of one is modified with the help of the other one.

Question 3: With what does convolution integral has some similar features?

1. Cross-correlation
2. Impulse response function
3. Schwartz functions
4. Faltung

Answer: Correct option 1

Explanation: The convolution integral has some similar features with the cross-correlation.

Question 4: The LTI system's output is described by using what?

1. Convolution integral
2. Non-zero input
3. Integrand
4. Superposition theory

Answer: Correct option 1

Explanation: The linear time-invariant (LTI) systems output is described with the help of convolution integral operations.

Question 5: The convolution integral of two functions f and g is written by using which operator?

1. &
2. *
3. #
4. @

Answer: Correct option 2

Explanation: Denoting the operator with symbol "*", the convolution of the two functions f and g is written as f*g

Related Concepts

• Analog signal processing
• Circulant matrix
• Generalized signal averaging
• Multidimensional discrete convolution