Convolution

A common engineering notational convention is:[2]

${\displaystyle f(t)*g(t)\mathrel {:=} \underbrace {\int _{-\infty }^{\infty }f(\tau )g(t-\tau )\,d\tau } _{(f*g)(t)},}$

which has to be interpreted carefully to avoid confusion. For instance, f(t)∗g(t − t₀) is equivalent to (f∗g)(t − t₀), but f(t − t₀)∗g(t − t₀) is in fact equivalent to (f∗g)(t − 2t₀).[3]

Given two functions ${\displaystyle f(t)}$ and ${\displaystyle g(t)}$ with bilateral Laplace transforms (two-sided Laplace transform)

${\displaystyle F(s)=\int _{-\infty }^{\infty }e^{-su}\ f(u)\ {\text{d}}u}$

and

${\displaystyle G(s)=\int _{-\infty }^{\infty }e^{-sv}\ g(v)\ {\text{d}}v}$

respectively, the convolution operation ${\displaystyle (f*g)(t)}$ can be defined as the inverse Laplace transform of the product of ${\displaystyle F(s)}$ and ${\displaystyle G(s)}$.[4][5] More precisely,

${\displaystyle {\begin{aligned}F(s)\cdot G(s)&=\int _{-\infty }^{\infty }e^{-su}\ f(u)\ {\text{d}}u\cdot \int _{-\infty }^{\infty }e^{-sv}\ g(v)\ {\text{d}}v\\&=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }e^{-s(u+v)}\ f(u)\ g(v)\ {\text{d}}u\ {\text{d}}v\end{aligned}}}$

Let ${\displaystyle t=u+v}$ such that

${\displaystyle {\begin{aligned}F(s)\cdot G(s)&=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }e^{-st}\ f(u)\ g(t-u)\ {\text{d}}u\ {\text{d}}t\\&=\int _{-\infty }^{\infty }e^{-st}\underbrace {\int _{-\infty }^{\infty }f(u)\ g(t-u)\ {\text{d}}u} _{(f*g)(t)}\ {\text{d}}t\\&=\int _{-\infty }^{\infty }e^{-st}(f*g)(t)\ {\text{d}}t\end{aligned}}}$

Note that ${\displaystyle F(s)\cdot G(s)}$ is the bilateral Laplace transform of ${\displaystyle (f*g)(t)}$. A similar derivation can be done using the unilateral Laplace transform (one-sided Laplace transform).

The convolution operation also describes the output (in terms of the input) of an important class of operations known as linear time-invariant (LTI). See LTI system theory for a derivation of convolution as the result of LTI constraints. In terms of the Fourier transforms of the input and output of an LTI operation, no new frequency components are created. The existing ones are only modified (amplitude and/or phase). In other words, the output transform is the pointwise product of the input transform with a third transform (known as a transfer function). See Convolution theorem for a derivation of that property of convolution. Conversely, convolution can be derived as the inverse Fourier transform of the pointwise product of two Fourier transforms.

When a function g_N is periodic, with period N, then for functions, f, such that f∗g_N exists, the convolution is also periodic and identical to:

${\displaystyle (f*g_{N})[n]\equiv \sum _{m=0}^{N-1}\left(\sum _{k=-\infty }^{\infty }{f}[m+kN]\right)g_{N}[n-m].}$

The summation on k is called a periodic summation of the function f.

If g_N is a periodic summation of another function, g, then f∗g_N is known as a circular convolution of f and g.

When the non-zero durations of both f and g are limited to the interval [0, N − 1],  f∗g_N reduces to these common forms:

${\displaystyle {\begin{aligned}\left(f*g_{N}\right)[n]&=\sum _{m=0}^{N-1}f[m]g_{N}[n-m]\\&=\sum _{m=0}^{n}f[m]g[n-m]+\sum _{m=n+1}^{N-1}f[m]g[N+n-m]\\[2pt]&=\sum _{m=0}^{N-1}f[m]g[(n-m)_{\bmod {N}}]\\[2pt]&\triangleq \left(f*_{N}g\right)[n]\end{aligned}}}$

(Eq.1)

The notation (f ∗_N g) for cyclic convolution denotes convolution over the cyclic group of integers modulo N.

Circular convolution arises most often in the context of fast convolution with a fast Fourier transform (FFT) algorithm.

In many situations, discrete convolutions can be converted to circular convolutions so that fast transforms with a convolution property can be used to implement the computation. For example, convolution of digit sequences is the kernel operation in multiplication of multi-digit numbers, which can therefore be efficiently implemented with transform techniques (Knuth 1997, §4.3.3.C; von zur Gathen & Gerhard 2003, §8.2).

Eq.1 requires N arithmetic operations per output value and N² operations for N outputs. That can be significantly reduced with any of several fast algorithms. Digital signal processing and other applications typically use fast convolution algorithms to reduce the cost of the convolution to O(N log N) complexity.

The most common fast convolution algorithms use fast Fourier transform (FFT) algorithms via the circular convolution theorem. Specifically, the circular convolution of two finite-length sequences is found by taking an FFT of each sequence, multiplying pointwise, and then performing an inverse FFT. Convolutions of the type defined above are then efficiently implemented using that technique in conjunction with zero-extension and/or discarding portions of the output. Other fast convolution algorithms, such as the Schönhage–Strassen algorithm or the Mersenne transform,[14] use fast Fourier transforms in other rings. The Winograd method is used as an alternative to the FFT.[15] It significantly speeds up 1D,[16] 2D,[17] and 3D[18] convolution.

If one sequence is much longer than the other, zero-extension of the shorter sequence and fast circular convolution is not the most computationally efficient method available.[19] Instead, decomposing the longer sequence into blocks and convolving each block allows for faster algorithms such as the overlap–save method and overlap–add method.[20] A hybrid convolution method that combines block and FIR algorithms allows for a zero input-output latency that is useful for real-time convolution computations.[21]

If f and g are compactly supported continuous functions, then their convolution exists, and is also compactly supported and continuous (Hörmander 1983, Chapter 1). More generally, if either function (say f) is compactly supported and the other is locally integrable, then the convolution f∗g is well-defined and continuous.

Convolution of f and g is also well defined when both functions are locally square integrable on R and supported on an interval of the form [a, +∞) (or both supported on [−∞, a]).

The convolution of f and g exists if f and g are both Lebesgue integrable functions in L¹(R^d), and in this case f∗g is also integrable (Stein & Weiss 1971, Theorem 1.3). This is a consequence of Tonelli's theorem. This is also true for functions in L¹, under the discrete convolution, or more generally for the convolution on any group.

Likewise, if f ∈ L¹(R^d)  and  g ∈ L^p(R^d)  where 1 ≤ p ≤ ∞,  then  f∗g ∈ L^p(R^d),  and

${\displaystyle \|{f}*g\|_{p}\leq \|f\|_{1}\|g\|_{p}.}$

In the particular case p = 1, this shows that L¹ is a Banach algebra under the convolution (and equality of the two sides holds if f and g are non-negative almost everywhere).

More generally, Young's inequality implies that the convolution is a continuous bilinear map between suitable L^p spaces. Specifically, if 1 ≤ p, q, r ≤ ∞ satisfy:

${\displaystyle {\frac {1}{p}}+{\frac {1}{q}}={\frac {1}{r}}+1,}$

then

${\displaystyle \left\Vert f*g\right\Vert _{r}\leq \left\Vert f\right\Vert _{p}\left\Vert g\right\Vert _{q},\quad f\in {\mathcal {L}}^{p},\ g\in {\mathcal {L}}^{q},}$

so that the convolution is a continuous bilinear mapping from L^p×L^q to L^r. The Young inequality for convolution is also true in other contexts (circle group, convolution on Z). The preceding inequality is not sharp on the real line: when 1 < p, q, r < ∞, there exists a constant B_p,q < 1 such that:

${\displaystyle \left\Vert f*g\right\Vert _{r}\leq B_{p,q}\left\Vert f\right\Vert _{p}\left\Vert g\right\Vert _{q},\quad f\in {\mathcal {L}}^{p},\ g\in {\mathcal {L}}^{q}.}$

The optimal value of B_p,q was discovered in 1975[22] and independently in 1976,[23] see Brascamp–Lieb inequality.

A stronger estimate is true provided 1 < p, q, r < ∞:

${\displaystyle \|f*g\|_{r}\leq C_{p,q}\|f\|_{p}\|g\|_{q,w}}$

where ${\displaystyle \|g\|_{q,w}}$ is the weak L^q norm. Convolution also defines a bilinear continuous map ${\displaystyle L^{p,w}\times L^{q,w}\to L^{r,w}}$ for ${\displaystyle 1<p,q,r<\infty }$, owing to the weak Young inequality:[24]

${\displaystyle \|f*g\|_{r,w}\leq C_{p,q}\|f\|_{p,w}\|g\|_{r,w}.}$

In addition to compactly supported functions and integrable functions, functions that have sufficiently rapid decay at infinity can also be convolved. An important feature of the convolution is that if f and g both decay rapidly, then f∗g also decays rapidly. In particular, if f and g are rapidly decreasing functions, then so is the convolution f∗g. Combined with the fact that convolution commutes with differentiation (see #Properties), it follows that the class of Schwartz functions is closed under convolution (Stein & Weiss 1971, Theorem 3.3).

If f is a smooth function that is compactly supported and g is a distribution, then f∗g is a smooth function defined by

${\displaystyle \int _{\mathbb {R} ^{d}}{f}(y)g(x-y)\,dy=(f*g)(x)\in C^{\infty }(\mathbb {R} ^{d}).}$

More generally, it is possible to extend the definition of the convolution in a unique way with ${\displaystyle \varphi }$ the same as f above, so that the associative law

${\displaystyle f*(g*\varphi )=(f*g)*\varphi }$

remains valid in the case where f is a distribution, and g a compactly supported distribution (Hörmander 1983, §4.2).

The convolution of any two Borel measures μ and ν of bounded variation is the measure ${\displaystyle \mu *\nu }$ defined by (Rudin 1962)

${\displaystyle \int _{\mathbf {R} ^{d}}f(x)\,d(\mu *\nu )(x)=\int _{\mathbf {R} ^{d}}\int _{\mathbf {R} ^{d}}f(x+y)\,d\mu (x)\,d\nu (y).}$

In particular,

${\displaystyle (\mu *\nu )(A)=\int _{\mathbf {R} ^{d}\times \mathbf {R} ^{d}}1_{A}(x+y)\,d(\mu \times \nu )(x,y),}$

where ${\displaystyle A\subset \mathbf {R} ^{d}}$ is a measurable set and ${\displaystyle 1_{A}}$ is the indicator function of ${\displaystyle A}$.

This agrees with the convolution defined above when μ and ν are regarded as distributions, as well as the convolution of L¹ functions when μ and ν are absolutely continuous with respect to the Lebesgue measure.

The convolution of measures also satisfies the following version of Young's inequality

${\displaystyle \|\mu *\nu \|\leq \|\mu \|\|\nu \|}$

where the norm is the total variation of a measure. Because the space of measures of bounded variation is a Banach space, convolution of measures can be treated with standard methods of functional analysis that may not apply for the convolution of distributions.

The convolution defines a product on the linear space of integrable functions. This product satisfies the following algebraic properties, which formally mean that the space of integrable functions with the product given by convolution is a commutative associative algebra without identity (Strichartz 1994, §3.3). Other linear spaces of functions, such as the space of continuous functions of compact support, are closed under the convolution, and so also form commutative associative algebras.

Commutativity

$${\displaystyle f*g=g*f}$$

Proof: By definition:

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

Changing the variable of integration to ${\displaystyle u=t-\tau }$ the result follows.

Associativity

$${\displaystyle f*(g*h)=(f*g)*h}$$

Proof: This follows from using Fubini's theorem (i.e., double integrals can be evaluated as iterated integrals in either order).

Distributivity

$${\displaystyle f*(g+h)=(f*g)+(f*h)}$$

Proof: This follows from linearity of the integral.

Associativity with scalar multiplication

$${\displaystyle a(f*g)=(af)*g}$$

for any real (or complex) number ${\displaystyle a}$.

Multiplicative identity

No algebra of functions possesses an identity for the convolution. The lack of identity is typically not a major inconvenience, since most collections of functions on which the convolution is performed can be convolved with a delta distribution (a unitary impulse, centered at zero) or, at the very least (as is the case of L¹) admit approximations to the identity. The linear space of compactly supported distributions does, however, admit an identity under the convolution. Specifically,

$${\displaystyle f*\delta =f}$$

where δ is the delta distribution.

Inverse element

Some distributions S have an inverse element S⁻¹ for the convolution which then must satisfy

$${\displaystyle S^{-1}*S=\delta }$$

from which an explicit formula for S⁻¹ may be obtained.

The set of invertible distributions forms an abelian group under the convolution.

Complex conjugation

$${\displaystyle {\overline {f*g}}={\overline {f}}*{\overline {g}}}$$

Relationship with differentiation

$${\displaystyle (f*g)'=f'*g=f*g'}$$

Proof:

${\displaystyle {\begin{aligned}(f*g)'&={\frac {d}{dt}}\int _{-\infty }^{\infty }f(\tau )g(t-\tau )\,d\tau \\&=\int _{-\infty }^{\infty }f(\tau ){\frac {\partial }{\partial t}}g(t-\tau )\,d\tau \\&=\int _{-\infty }^{\infty }f(\tau )g'(t-\tau )\,d\tau =f*g'.\end{aligned}}}$

Relationship with integration

If ${\textstyle F(t)=\int _{-\infty }^{t}f(\tau )d\tau ,}$ and ${\textstyle G(t)=\int _{-\infty }^{t}g(\tau )\,d\tau ,}$ then

$${\displaystyle (F*g)(t)=(f*G)(t)=\int _{-\infty }^{t}(f*g)(\tau )\,d\tau .}$$

If f and g are integrable functions, then the integral of their convolution on the whole space is simply obtained as the product of their integrals:[25]

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

This follows from Fubini's theorem. The same result holds if f and g are only assumed to be nonnegative measurable functions, by Tonelli's theorem.

In the one-variable case,

${\displaystyle {\frac {d}{dx}}(f*g)={\frac {df}{dx}}*g=f*{\frac {dg}{dx}}}$

where ${\displaystyle {\frac {d}{dx}}}$ is the derivative. More generally, in the case of functions of several variables, an analogous formula holds with the partial derivative:

${\displaystyle {\frac {\partial }{\partial x_{i}}}(f*g)={\frac {\partial f}{\partial x_{i}}}*g=f*{\frac {\partial g}{\partial x_{i}}}.}$

A particular consequence of this is that the convolution can be viewed as a "smoothing" operation: the convolution of f and g is differentiable as many times as f and g are in total.

These identities hold under the precise condition that f and g are absolutely integrable and at least one of them has an absolutely integrable (L¹) weak derivative, as a consequence of Young's convolution inequality. For instance, when f is continuously differentiable with compact support, and g is an arbitrary locally integrable function,

${\displaystyle {\frac {d}{dx}}(f*g)={\frac {df}{dx}}*g.}$

These identities also hold much more broadly in the sense of tempered distributions if one of f or g is a rapidly decreasing tempered distribution, a compactly supported tempered distribution or a Schwartz function and the other is a tempered distribution. On the other hand, two positive integrable and infinitely differentiable functions may have a nowhere continuous convolution.

In the discrete case, the difference operator D f(n) = f(n + 1) − f(n) satisfies an analogous relationship:

${\displaystyle D(f*g)=(Df)*g=f*(Dg).}$

The convolution theorem states that[26]

${\displaystyle {\mathcal {F}}\{f*g\}={\mathcal {F}}\{f\}\cdot {\mathcal {F}}\{g\}}$

where ${\displaystyle {\mathcal {F}}\{f\}}$ denotes the Fourier transform of ${\displaystyle f}$.

Versions of this theorem also hold for the Laplace transform, two-sided Laplace transform, Z-transform and Mellin transform.

If ${\displaystyle {\mathcal {W}}}$ is the Fourier transform matrix, then

${\displaystyle {\mathcal {W}}\left(C^{(1)}x\ast C^{(2)}y\right)=\left({\mathcal {W}}C^{(1)}\bullet {\mathcal {W}}C^{(2)}\right)(x\otimes y)={\mathcal {W}}C^{(1)}x\circ {\mathcal {W}}C^{(2)}y}$,

where ${\displaystyle \bullet }$ is face-splitting product,[27][28][29][30][31] ${\displaystyle \otimes }$ denotes Kronecker product, ${\displaystyle \circ }$ denotes Hadamard product (this result is an evolving of count sketch properties[32]).

The convolution commutes with translations, meaning that

${\displaystyle \tau _{x}(f*g)=(\tau _{x}f)*g=f*(\tau _{x}g)}$

where τ_xf is the translation of the function f by x defined by

${\displaystyle (\tau _{x}f)(y)=f(y-x).}$

If f is a Schwartz function, then τ_xf is the convolution with a translated Dirac delta function τ_xf = f ∗ τ_x δ. So translation invariance of the convolution of Schwartz functions is a consequence of the associativity of convolution.

Furthermore, under certain conditions, convolution is the most general translation invariant operation. Informally speaking, the following holds

Suppose that S is a bounded linear operator acting on functions which commutes with translations: S(τ_xf) = τ_x(Sf) for all x. Then S is given as convolution with a function (or distribution) g_S; that is Sf = g_S ∗ f.

Thus some translation invariant operations can be represented as convolution. Convolutions play an important role in the study of time-invariant systems, and especially LTI system theory. The representing function g_S is the impulse response of the transformation S.

A more precise version of the theorem quoted above requires specifying the class of functions on which the convolution is defined, and also requires assuming in addition that S must be a continuous linear operator with respect to the appropriate topology. It is known, for instance, that every continuous translation invariant continuous linear operator on L¹ is the convolution with a finite Borel measure. More generally, every continuous translation invariant continuous linear operator on L^p for 1 ≤ p < ∞ is the convolution with a tempered distribution whose Fourier transform is bounded. To wit, they are all given by bounded Fourier multipliers.