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Convolution

In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ($$f$$ and $$g$$) that produces a third function ($$f*g$$). The term convolution refers to both the result function and to the process of computing it. It is defined as the integral of the product of the two functions after one is reflected about the y-axis and shifted. The integral is evaluated for all values of shift, producing the convolution function. The choice of which function is reflected and shifted before the integral does not change the integral result (see commutativity). Graphically, it expresses how the 'shape' of one function is modified by the other.

Some features of convolution are similar to cross-correlation: for real-valued functions, of a continuous or discrete variable, convolution ($$f*g$$) differs from cross-correlation ($$f \star g$$) only in that either $$f(x)$$ or $$g(x)$$ is reflected about the y-axis in convolution; thus it is a cross-correlation of $$g(-x)$$ and $$f(x)$$, or $$f(-x)$$ and $$g(x)$$. For complex-valued functions, the cross-correlation operator is the adjoint of the convolution operator.

Convolution has applications that include probability, statistics, acoustics, spectroscopy, signal processing and image processing, geophysics, engineering, physics, computer vision and differential equations.

The convolution can be defined for functions on Euclidean space and other groups (as algebraic structures). For example, periodic functions, such as the discrete-time Fourier transform, can be defined on a circle and convolved by periodic convolution. (See row 18 at .) A discrete convolution can be defined for functions on the set of integers.

Generalizations of convolution have applications in the field of numerical analysis and numerical linear algebra, and in the design and implementation of finite impulse response filters in signal processing.

Computing the inverse of the convolution operation is known as deconvolution.

Definition
The convolution of $$f$$ and $$g$$ is written $$f * g$$, denoting the operator with the symbol $$*$$. It is defined as the integral of the product of the two functions after one is reflected about the y-axis and shifted. As such, it is a particular kind of integral transform:

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

An equivalent definition is (see commutativity):

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

While the symbol $$t$$ is used above, it need not represent the time domain. At each $$t$$, the convolution formula can be described as the area under the function $$f(\tau)$$ weighted by the function $$g(-\tau)$$ shifted by the amount $$t$$. As $$t$$ changes, the weighting function $$g(t-\tau)$$ emphasizes different parts of the input function $$f(\tau)$$; If $$t$$ is a positive value, then $$g(t-\tau)$$ is equal to $$g(-\tau)$$ that slides or is shifted along the $$\tau$$-axis toward the right (toward $$+\infty$$) by the amount of $$t$$, while if $$t$$ is a negative value, then $$g(t-\tau)$$ is equal to $$g(-\tau)$$ that slides or is shifted toward the left (toward $$-\infty$$) by the amount of $$|t|$$.

For functions $$f$$, $$g$$ supported on only $$[0,\infty)$$ (i.e., zero for negative arguments), the integration limits can be truncated, resulting in:

 * $$(f * g)(t) = \int_{0}^{t} f(\tau) g(t - \tau)\, d\tau \quad \ \text{for } f, g : [0, \infty) \to \mathbb{R}.$$

For the multi-dimensional formulation of convolution, see domain of definition (below).

Notation
A common engineering notational convention is:

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

which has to be interpreted carefully to avoid confusion. For instance, $$f(t) * g(t-t_0)$$ is equivalent to $$(f*g)(t-t_0)$$, but $$f(t-t_0) * g(t-t_0)$$ is in fact equivalent to $$(f * g)(t-2t_0)$$.

Relations with other transforms
Given two functions $$ f(t) $$ and $$ g(t) $$ with bilateral Laplace transforms (two-sided Laplace transform)

 * $$ F(s) = \int_{-\infty}^\infty e^{-su} \ f(u) \ \text{d}u $$

and

 * $$ G(s) = \int_{-\infty}^\infty e^{-sv} \ g(v) \ \text{d}v $$

respectively, the convolution operation $$ (f * g)(t) $$ can be defined as the inverse Laplace transform of the product of $$ F(s) $$ and $$ G(s) $$. More precisely,

\begin{align} 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{align} $$

Let $$ t = u + v $$ such that

\begin{align} 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{align} $$

Note that $$ F(s) \cdot G(s) $$ is the bilateral Laplace transform of $$ (f * g)(t) $$. 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.

Historical developments
One of the earliest uses of the convolution integral appeared in D'Alembert's derivation of Taylor's theorem in Recherches sur différents points importants du système du monde, published in 1754.

Also, an expression of the type:

 * $$\int f(u)\cdot g(x - u) \, du$$

is used by Sylvestre François Lacroix on page 505 of his book entitled Treatise on differences and series, which is the last of 3 volumes of the encyclopedic series: Traité du calcul différentiel et du calcul intégral, Chez Courcier, Paris, 1797–1800. Soon thereafter, convolution operations appear in the works of Pierre Simon Laplace, Jean-Baptiste Joseph Fourier, Siméon Denis Poisson, and others. The term itself did not come into wide use until the 1950s or 1960s. Prior to that it was sometimes known as Faltung (which means folding in German), composition product, superposition integral, and Carson's integral. Yet it appears as early as 1903, though the definition is rather unfamiliar in older uses.

The operation:

 * $$\int_0^t \varphi(s)\psi(t - s) \, ds,\quad 0 \le t < \infty,$$

is a particular case of composition products considered by the Italian mathematician Vito Volterra in 1913.

Circular convolution
When a function $+∞$ is periodic, with period $t$, then for functions, $t$, such that $-∞$ exists, the convolution is also periodic and identical to:

 * $$(f * g_T)(t) \equiv \int_{t_0}^{t_0+T} \left[\sum_{k=-\infty}^\infty f(\tau + kT)\right] g_T(t - \tau)\, d\tau,$$

where $−∞$ is an arbitrary choice. The summation is called a periodic summation of the function $t$.

When $+∞$ is a periodic summation of another function, $t$, then $gT$ is known as a circular or cyclic convolution of $t$ and $t$.

And if the periodic summation above is replaced by $f ∗ gT$, the operation is called a periodic convolution of $t_{0}$ and $gT$.

Discrete convolution
For complex-valued functions $f ∗ gT$ defined on the set Z of integers, the discrete convolution of $t$ and $f$ is given by:

 * $$(f * g)[n] = \sum_{m=-\infty}^\infty f[m] g[n - m],$$

or equivalently (see commutativity) by:

 * $$(f * g)[n] = \sum_{m=-\infty}^\infty f[n-m] g[m].$$

The convolution of two finite sequences is defined by extending the sequences to finitely supported functions on the set of integers. When the sequences are the coefficients of two polynomials, then the coefficients of the ordinary product of the two polynomials are the convolution of the original two sequences. This is known as the Cauchy product of the coefficients of the sequences.

Thus when $g$ has finite support in the set $$\{-M,-M+1,\dots,M-1,M\}$$ (representing, for instance, a finite impulse response), a finite summation may be used:
 * $$(f * g)[n]=\sum_{m=-M}^M f[n-m]g[m].$$

Circular discrete convolution
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:

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

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:

$T$

The notation $$f *_N g$$ for cyclic convolution denotes convolution over the cyclic group of integers modulo $fT$.

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

Fast convolution algorithms
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.

$f$ requires $f$ arithmetic operations per output value and $fT$ operations for $g$ 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($f$ log $g$) 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, use fast Fourier transforms in other rings. The Winograd method is used as an alternative to the FFT. It significantly speeds up 1D, 2D, and 3D 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. 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. 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.

Domain of definition
The convolution of two complex-valued functions on $gT$ is itself a complex-valued function on $f, g$, defined by:

 * $$(f * g )(x) = \int_{\mathbf{R}^d} f(y)g(x-y)\,dy = \int_{\mathbf{R}^d} f(x-y)g(y)\,dy,$$

and is well-defined only if $f$ and $g$ decay sufficiently rapidly at infinity in order for the integral to exist. Conditions for the existence of the convolution may be tricky, since a blow-up in $g$ at infinity can be easily offset by sufficiently rapid decay in $$. The question of existence thus may involve different conditions on $$ and $N$:

Compactly supported functions
If $N$ and $N$ are compactly supported continuous functions, then their convolution exists, and is also compactly supported and continuous. More generally, if either function (say $N$) is compactly supported and the other is locally integrable, then the convolution $N$ is well-defined and continuous.

Convolution of $f$ and $g$ is also well defined when both functions are locally square integrable on $N^{2}$ and supported on an interval of the form $R^{d}$ (or both supported on $R^{d}$).

Integrable functions
The convolution of $g$ and $f$ exists if $f$ and $g$ are both Lebesgue integrable functions in $f∗g$($R$), and in this case $[a, +∞)$ is also integrable. This is a consequence of Tonelli's theorem. This is also true for functions in $[−∞, a]$, under the discrete convolution, or more generally for the convolution on any group.

Likewise, if $L^{1}$($R^{d}$) and  $f∗g$($L^{1}$)  where $f ∈ L^{1}$,  then  $R^{d}$($g ∈ L^{p}$),  and

 * $$\|{f}* g\|_p\le \|f\|_1\|g\|_p.$$

In the particular case $R^{d}$, this shows that $1 ≤ p ≤ ∞$ 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 $f*g ∈ L^{p}$ spaces. Specifically, if $R^{d}$ satisfy:

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

then

 * $$\left\Vert f*g\right\Vert_r\le\left\Vert f\right\Vert_p\left\Vert g\right\Vert_q,\quad f\in L^p,\ g\in L^q,$$

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

 * $$\left\Vert f*g\right\Vert_r\le B_{p,q}\left\Vert f\right\Vert_p\left\Vert g\right\Vert_q,\quad f\in L^p,\ g\in L^q.$$

The optimal value of $L^{r}$ was discovered in 1975 and independently in 1976, see Brascamp–Lieb inequality.

A stronger estimate is true provided $Z$:
 * $$\|f * g\|_r\le C_{p,q}\|f\|_p\|g\|_{q,w}$$

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

Functions of rapid decay
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 ), it follows that the class of Schwartz functions is closed under convolution.

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

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

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

 * $$f* (g* \varphi) = (f* g)* \varphi$$

remains valid in the case where f is a distribution, and g a compactly supported distribution.

Measures
The convolution of any two Borel measures μ and ν of bounded variation is the measure $$\mu*\nu$$ defined by
 * $$\int_{\mathbf{R}^d} f(x) \, d(\mu*\nu)(x) = \int_{\mathbf{R}^d}\int_{\mathbf{R}^d}f(x+y)\,d\mu(x)\,d\nu(y).$$

In particular,
 * $$(\mu*\nu)(A) = \int_{\mathbf{R}^d\times\mathbf R^d}1_A(x+y)\, d(\mu\times\nu)(x,y),$$

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

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
 * $$\|\mu* \nu\|\le \|\mu\|\|\nu\| $$

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.

Algebraic properties
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. 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: $$f * g = g * f $$ Proof: By definition: $$(f * g)(t) = \int^\infty_{-\infty} f(\tau)g(t - \tau)\, d\tau$$ Changing the variable of integration to $$u = t - \tau$$ the result follows.

 * Associativity: $$f * (g * h) = (f * g) * h$$ Proof: This follows from using Fubini's theorem (i.e., double integrals can be evaluated as iterated integrals in either order).

 * Distributivity: $$f * (g + h) = (f * g) + (f * h)$$ Proof: This follows from linearity of the integral.

 * Associativity with scalar multiplication: $$a (f * g) = (a f) * g$$ for any real (or complex) number $$a$$.

 * 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, $$f * \delta = f$$ where δ is the delta distribution.

 * Inverse element: Some distributions S have an inverse element S−1 for the convolution which then must satisfy $$S^{-1} * S = \delta$$ 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: $$\overline{f * g} = \overline{f} * \overline{g}$$

 * Time reversal: If  $$q(t) = r(t)*s(t),$$  then  $$q(-t) = r(-t)*s(-t).$$

Proof (using convolution theorem):

$$q(t) \ \stackrel{\mathcal{F}}{\Longleftrightarrow}\ \ Q(f) = R(f)S(f)$$

$$q(-t) \ \stackrel{\mathcal{F}}{\Longleftrightarrow}\ \ Q(-f) = R(-f)S(-f)$$

$$ \begin{align} q(-t) &= \mathcal{F}^{-1}\bigg\{R(-f)S(-f)\bigg\}\\ &= \mathcal{F}^{-1}\bigg\{R (-f)\bigg\} * \mathcal{F}^{-1}\bigg\{S(-f)\bigg\}\\ &= r(-t) * s(-t) \end{align} $$
 * Relationship with differentiation: $$(f * g)' = f' * g = f * g'$$ Proof:

\begin{align} (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{align} $$

 * Relationship with integration: If $F(t) = \int^t_{-\infty} f(\tau) d\tau,$ and $G(t) = \int^t_{-\infty} g(\tau) \, d\tau,$  then $$(F * g)(t) = (f * G)(t) = \int^t_{-\infty}(f * g)(\tau)\,d\tau.$$

Integration
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:
 * $$\int_{\mathbf{R}^d}(f * g)(x) \, dx=\left(\int_{\mathbf{R}^d}f(x) \, dx\right) \left(\int_{\mathbf{R}^d}g(x) \, dx\right).$$

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.

Differentiation
In the one-variable case,
 * $$\frac{d}{dx}(f * g) = \frac{df}{dx} * g = f * \frac{dg}{dx}$$

where $$\frac{d}{dx}$$ is the derivative. More generally, in the case of functions of several variables, an analogous formula holds with the partial derivative:
 * $$\frac{\partial}{\partial x_i}(f * g) = \frac{\partial f}{\partial x_i} * g = f * \frac{\partial g}{\partial x_i}.$$

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 for example under the 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,
 * $$\frac{d}{dx}(f* g) = \frac{df}{dx} * g.$$

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:
 * $$D(f * g) = (Df) * g = f * (Dg).$$

Convolution theorem
The convolution theorem states that
 * $$ \mathcal{F}\{f * g\} = \mathcal{F}\{f\}\cdot \mathcal{F}\{g\}$$

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

Convolution in other types of transformations
Versions of this theorem also hold for the Laplace transform, two-sided Laplace transform, Z-transform and Mellin transform.

Convolution on matrices
If $$\mathcal W$$ is the Fourier transform matrix, then
 * $$\mathcal W\left(C^{(1)}x \ast C^{(2)}y\right) = \left(\mathcal W C^{(1)} \bull \mathcal W C^{(2)}\right)(x \otimes y) = \mathcal W C^{(1)}x \circ \mathcal W C^{(2)}y$$,

where $$ \bull $$ is face-splitting product,    $$ \otimes $$ denotes Kronecker product, $$ \circ $$ denotes Hadamard product (this result is an evolving of count sketch properties ).

This can be generalized for appropriate matrices $$\mathbf{A},\mathbf{B}$$:

 * $$\mathcal W\left((\mathbf{A}x) \ast (\mathbf{B}y)\right) = \left((\mathcal W \mathbf{A}) \bull (\mathcal W \mathbf{B})\right)(x \otimes y) = (\mathcal W \mathbf{A}x) \circ (\mathcal W \mathbf{B}y)$$

from the properties of the face-splitting product.

Translational equivariance
The convolution commutes with translations, meaning that
 * $$\tau_x (f * g) = (\tau_x f) * g = f * (\tau_x g)$$

where τxf is the translation of the function f by x defined by
 * $$(\tau_x f)(y) = f(y - x).$$

If f is a Schwartz function, then τxf is the convolution with a translated Dirac delta function τxf = f ∗ τx 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(τxf) = τx(Sf) for all x. Then S is given as convolution with a function (or distribution) gS; that is Sf = gS ∗ 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 gS 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 Lp 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.

Convolutions on groups
If G is a suitable group endowed with a measure λ, and if f and g are real or complex valued integrable functions on G, then we can define their convolution by

 * $$(f * g)(x) = \int_G f(y) g\left(y^{-1}x\right)\,d\lambda(y).$$

It is not commutative in general. In typical cases of interest G is a locally compact Hausdorff topological group and λ is a (left-) Haar measure. In that case, unless G is unimodular, the convolution defined in this way is not the same as $\int f\left(xy^{-1}\right)g(y) \, d\lambda(y)$. The preference of one over the other is made so that convolution with a fixed function g commutes with left translation in the group:

 * $$L_h(f* g) = (L_hf)* g.$$

Furthermore, the convention is also required for consistency with the definition of the convolution of measures given below. However, with a right instead of a left Haar measure, the latter integral is preferred over the former.

On locally compact abelian groups, a version of the convolution theorem holds: the Fourier transform of a convolution is the pointwise product of the Fourier transforms. The circle group T with the Lebesgue measure is an immediate example. For a fixed g in L1(T), we have the following familiar operator acting on the Hilbert space L2(T):

 * $$T {f}(x) = \frac{1}{2 \pi} \int_{\mathbf{T}} {f}(y) g( x - y) \, dy.$$

The operator T is compact. A direct calculation shows that its adjoint T*  is convolution with

 * $$\bar{g}(-y).$$

By the commutativity property cited above, T is normal: T* T = TT*. Also, T commutes with the translation operators. Consider the family S of operators consisting of all such convolutions and the translation operators. Then S is a commuting family of normal operators. According to spectral theory, there exists an orthonormal basis {hk} that simultaneously diagonalizes S. This characterizes convolutions on the circle. Specifically, we have

 * $$h_k (x) = e^{ikx}, \quad k \in \mathbb{Z},\;$$

which are precisely the characters of T. Each convolution is a compact multiplication operator in this basis. This can be viewed as a version of the convolution theorem discussed above.

A discrete example is a finite cyclic group of order n. Convolution operators are here represented by circulant matrices, and can be diagonalized by the discrete Fourier transform.

A similar result holds for compact groups (not necessarily abelian): the matrix coefficients of finite-dimensional unitary representations form an orthonormal basis in L2 by the Peter–Weyl theorem, and an analog of the convolution theorem continues to hold, along with many other aspects of harmonic analysis that depend on the Fourier transform.

Convolution of measures
Let G be a (multiplicatively written) topological group. If μ and ν are finite Borel measures on G, then their convolution μ∗ν is defined as the pushforward measure of the group action and can be written as
 * $$(\mu * \nu)(E) = \iint 1_E(xy) \,d\mu(x) \,d\nu(y)$$

for each measurable subset E of G. The convolution is also a finite measure, whose total variation satisfies
 * $$\|\mu * \nu\| \le \left\|\mu\right\| \left\|\nu\right\|.$$

In the case when G is locally compact with (left-)Haar measure λ, and μ and ν are absolutely continuous with respect to a λ, so that each has a density function, then the convolution μ∗ν is also absolutely continuous, and its density function is just the convolution of the two separate density functions.

If μ and ν are probability measures on the topological group (R,+), then the convolution μ∗ν is the probability distribution of the sum X + Y of two independent random variables X and Y whose respective distributions are μ and ν.

Infimal convolution
In convex analysis, the infimal convolution of proper (not identically $$+\infty$$) convex functions $$f_1,\dots,f_m$$ on $$\mathbb R^n$$ is defined by: $$(f_1*\cdots*f_m)(x)=\inf_x \{ f_1(x_1)+\cdots+f_m(x_m) | x_1+\cdots+x_m = x\}.$$ It can be shown that the infimal convolution of convex functions is convex. Furthermore, it satisfies an identity analogous to that of the Fourier transform of a traditional convolution, with the role of the Fourier transform is played instead by the Legendre transform: $$\varphi^*(x) = \sup_y ( x\cdot y - \varphi(y)).$$ We have: $$(f_1*\cdots *f_m)^*(x) = f_1^*(x) + \cdots + f_m^*(x).$$

Bialgebras
Let (X, Δ, ∇, ε, η) be a bialgebra with comultiplication Δ, multiplication ∇, unit η, and counit ε. The convolution is a product defined on the endomorphism algebra End(X) as follows. Let φ, ψ ∈ End(X), that is, φ, ψ: X → X are functions that respect all algebraic structure of X, then the convolution φ∗ψ is defined as the composition

 * $$X \mathrel{\xrightarrow{\Delta}} X \otimes X \mathrel{\xrightarrow{\phi\otimes\psi}} X \otimes X \mathrel{\xrightarrow{\nabla}} X.$$

The convolution appears notably in the definition of Hopf algebras. A bialgebra is a Hopf algebra if and only if it has an antipode: an endomorphism S such that
 * $$S * \operatorname{id}_X = \operatorname{id}_X * S = \eta\circ\varepsilon.$$

Applications

Convolution and related operations are found in many applications in science, engineering and mathematics.
 * Convolutional neural networks apply multiple cascaded convolution kernels with applications in machine vision and artificial intelligence. Though these are actually cross-correlations rather than convolutions in most cases.
 * In non-neural-network-based image processing
 * In digital image processing convolutional filtering plays an important role in many important algorithms in edge detection and related processes (see Kernel (image processing))
 * In optics, an out-of-focus photograph is a convolution of the sharp image with a lens function. The photographic term for this is bokeh.
 * In image processing applications such as adding blurring.
 * In digital data processing
 * In analytical chemistry, Savitzky–Golay smoothing filters are used for the analysis of spectroscopic data. They can improve signal-to-noise ratio with minimal distortion of the spectra
 * In statistics, a weighted moving average is a convolution.
 * In acoustics, reverberation is the convolution of the original sound with echoes from objects surrounding the sound source.
 * In digital signal processing, convolution is used to map the impulse response of a real room on a digital audio signal.
 * In electronic music convolution is the imposition of a spectral or rhythmic structure on a sound. Often this envelope or structure is taken from another sound. The convolution of two signals is the filtering of one through the other.
 * In electrical engineering, the convolution of one function (the input signal) with a second function (the impulse response) gives the output of a linear time-invariant system (LTI). At any given moment, the output is an accumulated effect of all the prior values of the input function, with the most recent values typically having the most influence (expressed as a multiplicative factor). The impulse response function provides that factor as a function of the elapsed time since each input value occurred.
 * In physics, wherever there is a linear system with a "superposition principle", a convolution operation makes an appearance. For instance, in spectroscopy line broadening due to the Doppler effect on its own gives a Gaussian spectral line shape and collision broadening alone gives a Lorentzian line shape. When both effects are operative, the line shape is a convolution of Gaussian and Lorentzian, a Voigt function.
 * In time-resolved fluorescence spectroscopy, the excitation signal can be treated as a chain of delta pulses, and the measured fluorescence is a sum of exponential decays from each delta pulse.
 * In computational fluid dynamics, the large eddy simulation (LES) turbulence model uses the convolution operation to lower the range of length scales necessary in computation thereby reducing computational cost.
 * In probability theory, the probability distribution of the sum of two independent random variables is the convolution of their individual distributions.
 * In kernel density estimation, a distribution is estimated from sample points by convolution with a kernel, such as an isotropic Gaussian.
 * In radiotherapy treatment planning systems, most part of all modern codes of calculation applies a convolution-superposition algorithm.
 * In structural reliability, the reliability index can be defined based on the convolution theorem.
 * The definition of reliability index for limit state functions with nonnormal distributions can be established corresponding to the joint distribution function. In fact, the joint distribution function can be obtained using the convolution theory.
 * In Smoothed-particle hydrodynamics, simulations of fluid dynamics are calculated using particles, each with surrounding kernels. For any given particle $$i$$, some physical quantity $$A_i$$ is calculated as a convolution of $$A_j$$ with a weighting function, where $$j$$ denotes the neighbors of particle $$i$$: those that are located within its kernel. The convolution is approximated as a summation over each neighbor.
 * In Fractional calculus convolution is instrumental in various definitions of fractional integral and fractional derivative.

title: Convolution


text:
Calico

Calico (in British usage since 1505) is a heavy plain-woven textile made from unbleached, and often not fully processed, cotton. It may also contain unseparated husk parts. The fabric is far coarser than muslin, but less coarse and thick than canvas or denim. However, it is still very cheap owing to its unfinished and undyed appearance.

The fabric was originally from the city of Calicut in southwestern India. It was made by the traditional weavers called cāliyans. The raw fabric was dyed and printed in bright hues, and calico prints became popular in Europe.

Origins
Calico originated in Calicut, from which the name of the textile came, in South India, now Kerala, during the 11th century, where the cloth was known as "chaliyan". It was mentioned in Indian literature by the 12th century when the polymath and writer Hemachandra described calico fabric prints with a lotus design. Calico was woven using Gujarati cotton from Surat for both the warp and weft. By the 15th century, calico from Gujarat made its appearance in Cairo, then capital of the Egypt Eyalet under the Ottoman Empire. Trade with Europe followed from the 17th century onwards.

Politics of cotton in the British Empire
In the 18th century, England was famous for its woollen and worsted cloth. That industry, centered in the east and south in towns such as Norwich, jealously protected their product. Cotton processing was tiny: in 1701, only 1,985,868 lb of cottonwool was imported into England, and by 1730 this had fallen to 1,545,472 lb. This was due to commercial legislation to protect the woollen industry. Cheap calico prints, imported by the East India Company from Hindustān (India), had become popular. In 1700 an Act of Parliament passed to prevent the importation of dyed or printed calicoes from India, China or Persia. This caused demand to switch to imported grey cloth instead—calico that had not been finished—dyed or printed. These were printed with popular patterns in southern England. Also, Lancashire businessmen produced grey cloth with linen warp and cotton weft, known as fustian, which they sent to London for finishing. Cottonwool imports recovered though, and by 1720 were almost back to their 1701 levels. Coventry woollen manufacturers claimed that the imports were taking jobs away from their workers. The Woollen, etc., Manufactures Act 1720 was passed, enacting fines against anyone caught wearing printed or stained calico muslins, but neckcloths and fustians were exempted. The Lancashire manufacturers exploited this exemption; coloured cotton weft with linen warp were specifically permitted by the 1736 Manchester Act.

In 1764, 3,870,392 lb of cottonwool was imported. This change in consumption patterns, as a result of the restriction on imported finished goods, was a key part of the process that reduced the Indian economy from sophisticated textile production to the mere supply of raw materials. These events occurred under colonial rule, which started after 1757, and were described by Nehru and also some more recent scholars as "de-industrialization".

Calico printing

Early Indian chintz, that is, glazed calico with a large floral pattern, was primarily produced using painting techniques. Later, the hues were applied by wooden blocks, and the cloth manufacturers in Britain printed calico using wooden block printing. Calico printers at work are depicted in one of the stained glass windows made by Stephen Adam for the Maryhill Burgh Halls, Glasgow. Confusingly, linen and silk printed this way were known as linen calicoes and silk calicoes. Early European calicoes (1680) were cheap plain weave white cotton fabric, or cream or unbleached cotton, with a design block-printed using a single alizarin dye fixed with two mordants, giving a red and black pattern. Polychromatic prints were possible, using two sets of blocks and an additional blue dye. The Indian taste was for dark printed backgrounds, while the European market preferred a pattern on a cream base. As the century progressed the European preference moved from the large chintz patterns to smaller, tighter patterns.

Thomas Bell patented a printing technique in 1783 that used copper rollers. In 1785, Livesey, Hargreaves and Company put the first machine that used this technique into operation in Walton-le-Dale, Lancashire. The production volume for printed cloth in Lancashire in 1750 was estimated at 50,000 pieces of 30 yd; in 1850, it was 20,000,000 pieces. The commercial method of calico printing using engraved rollers was invented in 1821 in New Mills, Derbyshire, in the United Kingdom. John Potts of Potts, Oliver and Potts used a copper-engraved master to produce rollers to transfer the inks. After 1888, block printing was only used for short-run specialized jobs. After 1880, profits from printing fell due to overcapacity and the firms started to form combines. In the first, three Scottish firms formed the United Turkey Red Co. Ltd in 1897, and the second, in 1899, was the much larger Calico Printers' Association 46 printing concerns and 13 merchants combined, representing 85% of the British printing capacity. Some of this capacity was removed and in 1901 Calico had 48% of the printing trade. In 1916, they and the other printers formed and joined a trade association, which then set minimum prices for each 'price section' of the industry.

The trade association remained in operation until 1954, when the arrangement was challenged by the government Monopolies Commission. Over the intervening period much trade had been lost overseas.

Terminology
In the UK, Australia and New Zealand:
 * Calico – simple, cheap equal weft and warp plain weave fabric in white, cream or unbleached cotton
 * Calico bag - a bag made of calico used by banks and other financial institutions
 * Muslin – a very fine, light plain weave cotton fabric
 * Muslin gauze – US: muslin – simple, cheap equal weft and warp plain weave fabric in white, cream or unbleached cotton and/or a very fine, light plain weave cotton fabric
 * Gauze – extremely soft and fine cotton fabric with a very open plain weave
 * Cheesecloth – US: gauze – any very light fabric, generally with a plain weave
 * Tote bag - sometimes made of calico

In the US:
 * Calico – cotton fabric with a small, all-over floral print
 * Muslin – UK: muslin gauze – simple, cheap equal weft and warp plain weave fabric in white, cream or unbleached cotton and/or a very fine, light plain weave cotton fabric
 * Muslin gauze – the very lightest, most open weave of muslin
 * Gauze – UK: cheesecloth – any very light fabric, generally with a plain weave
 * Cheesecloth – extremely soft and fine cotton fabric with a very open plain weave

Printed calico was imported into the United States from Lancashire in the 1780s, and here a linguistic separation occurred. While Europe maintained the word calico for the fabric, in the States it was used to refer to the printed design.

These colourful, small-patterned printed fabrics gave rise to the use of the word calico to describe a cat coat colour: calico cat. The patterned fabric also gave its name to two species of North American crabs; see Ovalipes ocellatus.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Calico


text:
Calorimetry

In chemistry and thermodynamics, calorimetry is the science or act of measuring changes in state variables of a body for the purpose of deriving the heat transfer associated with changes of its state due, for example, to chemical reactions, physical changes, or phase transitions under specified constraints. Calorimetry is performed with a calorimeter. Scottish physician and scientist Joseph Black, who was the first to recognize the distinction between heat and temperature, is said to be the founder of the science of calorimetry.

Indirect calorimetry calculates heat that living organisms produce by measuring either their production of carbon dioxide and nitrogen waste (frequently ammonia in aquatic organisms, or urea in terrestrial ones), or from their consumption of oxygen. Lavoisier noted in 1780 that heat production can be predicted from oxygen consumption this way, using multiple regression. The dynamic energy budget theory explains why this procedure is correct. Heat generated by living organisms may also be measured by direct calorimetry, in which the entire organism is placed inside the calorimeter for the measurement.

A widely used modern instrument is the differential scanning calorimeter, a device which allows thermal data to be obtained on small amounts of material. It involves heating the sample at a controlled rate and recording the heat flow either into or from the specimen.

Basic classical calculation with respect to volume
Calorimetry requires that a reference material that changes temperature have known definite thermal constitutive properties. The classical rule, recognized by Clausius and Kelvin, is that the pressure exerted by the calorimetric material is fully and rapidly determined solely by its temperature and volume; this rule is for changes that do not involve phase change, such as melting of ice. There are many materials that do not comply with this rule, and for them, the present formula of classical calorimetry does not provide an adequate account. Here the classical rule is assumed to hold for the calorimetric material being used, and the propositions are mathematically written:

The thermal response of the calorimetric material is fully described by its pressure $$p\ $$ as the value of its constitutive function $$p(V,T)\ $$ of just the volume $$V\ $$ and the temperature $$T\ $$. All increments are here required to be very small. This calculation refers to a domain of volume and temperature of the body in which no phase change occurs, and there is only one phase present. An important assumption here is continuity of property relations. A different analysis is needed for phase change

When a small increment of heat is gained by a calorimetric body, with small increments, $$\delta V\ $$ of its volume, and $$\delta T\ $$ of its temperature, the increment of heat, $$\delta Q\ $$, gained by the body of calorimetric material, is given by

 * $$\delta Q\ =C^{(V)}_T(V,T)\, \delta V\,+\,C^{(T)}_V(V,T)\,\delta T$$

where

 * $$C^{(V)}_T(V,T)\ $$ denotes the latent heat with respect to volume, of the calorimetric material at constant controlled temperature $$T$$. The surroundings' pressure on the material is instrumentally adjusted to impose a chosen volume change, with initial volume $$V\ $$. To determine this latent heat, the volume change is effectively the independently instrumentally varied quantity. This latent heat is not one of the widely used ones, but is of theoretical or conceptual interest.
 * $$C^{(T)}_V(V,T)\ $$ denotes the heat capacity, of the calorimetric material at fixed constant volume $$V\ $$, while the pressure of the material is allowed to vary freely, with initial temperature $$T\ $$. The temperature is forced to change by exposure to a suitable heat bath. It is customary to write $$C^{(T)}_V(V,T)\ $$ simply as $$C_V(V,T)\ $$, or even more briefly as $$C_V\ $$. This latent heat is one of the two widely used ones.

The latent heat with respect to volume is the heat required for unit increment in volume at constant temperature. It can be said to be 'measured along an isotherm', and the pressure the material exerts is allowed to vary freely, according to its constitutive law $$p=p(V,T)\ $$. For a given material, it can have a positive or negative sign or exceptionally it can be zero, and this can depend on the temperature, as it does for water about 4 C.   The concept of latent heat with respect to volume was perhaps first recognized by Joseph Black in 1762. The term 'latent heat of expansion' is also used. The latent heat with respect to volume can also be called the 'latent energy with respect to volume'. For all of these usages of 'latent heat', a more systematic terminology uses 'latent heat capacity'.

The heat capacity at constant volume is the heat required for unit increment in temperature at constant volume. It can be said to be 'measured along an isochor', and again, the pressure the material exerts is allowed to vary freely. It always has a positive sign. This means that for an increase in the temperature of a body without change of its volume, heat must be supplied to it. This is consistent with common experience.

Quantities like $$\delta Q\ $$ are sometimes called 'curve differentials', because they are measured along curves in the $$(V,T)\ $$ surface.

Classical theory for constant-volume (isochoric) calorimetry
Constant-volume calorimetry is calorimetry performed at a constant volume. This involves the use of a constant-volume calorimeter. Heat is still measured by the above-stated principle of calorimetry.

This means that in a suitably constructed calorimeter, called a bomb calorimeter, the increment of volume $$\delta V\ $$ can be made to vanish, $$\delta V=0\ $$. For constant-volume calorimetry:

 * $$\delta Q = C_V \delta T\ $$

where

 * $$\delta T\ $$ denotes the increment in temperature and
 * $$C_V\ $$ denotes the heat capacity at constant volume.

Classical heat calculation with respect to pressure
From the above rule of calculation of heat with respect to volume, there follows one with respect to pressure.

In a process of small increments, $$\delta p\ $$ of its pressure, and $$\delta T\ $$ of its temperature, the increment of heat, $$\delta Q\ $$, gained by the body of calorimetric material, is given by

 * $$\delta Q\ =C^{(p)}_T(p,T)\, \delta p\,+\,C^{(T)}_p(p,T)\,\delta T$$

where

 * $$C^{(p)}_T(p,T)\ $$ denotes the latent heat with respect to pressure, of the calorimetric material at constant temperature, while the volume and pressure of the body are allowed to vary freely, at pressure $$p\ $$ and temperature $$T\ $$;
 * $$C^{(T)}_p(p,T)\ $$ denotes the heat capacity, of the calorimetric material at constant pressure, while the temperature and volume of the body are allowed to vary freely, at pressure $$p\ $$ and temperature $$T\ $$. It is customary to write $$C^{(T)}_p(p,T)\ $$ simply as $$C_p(p,T)\ $$, or even more briefly as $$C_p\ $$.

The new quantities here are related to the previous ones:

 * $$C^{(p)}_T(p,T)=\frac{C^{(V)}_T(V,T)}{\left.\cfrac{\partial p}{\partial V}\right|_{(V,T)}} $$

 * $$C^{(T)}_p(p,T)=C^{(T)}_V(V,T)-C^{(V)}_T(V,T) \frac{\left.\cfrac{\partial p}{\partial T}\right|_{(V,T)}}{\left.\cfrac{\partial p}{\partial V}\right|_{(V,T)}} $$

where
 * $$\left.\frac{\partial p}{\partial V}\right|_{(V,T)}$$ denotes the partial derivative of $$p(V,T)\ $$ with respect to $$V\ $$ evaluated for $$(V,T)\ $$

and

 * $$\left.\frac{\partial p}{\partial T}\right|_{(V,T)}$$ denotes the partial derivative of $$p(V,T)\ $$ with respect to $$T\ $$ evaluated for $$(V,T)\ $$.

The latent heats $$C^{(V)}_T(V,T)\ $$ and $$C^{(p)}_T(p,T)\ $$ are always of opposite sign.

It is common to refer to the ratio of specific heats as

 * $$\gamma(V,T)=\frac{C^{(T)}_p(p,T)}{C^{(T)}_V(V,T)}$$ often just written as $$\gamma=\frac{C_p}{C_V}$$.

Calorimetry through phase change, equation of state shows one jump discontinuity
An early calorimeter was that used by Laplace and Lavoisier, as shown in the figure above. It worked at constant temperature, and at atmospheric pressure. The latent heat involved was then not a latent heat with respect to volume or with respect to pressure, as in the above account for calorimetry without phase change. The latent heat involved in this calorimeter was with respect to phase change, naturally occurring at constant temperature. This kind of calorimeter worked by measurement of mass of water produced by the melting of ice, which is a phase change.

Cumulation of heating
For a time-dependent process of heating of the calorimetric material, defined by a continuous joint progression $$P(t_1,t_2)\ $$ of $$V(t)\ $$ and $$T(t)\ $$, starting at time $$t_1\ $$ and ending at time $$t_2\ $$, there can be calculated an accumulated quantity of heat delivered, $$\Delta Q(P(t_1,t_2))\, $$. This calculation is done by mathematical integration along the progression with respect to time. This is because increments of heat are 'additive'; but this does not mean that heat is a conservative quantity. The idea that heat was a conservative quantity was invented by Lavoisier, and is called the 'caloric theory'; by the middle of the nineteenth century it was recognized as mistaken. Written with the symbol $$\Delta\ $$, the quantity $$\Delta Q(P(t_1,t_2))\, $$ is not at all restricted to be an increment with very small values; this is in contrast with $$\delta Q\ $$.

One can write

 * $$\Delta Q(P(t_1,t_2))\ $$

 * $$=\int_{P(t_1,t_2)} \dot Q(t)dt$$

 * $$=\int_{P(t_1,t_2)} C^{(V)}_T(V,T)\, \dot V(t)\, dt\,+\,\int_{P(t_1,t_2)}C^{(T)}_V(V,T)\,\dot T(t)\,dt $$.

This expression uses quantities such as $$\dot Q(t)\ $$ which are defined in the section below headed 'Mathematical aspects of the above rules'.

Mathematical aspects of the above rules
The use of 'very small' quantities such as $$\delta Q\ $$ is related to the physical requirement for the quantity $$p(V,T)\ $$ to be 'rapidly determined' by $$V\ $$ and $$T\ $$; such 'rapid determination' refers to a physical process. These 'very small' quantities are used in the Leibniz approach to the infinitesimal calculus. The Newton approach uses instead 'fluxions' such as $$\dot V(t) = \left.\frac{dV}{dt}\right|_t$$, which makes it more obvious that $$p(V,T)\ $$ must be 'rapidly determined'.

In terms of fluxions, the above first rule of calculation can be written

 * $$\dot Q(t)\ =C^{(V)}_T(V,T)\, \dot V(t)\,+\,C^{(T)}_V(V,T)\,\dot T(t)$$

where

 * $$t\ $$ denotes the time

 * $$\dot Q(t)\ $$ denotes the time rate of heating of the calorimetric material at time $$t\ $$

 * $$\dot V(t)\ $$ denotes the time rate of change of volume of the calorimetric material at time $$t\ $$

 * $$\dot T(t)\ $$ denotes the time rate of change of temperature of the calorimetric material.

The increment $$\delta Q\ $$ and the fluxion $$\dot Q(t)\ $$ are obtained for a particular time $$t\ $$ that determines the values of the quantities on the righthand sides of the above rules. But this is not a reason to expect that there should exist a mathematical function $$Q(V,T)\ $$. For this reason, the increment $$\delta Q\ $$ is said to be an 'imperfect differential' or an 'inexact differential'. Some books indicate this by writing $$q\ $$ instead of $$\delta Q\ $$. Also, the notation đQ is used in some books. Carelessness about this can lead to error.

The quantity $$\Delta Q(P(t_1,t_2))\ $$ is properly said to be a functional of the continuous joint progression $$P(t_1,t_2)\ $$ of $$V(t)\ $$ and $$T(t)\ $$, but, in the mathematical definition of a function, $$\Delta Q(P(t_1,t_2))\ $$ is not a function of $$(V,T)\ $$. Although the fluxion $$\dot Q(t)\ $$ is defined here as a function of time $$t\ $$, the symbols $$Q\ $$ and $$Q(V,T)\ $$ respectively standing alone are not defined here.

Physical scope of the above rules of calorimetry
The above rules refer only to suitable calorimetric materials. The terms 'rapidly' and 'very small' call for empirical physical checking of the domain of validity of the above rules.

The above rules for the calculation of heat belong to pure calorimetry. They make no reference to thermodynamics, and were mostly understood before the advent of thermodynamics. They are the basis of the 'thermo' contribution to thermodynamics. The 'dynamics' contribution is based on the idea of work, which is not used in the above rules of calculation.

Experimentally conveniently measured coefficients
Empirically, it is convenient to measure properties of calorimetric materials under experimentally controlled conditions.

Pressure increase at constant volume
For measurements at experimentally controlled volume, one can use the assumption, stated above, that the pressure of the body of calorimetric material is can be expressed as a function of its volume and temperature.

For measurement at constant experimentally controlled volume, the isochoric coefficient of pressure rise with temperature, is defined by

 * $$\alpha _V(V,T)\ = \frac{1}{p(V,T)}{\left.\cfrac{\partial p}{\partial V}\right|_{(V,T)}} $$

Expansion at constant pressure
For measurements at experimentally controlled pressure, it is assumed that the volume $$V\ $$ of the body of calorimetric material can be expressed as a function $$V(T,p)\ $$ of its temperature $$T\ $$ and pressure $$p\ $$. This assumption is related to, but is not the same as, the above used assumption that the pressure of the body of calorimetric material is known as a function of its volume and temperature; anomalous behaviour of materials can affect this relation.

The quantity that is conveniently measured at constant experimentally controlled pressure, the isobar volume expansion coefficient, is defined by

 * $$\beta _p(T,p)\ = \frac{1}{V(T,p)}{\left.\cfrac{\partial V}{\partial T}\right|_{(T,p)}}  $$

Compressibility at constant temperature
For measurements at experimentally controlled temperature, it is again assumed that the volume $$V\ $$ of the body of calorimetric material can be expressed as a function $$V(T,p)\ $$ of its temperature $$T\ $$ and pressure $$p\ $$, with the same provisos as mentioned just above.

The quantity that is conveniently measured at constant experimentally controlled temperature, the isothermal compressibility, is defined by

 * $$\kappa _T(T,p)\ = -\frac{1}{V(T,p)}{\left.\cfrac{\partial V}{\partial p}\right|_{(T,p)}} $$

Relation between classical calorimetric quantities
Assuming that the rule $$p=p(V,T)\ $$ is known, one can derive the function of $$\frac{\partial p}{\partial T}\ $$ that is used above in the classical heat calculation with respect to pressure. This function can be found experimentally from the coefficients $$\beta _p(T,p)\ $$ and $$\kappa _T(T,p)\ $$ through the mathematically deducible relation

 * $$\frac{\partial p}{\partial T}=\frac{\beta _p(T,p)}{\kappa _T(T,p)}$$.

Connection between calorimetry and thermodynamics
Thermodynamics developed gradually over the first half of the nineteenth century, building on the above theory of calorimetry which had been worked out before it, and on other discoveries. According to Gislason and Craig (2005): "Most thermodynamic data come from calorimetry..." According to Kondepudi (2008): "Calorimetry is widely used in present day laboratories."

In terms of thermodynamics, the internal energy $$U\ $$ of the calorimetric material can be considered as the value of a function $$U(V,T)\ $$ of $$(V,T)\ $$, with partial derivatives $$\frac{\partial U}{\partial V}\ $$ and $$\frac{\partial U}{\partial T}\ $$.

Then it can be shown that one can write a thermodynamic version of the above calorimetric rules:

 * $$\delta Q\ =\left [p(V,T)\,+\,\left.\frac{\partial U}{\partial V}\right|_{(V,T)}\right ]\, \delta V\,+\,\left.\frac{\partial U}{\partial T}\right|_{(V,T)}\,\delta T$$

with

 * $$C^{(V)}_T(V,T)=p(V,T)\,+\,\left.\frac{\partial U}{\partial V}\right|_{(V,T)}\ $$

and

 * $$C^{(T)}_V(V,T)=\left.\frac{\partial U}{\partial T}\right|_{(V,T)}\ $$.

Again, further in terms of thermodynamics, the internal energy $$U\ $$ of the calorimetric material can sometimes, depending on the calorimetric material, be considered as the value of a function $$U(p,T)\ $$ of $$(p,T)\ $$, with partial derivatives $$\frac{\partial U}{\partial p}\ $$ and $$\frac{\partial U}{\partial T}\ $$, and with $$V\ $$ being expressible as the value of a function $$V(p,T)\ $$ of $$(p,T)\ $$, with partial derivatives $$\frac{\partial V}{\partial p}\ $$ and $$\frac{\partial V}{\partial T}\ $$.

Then, according to Adkins (1975), it can be shown that one can write a further thermodynamic version of the above calorimetric rules:

 * $$\delta Q\ =\left [\left. \frac{\partial U}{\partial p}\right |_{(p,T)}\,+\,p \left.\frac{\partial V}{\partial p}\right |_{(p,T)}\right ]\delta p\,+\,\left [ \left.\frac{\partial U}{\partial T}\right|_{(p,T)}\,+\,p \left.\frac{\partial V}{\partial T}\right |_{(p,T)}\right ]\delta T$$

with

 * $$C^{(p)}_T(p,T)=\left.\frac{\partial U}{\partial p}\right|_{(p,T)}\,+\,p\left.\frac{\partial V}{\partial p}\right|_{(p,T)}\ $$

and

 * $$C^{(T)}_p(p,T)=\left.\frac{\partial U}{\partial T}\right|_{(p,T)}\,+\,p\left.\frac{\partial V}{\partial T}\right|_{(p,T)}\ $$.

Beyond the calorimetric fact noted above that the latent heats $$C^{(V)}_T(V,T)\ $$ and $$C^{(p)}_T(p,T)\ $$ are always of opposite sign, it may be shown, using the thermodynamic concept of work, that also

 * $$C^{(V)}_T(V,T)\,\left.\frac{\partial p}{\partial T}\right|_{(V,T)} \geq 0\,.$$

Special interest of thermodynamics in calorimetry: the isothermal segments of a Carnot cycle
Calorimetry has a special benefit for thermodynamics. It tells about the heat absorbed or emitted in the isothermal segment of a Carnot cycle.

A Carnot cycle is a special kind of cyclic process affecting a body composed of material suitable for use in a heat engine. Such a material is of the kind considered in calorimetry, as noted above, that exerts a pressure that is very rapidly determined just by temperature and volume. Such a body is said to change reversibly. A Carnot cycle consists of four successive stages or segments:

(1) a change in volume from a volume $$V_a\ $$ to a volume $$V_b\ $$ at constant temperature $$T^+\ $$ so as to incur a flow of heat into the body (known as an isothermal change)

(2) a change in volume from $$V_b\ $$ to a volume $$V_c\ $$ at a variable temperature just such as to incur no flow of heat (known as an adiabatic change)

(3) another isothermal change in volume from $$V_c\ $$ to a volume $$V_d\ $$ at constant temperature $$T^-\ $$ such as to incur a flow or heat out of the body and just such as to precisely prepare for the following change

(4) another adiabatic change of volume from $$V_d\ $$ back to $$V_a\ $$ just such as to return the body to its starting temperature $$T^+\ $$.

In isothermal segment (1), the heat that flows into the body is given by

 * $$\Delta Q(V_a,V_b;T^+)\,=\,\,\,\,\,\,\,\,\int_{V_a}^{V_b} C^{(V)}_T(V,T^+)\, dV\ $$

and in isothermal segment (3) the heat that flows out of the body is given by

 * $$-\Delta Q(V_c,V_d;T^-)\,=\,-\int_{V_c}^{V_d} C^{(V)}_T(V,T^-)\, dV\ $$.

Because the segments (2) and (4) are adiabats, no heat flows into or out of the body during them, and consequently the net heat supplied to the body during the cycle is given by

 * $$\Delta Q(V_a,V_b;T^+;V_c,V_d;T^-)\,=\,\Delta Q(V_a,V_b;T^+)\,+\,\Delta Q(V_c,V_d;T^-)\,=\,\int_{V_a}^{V_b} C^{(V)}_T(V,T^+)\, dV\,+\,\int_{V_c}^{V_d} C^{(V)}_T(V,T^-)\, dV\ $$.

This quantity is used by thermodynamics and is related in a special way to the net work done by the body during the Carnot cycle. The net change of the body's internal energy during the Carnot cycle, $$\Delta U(V_a,V_b;T^+;V_c,V_d;T^-)\ $$, is equal to zero, because the material of the working body has the special properties noted above.

Relation of latent heat with respect to volume, and the equation of state
The quantity $$C^{(V)}_T(V,T)\ $$, the latent heat with respect to volume, belongs to classical calorimetry. It accounts for the occurrence of energy transfer by work in a process in which heat is also transferred; the quantity, however, was considered before the relation between heat and work transfers was clarified by the invention of thermodynamics. In the light of thermodynamics, the classical calorimetric quantity is revealed as being tightly linked to the calorimetric material's equation of state $$p=p(V,T)\ $$. Provided that the temperature $$T\, $$ is measured in the thermodynamic absolute scale, the relation is expressed in the formula

 * $$C^{(V)}_T(V,T)=T \left.\frac{\partial p}{\partial T}\right|_{(V,T)}\ $$.

Difference of specific heats
Advanced thermodynamics provides the relation

 * $$C_p(p,T)-C_V(V,T)=\left [p(V,T)\,+\,\left.\frac{\partial U}{\partial V}\right|_{(V,T)}\right ]\, \left.\frac{\partial V}{\partial T}\right|_{(p,T)}$$.

From this, further mathematical and thermodynamic reasoning leads to another relation between classical calorimetric quantities. The difference of specific heats is given by

 * $$C_p(p,T)-C_V(V,T)=\frac{TV\,\beta _p^2(T,p)}{\kappa _T(T,p)}$$.

Practical constant-volume calorimetry (bomb calorimetry) for thermodynamic studies
Constant-volume calorimetry is calorimetry performed at a constant volume. This involves the use of a constant-volume calorimeter.

No work is performed in constant-volume calorimetry, so the heat measured equals the change in internal energy of the system. The heat capacity at constant volume is assumed to be independent of temperature.

Heat is measured by the principle of calorimetry.

 * $$q = C_V \Delta T = \Delta U \,,$$

where

 * ΔU is change in internal energy,
 * ΔT is change in temperature and
 * CV is the heat capacity at constant volume.

In constant-volume calorimetry the pressure is not held constant. If there is a pressure difference between initial and final states, the heat measured needs adjustment to provide the enthalpy change. One then has
 * $$\Delta H = \Delta U + \Delta (PV) = \Delta U + V \Delta P \,,$$

where

 * ΔH is change in enthalpy and
 * V is the unchanging volume of the sample chamber.

title: Calorimetry