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Is there any formula or expansion for

$$a(t)*[b(t) \cdot c(t)]$$

$$a(t) \cdot[b(t)*c(t)]$$

where $*$ denotes the convolution?

By expansion I mean something like $a(t)\cdot[b(t)+c(t)]=a(t)b(t)+a(t)c(t)$.

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    $\begingroup$ Have you tried writing down the convolution integral? Then you'll see it right away (namely that there's not much you can do). $\endgroup$ – Matt L. Jul 2 at 15:08
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    $\begingroup$ In Frequency domain, it will translate to $A\cdot(B*C)$, if that helps any. $\endgroup$ – Florian Jul 2 at 15:44
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    $\begingroup$ @Florian "...translate to $A\cdot(B*C)$...." and then we all can ponder the dual question "Does multiplication distribute over convolution?" $\endgroup$ – Dilip Sarwate Jul 3 at 20:45
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Addition is (by nature) additive, and multiply and convolution are both somehow of multiplicative nature. Thus, I suspect there is no generic distributivity (apart for Boole or Boolean algebras, where the restriction on allowed values shrinks the problem, see below).

However, there is a sort of commutativity of multiply and convolution, as follows. It is interesting to first look at such operator properties from an algebraic point of view. Properties are for instance, for a generic operator binary $\bigcirc$:

  • commutativity: $a \bigcirc b = b \bigcirc a $
  • associativity: $a \bigcirc( b \bigcirc c) = (a \bigcirc b) \bigcirc c $
  • alternativity: $a \bigcirc( a \bigcirc b) = (a \bigcirc a) \bigcirc b $ or $(a \bigcirc b) \bigcirc b = a \bigcirc (b \bigcirc b)$
  • flexibility: $(a \bigcirc b) \bigcirc a = a \bigcirc (b \bigcirc a)$

Distributivity of a binary operator $\diamond$ over operator $\bigcirc$ is more involved: it can be left-distributed: $$a \diamond (b \bigcirc c) = (a \diamond b) \bigcirc (a \diamond c)$$ or right-distributed: $$(a \bigcirc b) \diamond c = (a \diamond c) \bigcirc (b\diamond c)$$

Division is right-distributive over add, not left-distributive. Distributivity of multiply over add is at play in rings and fields. But the converse does not apply. There are mathematical structures where two binary operators distribute over each other, like Boolean or switching algebras. Apart from that, I cannot remember of standard cases of multiply-like $\ast$ or $\cdot$ that would distribute. However, I will details sme specific cases, ending with the Hilbert transform and Bedrosian theorem.

In the case of multiplication and convolution, or $a$ scalar, this is associative, as :

$$a.[b(t)\cdot c(t)] = [a.b(t)]\cdot c(t)= b(t)\cdot [a.c(t)]$$ and $$a.[b(t)\ast c(t)] = [a.b(t)]\ast c(t)= b(t)\ast [a.c(t)]$$

You see that you cannot directly"distribute it", as this would become "bilinear". Convolution and point-wise product are linear. So, if you consider that $c(t)$ (or $b(t)$ by symmetry), there exists a linear operator $\Lambda$ such that:

$$\Lambda(c(t)) = a(t)\ast[b(t)\cdot c(t)] $$

Whether $\Lambda$ has a simple closed form expression is a difficult matter. As a side note, where $\mathcal{H} $ denotes the Hilbert transform, the Bedrosian theorem (A product theorem for Hilbert transforms states that (under some conditions):

$$\mathcal{H}(a(t)e^{i\theta t}) = a(t)(\mathcal{H}(e^{i\theta t}))\,.$$

where $a(t)$ is an (low-pass) envelope, and $e^{i\theta t}$ a modulation. More generally, if a low-pass signal $x_\flat$ and a high-pass $x_\sharp$ have non-overlapping spectra, then under Bedrosian:

$$\mathcal{H}(x_\flat(t)\cdot x_\sharp(t))=x_\flat(t)\cdot \mathcal{H}(x_\sharp(t))$$

Now, let us remind that the Hilbert transform can be seen as the convolution with the distribution ($\operatorname {p.v.}$ is the Cauchy principal value):

$$h(t) =\operatorname {p.v.} {\frac {1}{\pi t}}$$

Thus:

$$h(t) \ast (x_\flat(t)\cdot x_\sharp(t)) = x_\flat(t) \cdot(h(t)\ast x_\sharp(t))$$

This is the only practical case I can remember of a switching of operators in signal processing, more a form of limited commutativity.

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