Gaussians

Skip to main content

\(\newcommand{\vf}[1]{\mathbf{\boldsymbol{\vec{#1}}}} \renewcommand{\Hat}[1]{\mathbf{\boldsymbol{\hat{#1}}}} \let\VF=\vf \let\HAT=\Hat \newcommand{\Prime}{{}\kern0.5pt'} \newcommand{\PARTIAL}[2]{{\partial^2#1\over\partial#2^2}} \newcommand{\Partial}[2]{{\partial#1\over\partial#2}} \newcommand{\tr}{{\mathrm tr}} \newcommand{\CC}{{\mathbb C}} \newcommand{\HH}{{\mathbb H}} \newcommand{\KK}{{\mathbb K}} \newcommand{\RR}{{\mathbb R}} \newcommand{\HR}{{}^*{\mathbb R}} \renewcommand{\AA}{\vf A} \newcommand{\BB}{\vf B} \newcommand{\CCv}{\vf C} \newcommand{\EE}{\vf E} \newcommand{\FF}{\vf F} \newcommand{\GG}{\vf G} \newcommand{\HHv}{\vf H} \newcommand{\II}{\vf I} \newcommand{\JJ}{\vf J} \newcommand{\KKv}{\vf Kv} \renewcommand{\SS}{\vf S} \renewcommand{\aa}{\VF a} \newcommand{\bb}{\VF b} \newcommand{\ee}{\VF e} \newcommand{\gv}{\VF g} \newcommand{\iv}{\vf\imath} \newcommand{\rr}{\VF r} \newcommand{\rrp}{\rr\Prime} \newcommand{\uu}{\VF u} \newcommand{\vv}{\VF v} \newcommand{\ww}{\VF w} \newcommand{\grad}{\vf\nabla} \newcommand{\zero}{\vf 0} \newcommand{\Ihat}{\Hat I} \newcommand{\Jhat}{\Hat J} \newcommand{\nn}{\Hat n} \newcommand{\NN}{\Hat N} \newcommand{\TT}{\Hat T} \newcommand{\ihat}{\Hat\imath} \newcommand{\jhat}{\Hat\jmath} \newcommand{\khat}{\Hat k} \newcommand{\nhat}{\Hat n} \newcommand{\rhat}{\HAT r} \newcommand{\shat}{\HAT s} \newcommand{\xhat}{\Hat x} \newcommand{\yhat}{\Hat y} \newcommand{\zhat}{\Hat z} \newcommand{\that}{\Hat\theta} \newcommand{\phat}{\Hat\phi} \newcommand{\LL}{\mathcal{L}} \newcommand{\DD}[1]{D_{\textrm{$#1$}}} \newcommand{\bra}[1]{\langle#1|} \newcommand{\ket}[1]{|#1/rangle} \newcommand{\braket}[2]{\langle#1|#2\rangle} \newcommand{\LargeMath}[1]{\hbox{\large$#1$}} \newcommand{\INT}{\LargeMath{\int}} \newcommand{\OINT}{\LargeMath{\oint}} \newcommand{\LINT}{\mathop{\INT}\limits_C} \newcommand{\Int}{\int\limits} \newcommand{\dint}{\mathchoice{\int\!\!\!\int}{\int\!\!\int}{}{}} \newcommand{\tint}{\int\!\!\!\int\!\!\!\int} \newcommand{\DInt}[1]{\int\!\!\!\!\int\limits_{#1~~}} \newcommand{\TInt}[1]{\int\!\!\!\int\limits_{#1}\!\!\!\int} \newcommand{\Bint}{\TInt{B}} \newcommand{\Dint}{\DInt{D}} \newcommand{\Eint}{\TInt{E}} \newcommand{\Lint}{\int\limits_C} \newcommand{\Oint}{\oint\limits_C} \newcommand{\Rint}{\DInt{R}} \newcommand{\Sint}{\int\limits_S} \newcommand{\Item}{\smallskip\item{$\bullet$}} \newcommand{\LeftB}{\vector(-1,-2){25}} \newcommand{\RightB}{\vector(1,-2){25}} \newcommand{\DownB}{\vector(0,-1){60}} \newcommand{\DLeft}{\vector(-1,-1){60}} \newcommand{\DRight}{\vector(1,-1){60}} \newcommand{\Left}{\vector(-1,-1){50}} \newcommand{\Down}{\vector(0,-1){50}} \newcommand{\Right}{\vector(1,-1){50}} \newcommand{\ILeft}{\vector(1,1){50}} \newcommand{\IRight}{\vector(-1,1){50}} \newcommand{\Partials}[3] {\displaystyle{\partial^2#1\over\partial#2\,\partial#3}} \newcommand{\Jacobian}[4]{\frac{\partial(#1,#2)}{\partial(#3,#4)}} \newcommand{\JACOBIAN}[6]{\frac{\partial(#1,#2,#3)}{\partial(#4,#5,#6)}} \newcommand{\ii}{\ihat} \newcommand{\jj}{\jhat} \newcommand{\kk}{\khat} \newcommand{\dS}{dS} \newcommand{\dA}{dA} \newcommand{\dV}{d\tau} \renewcommand{\ii}{\xhat} \renewcommand{\jj}{\yhat} \renewcommand{\kk}{\zhat} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \definecolor{fillinmathshade}{gray}{0.9} \newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}} \)

Section 18.1 Gaussians

A Gaussian is a function of the form

\begin{equation} f(x)=N e^{-\frac{(x-x_0)^2}{2\sigma^2}}.\tag{18.1.1} \end{equation}

Sensemaking 18.1. Derivatives of the Gaussian.

What do the first and second derivatives of the delta function tell you about the shape of the graph?

The sign of the first derivative tells you whether the function is increasing (positive derivative) or decreasing (negative derivative). The sign of the second derivative tells you whether the function is concave up (positive second derivative) or concave down (negative second derivative).

The graph below shows the role of the parameters \(N\text{,}\) \(x_0\text{,}\) and \(\sigma\) on the shape of the graph.

Figure 18.1. The graph of a Gaussian with parameters \(N\text{,}\) \(x_0\text{,}\) and \(\sigma\text{.}\)

Activity 18.2. The relationship between the algebraic form of a Gaussian and its shape..

EXPLAIN the relationship between the algebraic form of the Gaussian function and how the parameters \(N\text{,}\) \(x_0\text{,}\) and \(\sigma\) control the shape of the graph.

\(N\) is an overall multiplicative factor, so increasing (or decreasing) \(N\) increases (or decreases) the overall amplitude (height) of the function.
\(x_0\) appears in the function in the form \(x-x_0\text{,}\) i.e. it appears as a shift in the value of the independent variable, so increasing (or decreasing) \(x_0\) results in a shift of the graph to the right (or left).
\(\sigma\) occurs in the denominator of a fraction, with the shifted independent variable \(x-x_0\) in the numerator. When \(\sigma\) increases (or decreases), the value of the fraction decreases (or increases). This fraction squared appears in anegative exponent, so as the value of the fraction decreases (or increases), the value of the exponential increases (or decreases) which makes the Gaussian wider (or narrower).