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I am doing some research on UWB radars which transmit a pulse given by: $$ g(t) = p(t) cos(2\pi f_{c}t) = A e^{(-\frac{t^{2}}{2\tau ^{2}})} cos(2\pi f_{c}t) $$ In some other papers, the transmitted pulse is also described as a higher order derivative of the Gaussian term $p(t)$, such that: $$ g(t) = \frac{d^{n}}{dt^{n}} \left( Ae^{(-\frac{t^{2}}{2\tau^2})} \right) $$ The Gaussian derivatives can be found recursively, which form the Hermite polynomials. Is there a way of showing how the two are equivalent analytically, i.e. without using graphical methods? I am aware of the fact that Hermite polynomials have asymptotic expansions in terms of a cosine function, but this does not help me for my derivation.

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You may be talking about a sine wave (carrier) of frequency $f_c$ modulated by an Hermite polynomial shape function. If a parameter $1/r$ of your UWB pulse waveform is commensurate with a carrier frequency $f_c$, this setup can also be considered an ultra wide band scenario. But if this is the case, the $\cos(2πf_ct)$ wave and the waves, which higher order Hermite polynomials resemble at low values of their argument, are absolutely unrelated things.

The other possibility is that you are talking about an Hermite polynomial approximation. As you are aware of the Hermite polynomial's asymptotic expansions, you should know that the cosine frequency $f_c$ in your first equation is not arbitrary (it must not be) and you should not write down a cosine of independent frequency $f_c$ in your waveform. The approximation of an n-th order Hermite polynomial is written as $$ \exp(-{\frac {x^{2}}{2}})\cdot H_{n}(x)\sim {\frac {2^{n}}{\sqrt {\pi }}}\Gamma \left({\frac {n+1}{2}}\right)\cos \left(x{\sqrt {2n}}-{\frac {n\pi }{2}}\right) \tag{1} $$ Because $H_n$ has n roots, eq.1 can be valid only in a bounded range of $x$ values. If you are tolerant of what may appear a circular-reasoning logical fallacy, I refer you to an improved approximation of $H_n$ in the Wikipedia article for you to be able to estimate the bound for $x$:

A better approximation, which accounts for the variation in frequency, is given by $$ e^{-{\frac {x^{2}}{2}}}\cdot H_{n}(x) \sim \left({\frac {2n}{e}}\right)^{\frac {n}{2}}{\sqrt {2}}\cos \left(x{\sqrt {2n+1-{\frac {x^{2}}{3}}}}-{\frac {n\pi }{2}}\right)\left(1-{\frac {x^{2}}{2n+1}}\right)^{-{\frac {1}{4}}}. $$

Id est, the eq.1 approximation is valid for $x^2/3 \ll (2n+1)$ .

Taking into account this constraint for $x$, you can prove that the recurrence relation $\widetilde {H_{n}}'(x)=2n\widetilde {H_{n-1}}(x)$ is asymptotically precise for eq.1's functions approximating precise Hermite polynomials $H_{n}, H_{n-1}$. The $\widetilde{H_0}$ approximation expression is a precise Hermite polynomial $H_0$, and one can conclude by induction that eq.1 holds in the indicated range of $x$ values for Hermite polynomials of all orders, with bounds expanding with each transition to higher order polynomials.

For a rigorous derivation of Hermite polynomial approximations, see, for example, The Asymptotic Forms Of The Hermite And Weber Functions By Nathan Schwid, page 358 (20/24), Theorem VIII, and Asymptotic analysis of the Hermite polynomials from their differential-difference equation by Diego Dominici, page 13/21, Theorem 5.

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  • $\begingroup$ Thank you so much for this detailed explanation. I will review this as well as all references you included and come back to you. $\endgroup$ – Charis Hadjipanayi Mar 23 at 13:32

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