exponential functionperiodic functionssignal processing
I have read that that period of sum/product of two periodic functions is the least common multiple of there individual time periods if the time periods are rationally related but I am not able to understand that for an exponential.
Example : Suppose $6T$ is the period of $e^{it}$ and $f(t) = e^{2it}$ and $g(t) = e^{3it}$ then
$h(t) = f(t)g(t) = e^{5it}$ whose period should be $\large\frac{6T}{5}$ but LCM gives period to be $6T.$
Any clarification?
P.S. : I am studying periodic time signals.
We make a few comments only.
$1.$ Note that $2\pi$ is a period of $\sin x$, or, equivalently, $1$ is a period of $\sin(2\pi x)$.
But $\sin x$ has many other periods, such as $4\pi$, $6\pi$, and so on. However, $\sin x$ has no (positive) period shorter than $2\pi$.
$2.$ If $p$ is a period of $f(x)$, and $H$ is any function, then $p$ is a period of $H(f(x))$. So in particular, $2\pi$ is a period of $\sin^2 x$. However, $\sin^2 x$ has a period which is smaller than $2\pi$, namely $\pi$. Note that $\sin(x+\pi)=-\sin x$, so $\sin^2(x+\pi)=\sin^2 x$. It turns out that $\pi$ is the shortest period of $\sin^2 x$.
$3.$ For sums and products, the general situation is complicated. Let $p$ be a period of $f(x)$ and let $q$ be a period of $g(x)$. Suppose that there are positive integers $a$ and $b$ such that $ap=bq=r$. Then $r$ is a period of $f(x)+g(x)$, and also of $f(x)g(x)$.
So for example, if $f(x)$ has $5\pi$ as a period, and $g(x)$ has $7\pi$ as a period, then $f(x)+g(x)$ and $f(x)g(x)$ each have $35\pi$ as a period. However, even if $5\pi$ is the shortest period of $f(x)$ and $7\pi$ is the shortest period of $g(x)$, the number $35\pi$ need not be the shortest period of $f(x)+g(x)$ or $f(x)g(x)$.
We already had an example of this phenomenon: the shortest period of $\sin x$ is $2\pi$, while the shortest period of $(\sin x)(\sin x)$ is $\pi$. Here is a more dramatic example. Let $f(x)=\sin x$, and $g(x)=-\sin x$. Each function has smallest period $2\pi$. But their sum is the $0$-function, which has every positive number $p$ as a period!
$4.$ If $p$ and $q$ are periods of $f(x)$ and $g(x)$ respectively, then any common multiple of $p$ and $q$ is a period of $H(f(x), g(x))$ for any function $H(u,v)$, in particular when $H$ is addition and when $H$ is multiplication. So the least common multiple of $p$ and $q$, if it exists, is a period of $H(f(x),g(x))$. However, it need not be the smallest period.
$5.$ Periods can exhibit quite strange behaviour. For example, let $f(x)=1$ when $x$ is rational, and let $f(x)=0$ when $x$ is irrational. Then every positive rational $r$ is a period of $f(x)$. In particular, $f(x)$ is periodic but has no shortest period.
$6.$ Quite often, the sum of two periodic functions is not periodic. For example, let $f(x)=\sin x+\cos 2\pi x$. The first term has period $2\pi$, the second has period $1$. The sum is not a period. The problem is that $1$ and $2\pi$ are incommensurable. There do not exist positive integers $a$ and $b$ such that $(a)(1)=(b)(2\pi)$.
It is a composition of two complex exponentials each with it's own fundamental frequency. $$e^{j2\pi f_0n}$$ This way the exponentials would have frequencies $$f_1 = \frac{1}{3}$$ $$f_2 = \frac{3}{8}$$ And because neither of both frequencies can be simplified anymore then the fundamental period is defined to be the denominator of the expressions above so $$ T_1 = 3 $$ $$ T_2 = 8 $$
Best Answer
If $f$ and $g$ have rationally related periods $p_f$ and $p_g$ then the functions $f+g$ or $f\cdot g$ are guaranteed to be periodic with period $\ell:={\rm lcm}(p_f,p_g)$, but the fundamental period of such compositions could as well be ${\ell\over n}$ for some integer $n\geq1$, depending on circumstances.
The exponential functions you are considering are very special functions. In order to find the fundamental period of $f\cdot g$ when $f(t):=e^{2it}$ and $g(t)=e^{3it}$ we have to look at the situation in detail.
The function $t\mapsto e^{it}$ has fundamental period $2\pi$ (and not $=6T$, as you wrote). It follows that $f$ and $g$ have fundamental periods ${2\pi\over2}$ and ${2\pi\over3}$, whose lcm comes to $\ell=2\pi$ again. Now $f(t)\cdot g(t)=e^{5it}$, so that $f\cdot g$ has fundamental period ${2\pi\over5}={\ell\over5}$.