How do I determine if the fundamental period $T_{x}$ exists and if so what it is?

Question

I recently came across this

$$ x(t) = \cos(6t) + \sin(8t) + e^{j2t} $$

signal that I want to find the fundamental period $T_{0}$ and fundamental frequency $\omega_{0} = 2\pi f_{0} = \frac{2\pi}{T_{0}}$ of but am having trouble doing so as I don't understand 1) how to determine if the fundamental period even exists and 2) if it exists, what is it?

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Here is my attempt at trying to figure out what the fundamental period of $x(t)$ is assuming that it exists (which is not always the case, but bear with me).

Let

$$a(t) = \cos(6t),\ T_{a}=\frac{2\pi}{6}=\frac{\pi}{3} \rightarrow a(t) = a(t + xT_{a}),\ x \in \mathbb{Z}$$,

$$b(t) = \sin(8t),\ T_{b}=\frac{2\pi}{8}=\frac{\pi}{4} \rightarrow b(t) = b(t + yT_{b}),\ y \in \mathbb{Z}$$,

$$c(t)=e^{j2t},\ T_{c}=\frac{2\pi}{2}=\pi \rightarrow c(t) = c(t + zT_{c}),\ z \in \mathbb{Z}$$

and

$$\begin{align} x(t) &= a(t) + b(t) + c(t) \\ &= a(t + xT_{a}) + b(t + yT_{b}) + c(t + zT_{c}) \end{align}$$.

Now for $x(t)$ to be periodic and have a fundamental period we would have to find a $T_{x}$ such that

$$\begin{align} x(t) &= x(t + T_{x}) \\ &= a(t + T_{x} + xT_{a}) + b(t + T_{x} + yT_{b}) + c(t + T_{x} + zT_{c}) \\ &= a(t + xT_{a}) + b(t + yT_{b}) + c(t + zT_{c})\end{align}$$

because if $T_{x}$ truly is the fundamental period of $x(t)$ then we'll end up right where we started which is the RHS.

Now we see that in order for the rightmost equality to hold we need $T_{x}$ to be a multiple of $T_{a}$, $T_{b}$ and $T_{c}$ which yields the following relationships:

$$T_{x} = xT_{a}$$ $$T_{x} = yT_{b}$$ $$T_{x} = zT_{c}$$ $$T_{x} = xT_{a} = yT_{b} = zT_{c} = \frac{x\pi}{3} = \frac{y\pi}{4} = z\pi$$

Now I am guessing that what remains is figuring out for what $(x,y,z)$ the equality holds and as such I have determined what $T_{x}$ is, but I have no idea how do this or if I actually need to do it. Perhaps I am missing something here towards the end and thinking about it wrong.

The last equation from above is familiar to me because in the case of only two signals being summed up we end up with the common relationship criterion used when wanting to determine if the sum of two periodic signals is periodic too:

$$ aT_{1} = bT_{2} \iff \frac{T_{1}}{T_{2}} = \frac{b}{a},\ a,b \in \mathbb{Z} $$

which basically says that for a fundamental period to exist for the resulting signal when summing up two signals with periods $T_{1}$ and $T_{2}$, the quotient of the periods needs to be rational. That is it can be expressed as the division of two integers. So in the case of summing up two functions I know that I can just do the above test to see if the quotient is rational to determine if a fundamental period exists or not. But now in the case of summing up three signals, I get stuck.

Furthermore in each case (summing up two signals vs three vs general case) I do not know how to actually determine the fundamental period given that I have managed to determine that one does exist.

Answer 1

All (finite-energy) periodic signals can be represented by Fourier series which consist of a sum of harmonics of a fundamental frequency $\omega_0$. Some of the various harmonics as well as the fundamental frequency (a.k.a.first harmonic) might well have zero amplitudes (i.e. be completely absent from the Fourier series representation) e.g. $\cos(2t) + \sin(3t) + \cos(5t)$ is a Fourier series in which the fundamental frequency $\omega_0$ is $1$ but the first harmonic is absent.

Your signal \begin{align} \cos(6t) + \sin(8t) + e^{j2t} &= \frac 12 e^{-j8t} + \frac 12 e^{-j6t} + e^{j2t} +\frac 12 e^{j6t} + \frac 12 e^{j8t} \\ &= \frac 12 e^{-j4\omega_0t} + \frac 12 e^{-j3\omega_0t} + e^{j\omega_0t} +\frac 12 e^{j3\omega_0t} + \frac 12 e^{j4\omega_0t} \end{align} where $\omega_0 = 2$ is indeed a Fourier series (with fundamental present) and so its least period is _____ as per standard Fourier series theory which I am sure that you know already (blank left to be filled by the OP).mmIn short, we have replaced the LCM stuff (which the OP claims to understand but not quite know what to do with) with finding the greatest common divisor (GCD) of the exponents/arguments of the various terms.

Turning to the question of whether a sum such as $e^{j\omega_1 t} + e^{j\omega_2 t}+ \cdots + e^{j\omega_n t}$ is a periodic signal, and if so, what is its fundamental period, see if you can express the sum as a Fourier series with some fundamental frequency (where the first harmonic may well be absent as in $\cos(2t) + \sin(3t) + \cos(5t)$). Note that we could have opted to think of $$\frac 12 e^{-j8t} + \frac 12 e^{-j6t} + e^{j2t} +\frac 12 e^{j6t} + \frac 12 e^{j8t}$$ as a Fourier series with fundamental frequency $\omega_0 = 1$ and a missing first harmonic (effectively ignoring the fact that the GCD of $\{2,6,8\}$ is $2$, not $1$), but this would have led to an incorrect conclusion as to the fundamental period of the function. As a final test of understanding, determine whether $\cos(t) + \sin(2\pi t)$ is a periodic signal or not.

Answer 2

Your almost there. Let's pick it up at $$T_{x} = xT_{a} = yT_{b} = zT_{c} = \frac{x\pi}{3} = \frac{y\pi}{4} = z\pi$$

The key here is that $x,y,z \in \mathbb{I}$ must be integers. Let's pull out $\frac{\pi}{12}$ as a common divisor. We get

$$T_{x} = \frac{\pi}{12}4x = \frac{\pi}{12}3y = \frac{\pi}{12}12z$$. Now the whole thing simplifies to $$T_{x}\frac{12}{\pi} = 4x = 3y = 12z$$

That is indeed an LCM problem, albeit a trivila one. At this point is obvious that the smallest set of $[x,y,z]$ is $[3,4,1]$, so we simply get

$$T_{x}\frac{12}{\pi} = 12 \rightarrow T_x = \pi$$