Harmonic components of a signal

[Jncik]

Homework Statement

I have trouble understanding what the components are

suppose we have this discrete time signal

$$x[n] = e^{j \frac{2\pi}{3} n}$$

find the number of harmonic components

Homework Equations

The Attempt at a Solution

in my book it says that we have a set of harmonically related signals, that is signals that have a fundamental frequency that is multiple of $$\omega_{0}$$ where this omega is the fundamental frequency of x[n] in this case

so the set of this signals is $$x_{k}[n] = e^{j k \frac{2\pi}{3} n}$$

now,

for k = 0 we have a constant

for k = +1 and -1 we have the first harmonic component since the frequency is the smae
for k = +2 and -2 we have the second harmonic component
for k = +3 and -3 the third
for k = +4 and -4 the fourth which has frequency equal to the one where k = 0

now I don't understand the questions "how many harmonic components the signal has"

assuming that for k = +4, -4 and k = 0 we have the same frequency we can count this as one

for this reason can I say that we have 4 harmonic components?

the first for k = +1 and -1, the second for k = +2 and -2, the third for k = +3 and -3 and finally the fourth for k = +4 and -4?

last question: would it be the same if I started counting from k = 0, to k = 3,-3?

 

[kurt.physics]

Yes, you can say that there are 4 harmonic components. The fundamental frequency is given by \omega_{0} = \frac{2\pi}{3}. This means that the frequencies of the harmonics are given by \omega_{k} = k\omega_{0}, for k = \pm 1, \pm 2, \pm 3, \pm 4. So for k = +1 and -1, we have the first harmonic component. For k = +2 and -2, we have the second harmonic component, and so on. For k = +4 and -4, we have the fourth harmonic component, which has the same frequency as the fundamental frequency (since 4\omega_{0} = 4\frac{2\pi}{3} = \frac{8\pi}{3} = \frac{2\pi}{3} = \omega_{0}).It wouldn't make a difference if you started counting from k = 0, to k = 3, -3. This would still give you 4 harmonic components, since the frequency of the fourth harmonic component is the same as the fundamental frequency.