# Term for integral multiples of fundamental period?

As we know that a signal has fundamental frequency and integral multiples of fundamental frequency are known as harmonics

I am curious about the term that will be used for integral multiples of fundamental time period?What is that term?

• How about "a whole number of cycles"? – Cedron Dawg Apr 7 '19 at 14:51
• "whole number of periods". BTW, strictly-speaking, any integer multiple of the period of a purely periodic signal can also be called a "period". it's just that if you don't choose the smallest possible period as your fundamental period, then you will have entire sequences of harmonics with zero energy. e.g. if you choose two fundamental periods for $T$, it is still true that $x(t+T)=x(t)$, but all of your even-numbered harmonics will be zero. – robert bristow-johnson Apr 8 '19 at 21:01
• i said that wrong (and cannot edit the comment). all of your odd-numbered harmonics will be zero. – robert bristow-johnson Apr 8 '19 at 21:07
• yeah. in the past, often i would include a bigger equation and unless the "Edit" button is lit, you can't copy the $\LaTeX$. – robert bristow-johnson Apr 9 '19 at 19:31

If $$T$$ is a positive real number such that $$x(t) = x(t+T) ~\text{for all}~t \in \mathbb R\tag{1}$$ for continuous-time signals, or if $$T$$ is a positive integer and $$x[n] = x[n+T]~\text{for all}~n \in \mathbb Z \tag{2}$$ for discrete-time signals, then $$x$$ is said to be periodic with period $$T$$ or to have period $$T$$. A signal $$x$$ that has period $$T$$ also has period $$2T$$, period $$3T$$, $$\ldots$$, but might (or might not) have smaller period such as $$T/2$$ or $$T/3$$ etc. The smallest value of $$T$$ for which $$(1)$$ or $$(2)$$ is called the fundamental period of the signal, and all other periods necessarily are integer multiples of the fundamental period.
So, to answer your question about the name for integer multiples of the fundamental period, they can be called a period of $$x$$ which is perfectly valid, or, if it is necessary to emphasize the fact that $$T$$ is not the fundamental period, one can say more prolixly, "$$T$$ is a period but not the fundamental period of $$x$$." The more one want to nitpick (or strive for clarity and completeness), the more wordy one must be.
Note that if $$T$$ is the fundamental period of a signal $$x$$ and if $$T^\prime = mT$$ (where $$m$$ is an integer greater than $$1$$) is a period, then we can express the continuous-time signal $$x(t)$$ of period $$T^\prime$$ as a Fourier series with fundamental frequency $$f_0^\prime = \frac{1}{T^\prime} = \frac{1}{mT}= \frac{1}{m}f_0$$ where $$f_0 = \frac 1T$$ is the fundamental frequency of $$x(t)$$ of fundamental period $$T$$. $$f_0^\prime$$ is called a subharmonic of $$f_0$$. Notice, however that the Fourier series that we get by regarding $$x(t)$$ as a signal of period $$T^\prime$$ is exactly the same as the Fourier series that we get by regarding $$x(t)$$ as a signal of period $$T$$. In $$x(t) = \sum_{n=-\infty}^\infty c_n \exp(j2\pi nf_0^\prime t) = \sum_{n=-\infty}^\infty c_n \exp(j2\pi \frac nm f_0t)\tag{3}$$ the $$c_n$$'s have value $$0$$ except when $$n$$ is an integer multiple $$km$$ of $$m$$ and so the right side of $$(3)$$ can be expressed as $$x(t) = \sum_{k=-\infty}^\infty c_{km} \exp(j2\pi k f_0t)$$ which is precisely the Fourier series for $$x(t)$$ regarded as a signal of period $$T$$. In short, while we have introduced the concept of subharmonics of $$f_0$$, $$x(t)$$ has no frequency content at these subharmonic frequencies.
A similar result holds for discrete-time signals $$x[n]$$ but I am too lazy to write it all out. Perhaps @OlliNiemitalo will resuscitate his answer and spell it all out for those the benefit of those who are interested.