# How is frequency defined for a periodic signal?

For sinsusoids, frequency is $$2\pi/T$$, but for general periodic signals, how is frequency defined? Is it $$1/T$$ or $$2\pi/T$$?

($$T$$ is the fundamental period of the signal)

In my world, frequency is inverse to period, i.e., a sinusoid with fundamental period $$T$$ has a frequency $$f = \frac 1T$$. Of course, angular frequencies are also common, in which case we have $$\omega = 2\pi f = \frac{2\pi}{T}$$, I guess this is what you were referring to.

If a signal is periodic but not purely sinusoidal, it does not have "just one" frequency. Instead, it will have a line spectrum, i.e., frequency content at integer multiples of its fundamental frequency. The fundamental frequency is, as above $$\frac 1T$$ if $$T$$ is the fundamental period, $$\frac{2\pi}{T}$$ if you prefer angular frequencies.

To be a bit more precise, if your signal $$x(t)$$ is periodic with period $$T$$ so that $$x(t) = x(t+T) \; \forall t$$, you can expand it into its Fourier series given by $$x(t) = \sum_{\ell = -\infty}^\infty \gamma_\ell {\rm e}^{\jmath \omega_0 \ell t},$$ where $$\omega_0 = \frac{2\pi}{T}$$ is the fundamental frequency and the coefficients $$\gamma_\ell$$ tell you how much energy there is in the different overtones of the fundamental.

*edit: Regarding your request for clarification "As there's "a time-period of a signal", there's no such thing as "a frequency of a signal"?" You might think so, but I guess that's not the answer here. See, if a signal has fundamental period $$T$$, it is also periodic with $$2T$$, $$3T$$, $$4T$$ and to so. This means while we could consider $$\frac 1T$$ its base frequency, we might also consider $$\frac{1}{2T}$$, $$\frac{1}{3T}$$, $$\frac{1}{4T}$$ and so on, right? But here is the thing: this is also true for the pure sinusoid. And yet, its spectrum only has a line at $$\pm \frac{1}{T}$$. Which is the base frequency if you consider the base period, it would be the first overtone if you were to consider twice the base period.

So, that's not what's happening. What's happening is that the spectrum tells us how similar the signal looks to harmonics of certain frequencies. If you're given a harmonic, you just need one to explain the signal. If you're given a non-harmonic but periodic signal, one harmonic is not enough, you'll need the overtones as well.

• As a clarification, let me ask you this: As there's " a time-period of a periodic signal", there's no such thing as " a frequency of a periodic signal"? (signal, here, is in a general context.. non-siniusoids) – Viki Jul 9 '19 at 7:24
• I added a comment regarding your request for clarification. – Florian Jul 9 '19 at 7:31
• Thank you for the clarfication. one last thing: Is angular frequency defined for periodic non-sinusoids? Not in terms of the fourier spectrum but as a single value for the signal. something like "the angular frequency of a signal?" – Viki Jul 9 '19 at 7:46
• As I tried to clarify, a periodic non-sinusoid does not have one (angular) frequency, but multiple. It does have one fundamental (angular) frequency though. – Florian Jul 9 '19 at 8:07
• Thank you for your inputs. But what I guess is that a periodic non-sinusoid has a single frequency but not a single angular frequency. The notion of having a single angular frequency is valid only for the sinusoids (or the cisoids). – Viki Jul 9 '19 at 8:14

For audio signals, a periodic signal can have a single "pitch frequency" (detectable repeat rate, or 1/period), even though it might consist of multiple sinusoidal harmonic frequency elements, and even with a missing fundamental spectral element at the pitch frequency. e.g. the pitch frequency can be different from the dominant spectral frequency or frequencies, in Hertz, of a periodic signal.

However, this usage of the term "frequency" may not be appropriate for general non-audio signals.

• Right! All along I was a bit confused between the pitch frequency and many multiple harmonic frequencies. In the light of what you said at the end, can the pitch frequency be different from the fundamental frequency? – Viki Jul 9 '19 at 19:35