# Why frequency of a DC signal is chosen as zero? If the period can be anything, isn't it same for frequency?

This is a page from Oppenheim.

We can assume a DC signal as a repetition of 10 strips per 10 seconds/10 strips per 5 seconds. Like that, if we choose different periods, we get different frequencies also. Then why is frequency chosen particularly to be zero?

• See that word "constant?" That means unchanging. No change = no frequency because there's nothing to repeat. – JRE Jun 1 at 16:30
• Are you mistaking DC (zero oscillation) for zero amplitude? And thinking that as a result you can choose any period since as long as it has an amplitude of zero there will always be zero oscillation? I would also disagree that the period of a constant signal is undefined (unless you consider infinite to be undefined). It is infinite which is the same as 0Hz. – DKNguyen Jun 1 at 18:54
• @DKNguyen I guess the question is asking, if we accept this notion of what the period of a constant signal is (that it is periodic with "any positive value of T"), then why can't we use the same argument for frequency. And I think the OP has a point. If we can't accept that frequency is undefined, we should also not accept that the period is undefined. To be logically consistent, either we accept that both are undefined, or not. Your comment is the only answer that addresses this notion of the period being undefined. So I agree, period infinite, frequency zero, problem solved. – auspicious99 Jun 2 at 9:35

If you want to treat this signal as a periodic one, then you can take its Fourier series.

Unlike most other periodic signals, you have free choice of what frequency to consider as its fundamental frequency --- you can calculate the series for any fundamental frequency

But regardless of which fundamental frequency you choose, you'll find that all the terms in the series except for the 0-frequency one have zero magnitude.

• Thank you. I was looking for something mathematical. – Lelouch Yagami Jun 1 at 16:59

Frequential analysis is normally taken in the context of integral transforms with complex exponentials - such as Fourier and Laplace - which can alternatively be explained as infinite sums of trigonometric functions, with varying phases and amplitudes.

The only frequency value $$\\omega\$$ that will allow you to represent a constant with a trigonometric function, such as $$\cos(\omega t)\$$, is the value of $$\w = 0\$$. Using any other frequency value will instead represent an oscillation in time-domain, therefore not a constant.

As you wrote, "Like that, if choose different periods, we get different frequencies also." So, I think the problem is the author's saying that a constant signal is periodic "with period $$\T\$$ for any positive value of $$\T\$$". The more normal way to understand the period of a constant signal is, that it is $$\\infty\$$. Otherwise, it can lead to a confusion regarding the corresponding frequency. By duality, frequency could similarly be any value we choose, and hence undefined, if we allow the period to be any positive value of $$\T\$$.

So, we should define the period of a constant signal to be $$\\infty\$$, and by duality, it follows that the frequency is unambiguously $$\0\$$.

• All the standard textbooks quote the period of a constant signal as undefined and frequency=0. – Lelouch Yagami Jun 3 at 6:05
• So, your question is answered, right? Between defining the period as $\infty$ or undefined, it is a philosophical discussion (like, is $\infty$ a valid value for a period?), but you would now not have an issue with frequency=0? – auspicious99 Jun 3 at 6:34

First the frequency difinition from wikipedia link

https://en.wikipedia.org/wiki/Frequency

Frequency is the number of occurrences of a repeating event per unit of time.

The period is the duration of time of one cycle in a repeating event, so the period is the reciprocal of the frequency.

The relation between the frequency and the period, T of a repeating event or oscillation is given by

f=1/T

--for the DC (real direct current from battery) there is an infinite repeating event (very big period)

so f=1/∞ means that f=0

The fragment you quoted doesn't mean that "the period can be anything". The statement "x is periodic with period T" means that x(t) = x(t+T) for all t. This is not the same statement as "the period of x is T", because being periodic with period T also implies being periodic with period 2T, 3T, etc. (this is easy enough to work out on your own). What we call the period of x is the smallest T for which x is periodic. But if x(t) = x(t+T) regardless of T, because x is a constant function, there is no smallest T to choose. Therefore the period of x is undefined, not arbitrary.

As for why that should correspond to a frequency of zero, consider the family of functions cos(2π f t). With decreasing f, period gets longer and the frequency gets lower. In the limit as f goes to 0, we have cos(0) = 1, which is a constant function. This is a reasonable argument for saying that constant functions have 0 frequency.