# Fundamental frequency

Can someone help me:

I have a signal which consist of three frequencies; for example, $f_1=800\,\text{Hz}$, $f_2=1050\,\text{Hz}$, $f_3=1600\,\text{Hz}$. How can i compute the fundamental frequency?

Also, is there a method called "frequency histogram method" which is used to compute fundamental frequency? Can someone explain it to me?

Thanks.

• the fundamental frequency is gcd of the frequencies – phanitej Jan 18 '15 at 11:04
• Beware that the psychoacoustic fundamental pitch frequency depends on the power of each frequency component and might not be the simple mathematical GCD. 800 plus an out of tune major third might be what is heard. – hotpaw2 Jan 18 '15 at 16:33
• The harmonic product spectrum method of pitch estimation uses wrapped histograms to determine and weight pitch candidates. – hotpaw2 Jan 18 '15 at 16:39
• @hotpaw2: This comes closer to a (slightly flat) 'perfect' fourth. – Matt L. Jan 20 '15 at 11:21

A signal which is the sum of sinusoids with different frequencies $f_i$ is only periodic if

$$f_i=k_if_0,\quad i=1,2,\ldots$$

is satisfied for some $f_0$ and integer values $k_i$. Or, in other words, the frequencies $f_i$ must be rational multiples of each other. In your example, the signal is periodic but its fundamental frequency is not $f_1=800\,\text{Hz}$ because $f_2\neq k_2f_1$ for integer $k_2$. The fundamental frequency is $f_0=50\,\text{Hz}$, even though this frequency does not explicitly occur in the signal. However, note that its period equals $1/f_0$.

(Note: answer edited after Matt L.'s comment below; the definition of fundamental frequency given here was wrong).

For a sum of sinusoids, the fundamental frequency is the greatest common divisor of all the involved frequencies, assuming it exists. In your example, the fundamental is 50 Hz.

When you're dealing with a signal that is the result of adding several periodic signal together, be careful to verify that the result is periodic too. In theory, if you add two period signals and the ratio of their periods is not rational, then the result is not periodic.

I've never heard about the "frequency histogram method". You can, however, calculate the signal's Fourier series and just pick the harmonic closest to zero as your fundamental.

• The fundamental frequency is $f_0=1/T$, where $T$ is the signal's fundamental period. In this case this can't be $800\,\text{Hz}$ because the signal is not periodic with period $1/800$. Please see my answer for more detail. – Matt L. Jan 17 '15 at 21:40
• @MattL., I got confused: I thought for a moment that the fundamental was the first non-zero harmonic. Of course you're right. – MBaz Jan 17 '15 at 22:15

f1=800,f2=1050,f3=1600 taking H.C.F.(highest common factor)=it will be 50 hz. and fundamental period will be 1/50. 1.if you want FUNDAMENTAL FREQUENCY then take HCF. 2.if you want FUNDAMENTAL PERIOD if two or more than two periods are given then just take LCM(lowest common multiple.