# How can shot noise be dependent on dc current, not instantaneous average current?

After looking at the formula of shot noise and its derivation, I thought current in shot noise refer to instantaneous average current, not dc current. By instantaneous average current, I mean current that would have been obtained if not for noises. DC current, as far as my usual understanding goes, refers to dirac delta spike magnitude in zero frequency when current is fourier-transformed, or average of current over time.

Derivations heavily suggest that current being used is instantaneous average current, but all sources I looked at say current in the formula is dc current.

Can anyone clarify what is going on here? For specific context, I am thinking of noise inside MOS transistors.

Edit:

So spectral shot noise density is said to be S(w) = 2q|I_d| where |I_d| is "dc" current and q is electron charge constant. Current I can be I_0 + I_1 cos(w_1 t) + I_2 cos(w_2 t) +.. with I_i being constants. and usually I_0 is said to be dc current. My question is whether I_0 is |I_d| in spectral shot noise density, or I itself is |I_d|.

• Can anyone clarify what is going on here? I think you need to provide more information first. – Andy aka Nov 24 '15 at 17:47
• I clarified a bit more. – Dagaka Mademois Nov 24 '15 at 18:15
• OK cool but can you link to a document that covers this a bit wider please? – Andy aka Nov 24 '15 at 18:32
• – Dagaka Mademois Nov 24 '15 at 18:40
• I've seen the first formula but your links don't seem to cover the 2nd formula. – Andy aka Nov 24 '15 at 18:55

Shot noise is defined by the magnitude of the average value of all current sources, both static and dynamic. You can find the average value of the shot noise current by adding the static current with the average of each of the individual current sinusoids. The total current that may exist is (assuming dynamic sources of current are sinusoids): $$I_0 + I_1cos(\omega_1t) + I_2cos(\omega_2t) + ...+ I_ncos(\omega_nt)$$

Where n is the largest sinusoid you're willing to look at (ideally infinite).

That being said, the current magnitude you're looking for is

$$|I_d| = |I_0 + \frac1T_1\int_{-\frac {T_1}2}^\frac {T_1}2I_1cos(\omega_1t)\ dt + \frac1T_2\int_{-\frac {T_2}2}^\frac {T_2}2I_2cos(\omega_1t)\ dt +...+\frac1T_n\int_{-\frac {T_n}2}^\frac{T_n}2I_ncos(\omega_nt) \ dt| \text{, where} \ T_1, T_2,...T_n \text{ is the period for sinusoid n}$$

More simply,

$$|I_d| = |I_0 + I_{1_{average}} + I_{2_{average}} +...+ I_{n_{average}} |$$

And finally the classical shot noise is,

$$S(\omega) = 2q|I_d|$$

Note that the sinusoids can be noise components, but we can still find the average, or average DC value of the sinusoids, which we did.