# Time Domain Behaviour of Thermal Noise

Let's consider the thermal noise (or Johnson–Nyquist noise): It is white noise, that means that its power spectral density does not depend on frequency.

Now my question is: Which is a typical time domain graph of this noise? From a mathematical point of view, I'd say it is a $$\delta(t)$$, since power spectral density = constant means Fourier Transform = constant, that means signal in time domain = $$\delta(t)$$.

But physically it is difficult for me to understand it. I'd say that thermal noise is a series of random fluctuations, like this one (and this is the behaviour that can be also found on the web):

• If you grab a section of noise and do a Fourier Transform, it will be anything but flat. The autocorrelation of white noise approaches a delta function and it is the transform of the autocorrelation that approaches a flat spectrum. White noise is a mathematical abstraction. Real noise is band limited – Stanley Pawlukiewicz Oct 22 at 12:42
• $\delta(t)$ is not the time-domain representation of thermal noise or white noise if you prefer that as the model. True white noise can never be observed in its pristine unblemished state in nature (or Nature for tha matter) because all observing instruments/recorders/NMR machines etc have finite bandwidth and thus one can at best observe band-lmited white noise whose autocorrelation function is $\sigma^2\operatorname{sinc}\left(\frac{t}{T}\right)$ where $T$ is very small. – Dilip Sarwate Oct 23 at 2:02