# White Gaussian Noise Spectrum and Power

White Gaussian noise has constant power spectral density $$N_0/2$$. I know that the area under the power spectral density curve between two points gives the power of the signal between these two points.

1. If I want to know the power of a certain frequency in the signal (not in a range of frequencies), can we say that the power of each frequency in the signal is exactly $$N_0/2$$?

2. The total of power of additive white Gaussian noise is infinity, what does this mean? Is it reasonable to assume that the noise added to the signal have an infinite power?

If I want to know the power of a certain frequency in the signal (not in a range of frequencies), can we say that the power of each frequency in the signal is exactly No/2?

No, BUT: You mean the right thing, you just say it wrongly:

The Power Spectral Density is constant – a single frequency doesn't have any power; it has a "power per bandwidth"! To arrive at a power, you need to integrate the density over a non-zero mass of frequencies.

(This is kind of an important distinction to make – only infinitely long periodic signals, e.g. sine waves, have power at a single frequency; everything else has a "power distributed over frequencies".)

The total of power of additive white Gaussian noise is infinity, what does this mean? Is it reasonable to assume that the noise added to the signal have an infinite power?

Yes. Notice that you're never dealing with a truly white Gaussian noise in continuous-time systems (luckily for the universe, I might add); it's always approximately white for some bandwidth. Everything else is physically impossible – but rarely matters. Example: the thermal noise you can measure over a resistor is the classical example of white Gaussian noise in systems. However, it's not really white – the power density decreases at very high frequencies. But that's totally irrelevant to your observation – your measurement doesn't go into the terahertzes.

In time-discrete systems, things look different: for a sampled time-continuous stochastic signal (noise) to be white, it's sufficient that the original time-continuous signal had a constant PSD over a bandwidth. So, there's no physical problem in the time-continuous world. Since a discrete signal is just a sequence of numbers, there's no concern for "physicality" anyways.

• The power spectral density of AWGN is constant at No/2 or No? What I read is that it is constant at No/2, while the definition of No is the amount of power per unit bandwidth watt/Hz. Why No is divided by two? If we don't divide by two, then the value of the constant PSD is No (Watt/Hz), which when multiplied by the total bandwidth will give the total power. – Noha Dec 8 '20 at 12:05
• I need to understand the following: why we can't define power for a single frequency component, unless there is a sinusoid at that frequency? infinitely long periodic signals, e.g. sine waves, have power at a single frequency, but any signal consists of a range of frequencies in the form of sinusoidal signals. Each sinusoid has a certain duration in the signal. Infinitely long sinusoidal signals have power, and also finite sinusoidal signals have power. – Noha Dec 8 '20 at 12:49
• $N_0$ vs $N_0/2$: complex or real signals; depends on your definition of bandwidth. So, watch out for how the texts define bandwidth. – Marcus Müller Dec 8 '20 at 13:30
• We can define power at a single frequency. It's 0 for white noise. It's because an integral over a single point of a bounded function is always 0, see my comments under Mark's answer. – Marcus Müller Dec 8 '20 at 13:30
• I understand that mathematically, but I can not imagine that. A sinusoid always has power even if finite in duration, and of course every frequency is represented as a sinusoid with certain duration in the signal. – Noha Dec 8 '20 at 14:08

White noise is a conceptual signal more than real world signal.
In the context of estimation it is the signal which can't be estimated based on its past.
In the context of Frequency domain it is the one with constant value (On average) for any of its bins.

Now, for continuous signals, it implies it has infinite energy, hence it is only a mathematical concept. As no such thing in real life.

1. Yes. Indeed.
Remark The way I interpret your question is: "What's the the value of the PSD (Which you refer as power) frequency". The answer to that is that many people dealing with White Noise try to understand if they can intuitively think it is built by infinite sum of Harmonic Signals. Which actually the definition of White Noise: It requires all basis functions in order to build it. Each with the same power (On average).
2. In case you'd see such signal it will indeed have infinite power. Yet you can only encounter Band Limited White Noise which is white within the frequencies it was sampled. See How to Simulate AWGN (Additive White Gaussian Noise) in Communication Systems for Specific Bandwidth.
• 1. Nope, indeed not: at any frequency $f_0$ of white noise, the power is 0, since $$\int\limits_{f_0}^{f_0} G(f)\,\mathrm df=0$$ for all bounded functions $G(f)$, so especially for a constant value PSD $G(f)=N_0/2$. – Marcus Müller Dec 5 '20 at 10:22
• I'll allow myself to disagree there: they specifically say "(not in a range of frequencies)"! – Marcus Müller Dec 5 '20 at 11:29
• @Mark: "... can we say that the power of each frequency in the signal is exactly $𝑁_0/2$?" The only answer to this is really "no", because that question is based on a misunderstanding on what power spectral density means. You can't define power for a single frequency component, unless there is a sinusoid at that frequency, which corresponds to a Dirac delta impulse in the power spectrum. But that is not the case for white noise. – Matt L. Dec 5 '20 at 11:35
• @Mark really, no! The value of the PSD is not a power, and the difference between it being a power density or a power is really important, especially considering how the question was phrased with, referring to power on a single exact frequency. Any single exact frequency of white noise has exactly 0 power! – Marcus Müller Dec 5 '20 at 21:10
• @Mark: The value of the PSD is $N_0/2$ at each frequency, correct. However, this does absolutely not mean that the power at each frequency equals $N_0/2$. No matter how many times you repeat that, it remains completely wrong. – Matt L. Dec 6 '20 at 12:41