Additive White Gaussian Noise (AWGN)

Computing noise power within a specified bandwidth

The performance of a digital communication system is quantified by the probability of bit detection errors in the presence of thermal noise. In the context of wireless communications, the main source of thermal noise is addition of random signals arising from the vibration of atoms in the receiver electronics.

It is called additive white Gaussian noise (AWGN) due to the following reasons:

[Additive] The noise is additive, i.e., the received signal is equal to the transmitted signal plus noise. This gives the most widely used equality in communication systems.

(1)   \begin{equation*}                 r = s + w             \end{equation*}

and is shown in Figure below. Remember that the above equation is highly simplified due to neglecting every single imperfection a Tx signal encounters, except the noise itself.

A received signal as an addition of transmitted signal and noise

Moreover, this noise is statistically independent of the signal.

[White] Just like the white color which is composed of all frequencies in the visible spectrum, white noise refers to the idea that it has uniform power across the whole frequency band. As a consequence, the spectral density of white noise is ideally flat for all frequencies ranging from -\infty to +\infty.

Constant spectral density of white noise

[Gaussian] The probability distribution of the noise samples is Gaussian, i.e., in time domain, the samples can acquire both positive and negative values and in addition, the values close to zero have a higher chance of occurrence while the values far away from zero are less likely to appear. This is shown in Figure below.

Gaussian distribution of noise

As a result, the time domain average of a large number of noise samples is zero. We will use this attribute over and over again throughout the system design.

Nyquist investigated the properties of thermal noise and showed that its power spectral density is equal to k \times T, where k is a constant and T is the temperature in Kelvin. As a consequence, the noise power is directly proportional to the equivalent temperature and hence the name thermal noise. Historically, this constant value indicated in noise spectral density Figure is denoted as N_0/2 Watts/Hz.

When we view the constant spectral density as a rectangular sequence (we do not discuss random sequences here, so this discussion is just for a general understanding), its iDFT must be a unit impulse. Furthermore, in the article on correlation, we saw that the iDFT of the spectral density is the auto-correlation function of the signal. Combining these two facts, an implication of a constant spectral density is that the autocorrelation of the noise in time domain is a unit impulse, i.e., it is zero for all non-zero time shifts.

In words, each noise sample in a sequence is uncorrelated with every other noise sample in the same sequence.

In reality, the ideal flat spectrum from -\infty to +\infty is true for frequencies of interest in wireless communications (a few kHz to hundreds of GHz) but not for higher frequencies. Nevertheless, every wireless communication system involves filtering that removes most of the noise energy outside the spectral band occupied by our desired signal. Consequently after filtering, it is not possible to distinguish whether the spectrum was ideally flat or partially flat outside the band of interest. To help in mathematical analysis of the underlying waveforms resulting in closed-form expressions — a holy grail of communication theory — it can be assumed flat before filtering as shown in noise spectral density Figure.

For a discrete signal with sampling rate F_S, the sampling theorem dictates that the bandwidth is constrained by an above mentioned lowpass filter within the range \pm F_S/2 to avoid aliasing. This filter is an ideal lowpass filter with

    \begin{equation*}         H(F) =             \begin{cases}             1 &  -F_S/2 < F < +F_S/2 \\             0 &  \textmd{elsewhere}             \end{cases}     \end{equation*}

The resulting in-band power is shown in red in Figure below, while the rest is filtered out.

Computing noise power within a specified bandwidth

As with all densities, the value N_0 is the amount of noise power P_w per unit bandwidth B.

(2)   \begin{equation*}         N_0 = \frac{P_w}{B}     \end{equation*}

Plugging B=F_S/2 in the above equation, the noise power in a sampled bandlimited system is given as

(3)   \begin{equation*}          P_w = N_0\cdot \frac{F_S}{2}     \end{equation*}

Thus, the noise power is directly proportional to the system bandwidth at the sampling stage.

Fooling the Randomness

As we will see in our discussions about communication systems, a communication signal has a lot of structure in its construction. However, due to Gaussian nature of noise acquiring both positive and negative values, the result of adding a large number of noise-only samples tends to zero.

    \begin{equation*}             \sum _n w[n] \rightarrow 0 \qquad \text{for large} ~ n         \end{equation*}

Therefore, when a noisy signal is received, the Rx exploits this structure through correlating with what is expected. In the process, it estimates various unknown parameters and detects the actual message while averaging over a large number of observations. This correlation brings order out and averaging mitigates the harsh effects of noise.

An AWGN channel is the most basic model of a communication system. Even simple practical systems suffer from various kinds of imperfections in addition to AWGN. Examples of systems operating largely in AWGN conditions are space communications with highly directional antennas and some point-to-point microwave links.

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