Signal to Noise Ratio (SNR)

1.6 Signal to Noise Ratio (SNR)

In communication systems, the signal to noise power ratio or simply signal to noise ratio (SNR) is an import figure of merit. It is very meaningful especially for analog systems, where the fidelity between the transmitted $s (t)$ and received $r (t)$ waveforms is crucial, in a scheme such as:

Tx \to s (t) \to Channel \to r (t) \to Rx .

Conceptually, a SNR is simply the power of the signal of interest divided by the power of the “undesired” signal. This undesired signal (which is often a parcel of the received signal) may be composed of noise, interference and, eventually, a result of channel distortion. Hence, SNR can be very broadly defined as the ratio between the two power values:

SNR \overset{def}{=} \frac{P_{signal of interest}}{P_{undesired signal}} .

(1.10)

Depending on the characteristics of the communication channel, estimating the two signals in Eq. (1.10) or directly estimating their power values may be nontrivial. To simplify the following discussion, unless otherwise stated:

SNR is evaluated at the receiver,
the channels are time-invariant and the random signals are stationary.

It is also useful to assume the received signal $r (t)$ can be written as $r (t) = r_{s} (t) + r_{u} (t)$ , where $r_{s} (t)$ and $r_{u} (t)$ are the parcels that represent the signal of interest and the undesired signal, respectively.

In this specific case, one can write Eq. (1.10) as

SNR \overset{def}{=} \frac{𝔼 [| r_{s} (t) |^{2}]}{𝔼 [| r_{u} (t) |^{2}]} = \frac{𝔼 [| r_{s} (t) |^{2}]}{𝔼 [| r (t) - r_{s} (t) |^{2}]} .

(1.11)

When performing simulations, the user typically has knowledge of both $r_{u} (t)$ and $r_{s} (t)$ such that the SNR can be easily calculated using Eq. (1.11). Another situation that enables using Eq. (1.11) is when $s (t)$ is a training signal known to the receiver, and $r_{s} (t)$ can then be recovered at the receiver given its knowledge about $r (t)$ . However, in practice, estimating the SNR may be nontrivial.

The next paragraphs discuss the SNR definition in the specific context of the AWGN channel, which was introduced in Example C.56.

1.6.1 SNR definition for AWGN

The SNR estimation is simplified when the channel is AWGN. In this case, the signal of interest $r_{s} (t)$ is the transmitted signal $s (t)$ itself and the undesired component $r_{u} (t)$ is WGN $ν (t)$ with power $σ^{2}$ . Therefore, the received signal can be written as $r (t) = s (t) + ν (t)$ .

Hence, the SNR for AWGN is given by

SNR = \frac{𝔼 [| r_{s} (t) |^{2}]}{𝔼 [| r_{u} (t) |^{2}]} = \frac{𝔼 [| s (t) |^{2}]}{𝔼 [| ν (t) |^{2}]} = \frac{𝔼 [| r (t) - ν (t) |^{2}]}{𝔼 [| ν (t) |^{2}]} = \frac{𝔼 [| r (t) |^{2}]}{𝔼 [| ν (t) |^{2}]} - 1 = \frac{P_{r}}{σ^{2}} - 1,

(1.12)

where $P_{r}$ is the power of $r (t)$ . For deriving Eq. (1.12), $r (t)$ and $ν (t)$ were assumed to be uncorrelated such that $𝔼 [| r (t) - ν (t) |^{2}] = 𝔼 [| r (t) |^{2}]] - 𝔼 [| ν (t) |^{2}]$ , as suggested by Application C.7.

Note that Eq. (1.12) requires an estimate of the noise power $σ^{2}$ at the receiver, which may not be trivial.

Note also that scaling the output of the AWGN channel does not alter the SNR. As it will be further detailed in Section 1.7.6, any gain imposed at the AWGN receiver affects both the signal of interest and the noise.

1.6.2 Signal to interference plus noise ratio (SINR) definition

When the received signal is affected not only by noise, but also by “human-made” interference, it is useful to write Eq. (1.10) to account for the interference power $P_{i}$ , composing the signal to interference plus noise ratio (SINR):

SINR \overset{def}{=} \frac{P_{s}}{P_{ν} + P_{i}},

(1.13)

where $P_{s}$ is the power of the signal of interest $r_{s} (t)$ and $P_{ν}$ is the noise (typically WGN) power.

1.6.3 SNR definition for non-dispersive flat-fading channel

The channel model can be made more realistic by incorporating a LTI system. A useful model is to have the transmitted signal convolved with the LTI system impulse response $h_{c} (t)$ and then contaminated by additive noise $ν (t)$ . The noise $ν (t)$ is added at the receiver, after the LTI system, such that the received signal is

r (t) = s (t) * h_{c} (t) + ν (t) .

(1.14)

This channel will be further discussed in Chapter 4 and is depicted in Figure 4.1.

Using Eq. (1.11) is tricky in this case because estimating $r_{s} (t)$ from $r (t)$ given by Eq. (1.14) may be far from trivial due to the effect of $h_{c} (t)$ . Hence, the following simplification is sometimes adopted: the channel impulse response is assumed to be simply $h_{c} (t) = κδ (t)$ , which corresponds to a non-dispersive channel that scales the input by a gain $κ$ . This is a special case of the flat-fading channel depicted in Figure 4.2. The received signal is

r (t) = κs (t) + ν (t)

and its parcel of interest is $r_{s} (t) = κs (t)$ . Note that the gain scales only $s (t)$ , not the noise. Eq. (1.12) can be used in this case or, alternatively, the SNR can be written as

SNR = \frac{𝔼 [| r_{s} (t) |^{2}]}{𝔼 [| ν (t) |^{2}]} = \frac{κ^{2} 𝔼 [| s (t) |^{2}]}{𝔼 [| ν (t) |^{2}]} .

(1.15)

1.6.4 Carrier-to-noise density ratio

When estimating thermal noise power, the receiver bandwidth BW is often unknown or difficult to determine accurately. In such cases, it is convenient to use the carrier-to-noise density ratio ( $C ∕ N_{0}$ ), which represents the SNR normalized to a 1 Hz reference bandwidth. The resulting quantity is expressed in $dB \times Hz$ .

1.6.5 General comments on SNR estimation

Even for relatively simple channel models, it is often not possible to directly apply Eq. (1.11) nor Eq. (1.15). Therefore, there are many SNR estimators that depend especially on the model adopted for the channel and, sometimes, on the signal statistics. Of course, the SNR can be defined in a way that makes easier its estimation. These aspects lead to the existence of distinct definitions of SNR in the telecommunications literature, which is sometimes confusing.

Definitions of SNR may also take into account the adopted bandwidth and the impedance of the component that signals were extracted from. For example, when the noise at the receiver is modeled as white with PSD level $N_{0} ∕ 2$ , knowing the bandwidth $BW$ is required to calculate the noise power $P = N_{0} BW$ . In this case, the adopted $BW$ is typically assumed to be the bandwidth of the signal of interest or signaling bandwidth, given that with a good filtering strategy, the out-of-band noise can be eliminated.

The SNR estimation can be “blind” or “aided”. For example, an “aided-estimation” may rely on pre-specified pilot or training signals such that the receiver knows the transmitted symbols and, consequently, can try generating a local copy of $s (t)$ . These known signals do not carry information and are simply “overhead”. Hence, there is significant interest on blind SNR estimation methods.