When Normalization is Dangerous

The signal-to-noise ratio (SNR) generally depends on the transmit power, channel gain, and noise power:

Since the spectral efficiency (bit/s/Hz) and many other performance metrics of interest depend on the SNR, and not the individual values of the three parameters, it is a common practice to normalize one or two of the parameters to unity. This habit makes it easier to interpret performance expressions, to select reasonable SNR ranges, and to avoid mistakes in analytical derivations.

There are, however, situations when the absolute value of the transmitted/received signal power matters, and not the relative value with respect to the noise power, as measured by the SNR. In these situations, it is easy to make mistakes if you use normalized parameters. I see this type of errors far too often, both as a reviewer and in published papers. I will give some specific examples below, but I won’t tell you who has made these mistakes, to not point the finger at anyone specifically.

Wireless energy transfer

Electromagnetic radiation can be used to transfer energy to wireless receivers. In such wireless energy transfer, it is the received signal energy that is harvested by the receiver, not the SNR. Since the noise power is extremely small, the SNR is (at least) a billion times larger than the received signal power. Hence, a normalization error can lead to crazy conclusions, such as being able to transfer energy at a rate of 1 W instead of 1 nW. The former is enough to keep a wireless transceiver on continuously, while the latter requires you to harvest energy for a long time period before you can turn the transceiver on for a brief moment.

Energy efficiency

The energy efficiency (EE) of a wireless transmission is measured in bit/Joule. The EE is computed as the ratio between the data rate (bit/s) and the power consumption (Watt=Joule/s). While the data rate depends on the SNR, the power consumption does not. The same SNR value can be achieved over a long propagation distance by using high transmit power or over a short distance by using a low transmit power. The EE will be widely different in these cases. If a “normalized transmit power” is used instead of the actual transmit power when computing the EE, one can get EEs that are one million times smaller than they should be. As a rule-of-thumb, if you compute things correctly, you will get EE numbers in the range of 10 kbit/Joule to 10 Mbit/Joule.

Noise power depends on the bandwidth

The noise power is proportional to the communication bandwidth. When working with a normalized noise power, it is easy to forget that a given SNR value only applies for one particular value of the bandwidth.

Some papers normalize the noise variance and channel gain, but then make the SNR equal to the unnormalized transmit power (measured in W). This may greatly overestimate the SNR, but the achievable rates might still be in the reasonable range if you operate the system in an interference-limited regime.

Some papers contain an alternative EE definition where the spectral efficiency (bit/s/Hz) is divided by the power consumption (Joule/s). This leads to the alternative EE unit bit/Joule/Hz. This definition is not formally wrong, but gives the misleading impression that one can multiply the EE value with any choice of bandwidth to get the desired number of bit/Joule. That is not the case since the SNR only holds for one particular value of the bandwidth.

Knowing when to normalize

In summary, even if it is convenient to normalize system parameters in wireless communications, you should only do it if you understand when normalization is possible and when it is not. Otherwise, you can make embarrassing mistakes, such as submitting a paper where the results are six orders of magnitude wrong. And, unfortunately, there are several such papers that have been published and these create a bad circle by tricking others into making the same mistakes.

7 thoughts on “When Normalization is Dangerous”

  1. This blog post is so important for me. Thank you very much!

    In my paper, if I use a normalized noise power and still use the sum spectral efficiency divided by the transmit power, how can I define or name this expression to make it correct. Whether the expression (9) without unit in paper [1] is worth learning.

    [1] Ngo H Q, Matthaiou M, Larsson E G. “Massive MIMO With Optimal Power and Training Duration Allocation.” IEEE Wireless Communications Letters, 2014, 3(6): 605-608.

    1. Expression in (9) represents the energy per bit. Such a quantity should have the unit Joule/bit, and for that to happen the power should be measured in J/s/Hz or J/channel use and the rate in bit/s/Hz or bit/channel use. I don’t recommend computing such a quantity unless the result gets this unit.

  2. Thanks a lot for the nice post and very useful information. I have a question about SNR, and would be so thankful if you could please kindly guide me. Let N be the number of antennas.
    When using maximum ratio combining, I know that the signal components add “coherently” whereas the noise components add “incoherently.” Hence, the signal amplitude grows with N whereas the noise increases with sqrt(N), so SNR improves with N. My questions are:
    1. What if the variance of the signal components is different? How does SNR increase by increasing N?
    2. How does the “correlation” affect this relationship? (I mean the case when the noise components are correlated with signal components.)

    Thanks a lot in advance for your kind help with this.

    1. 1. You will get the sum of the signal’s variances.

      2. It depends on how the correlation between the noise and signal is defined, but this is basically what happens under pilot contamination; the noise will grow at the same speed as the signal term.

  3. Very valuable post.

    I wonder how the SNRs in different systems such as SISO, MISO/SIMO, MIMO, … increase with the number of antennas? Let there be M transmit antennas and N receivers are present. Then how can we define SNR at one receiver, the overall received SNR and how does the SNR increase with the number of antennas?

    1. In a multiantenna system you can define the SNR in different ways. The basic definition is the SNR per antenna that would be obtained in an equivalent single-antenna system. After precoding, the resulting SNR will typically be M times larger. But if there are multiple receivers that should get independent signals, you will have to divide the power between these receivers. That reduces the SNR of each transmission.

      I recommend you to read the first chapter of my book “Massive MIMO Networks” which describe these things in detail: https://www.nowpublishers.com/article/DownloadSummary/SIG-093

Leave a Reply

Your email address will not be published. Required fields are marked *