Circuit element models
The concept of effective noise temperature is a convenient model that allows the internal noise of a circuit element to be represented by the designer as an input source of noise temperature to an idealized version of that circuit.

Figure 2a below illustrates this concept applied to an amplifier or an attenuator and summarizes the well-known relationships:

EQ1:
EQ2:

where TR and TL are the effective temperature of an amplifier (receiver) and attenuator (lossy line), respectively. Also, F and L represent noise figure and loss factor, respectively. Figure 2b shows the model being applied to a pair of cascaded circuit elements (a lossy line plus an amplifier), where the gain of the lossy line can be expressed as 1/L.

The resulting composite noise temperature, T_comp, is expressed as:

EQ3:

Measurement points
Whenever an SNR measurement is made at some location in the receiver, then T is represented by the local noise temperature (T_local) at that location. As shown, T_local (whose effect can be measured at a chosen observation or reference point) accounts for the source noise power (Figure 3). The effect of a load is ignored since it cancels out in the computation of SNR.

Measurement locations
Fig. 3 shows three measurements of T_local and P_r/N₀ at points A, B and C within the receiving system. Use of the superscripts A, B and C with any parameters indicates that measurements are made at these respective locations. For each of these points we can describe the following relationships:

Point A: See equations 4a and 4b below. Where P_r^A is received signal power (stemming from a waveform measurement at point A), N₀^A is noise-power spectral density as inferred from a measurement at point A and T_Ant is the antenna temperature (the source noise temperature in Fig. 3).

EQ4a:
EQ4b:

Note that an SNR measurement is often accomplished in a three-step process. First, an information signal is applied to the communication system, and the received waveform power is measured at the output of the receiving antenna. The power in this received waveform is directly proportional to the power in the signal plus noise.

Next, the signal is removed, and a received noise-power-only measurement is made. The final step consists of subtracting the noise power from the first measurement, and forming a signal-power-to-noise-power ratio, or SNR.

Point B: See Equations 5a and 5b below, where P_r^B is the received signal power (stemming from a waveform measurement at point B), and N₀^B is noise-power spectral density inferred from a measurement at point B. (The effective temperature of the lossy line, T_L, has already been given in Equation 2.) Note from Equation 5a and Fig. 3 that (1/L) (T_Ant + T_L) represents the source noise temperature at point B.

EQ5a:
EQ5b:

Point C: See Equations 6a and 6b below, where point C represents the sampled output of the matched filter (whose output is a baseband pulse) and P_r^C is the received signal power (stemming from a measurement at point C). The noise-power spectral density N₀^C at point C can be measured directly (in the no-signal case) by measuring the noise power N at point C. This is so because for thermal noise with one-sided spectral density N₀ W/Hz, the output noise power of a matched filter is equal to N₀ watts.¹. From Equation 6a and Fig. 3, (G/L) (T_Ant + T_L) + GT_R represents the source noise temperature at point C. The amplification (power gain) of the receiver block is denoted G, and the effective temperature of this block, T_R, is given in Equation 1.

EQ6a:
EQ6b:

The measurements indicated by Equations 4-6 illustrate an SNR degradation as we progress from point A to point C (which agrees with our intuition). For each of the SNR measurements, we can ignore all circuitry to the right of the measurement point, since any load must affect the numerator and denominator of the SNR in the same way.

The predetection point
The demodulation/detection function of the receiver in Figure 1 can be described as a two-step process. In step 1, during each symbol duration, a correlator or matched filter recovers a baseband pulse representing a digital symbol, which is then sampled. The output of the receiver's sampler (point C), termed the predetection point, yields a test statistic consisting of two components—a received symbol and noise. The test statistic has a voltage value that is directly proportional to the energy contained in both the symbol and the noise, and thus contains the essence of the SNR metric.

In step 2, a decision (detection) is made regarding the discrete meaning of that symbol. The result is an information digit (a bit, for binary modulation). The accuracy of the detection step is a function of the predetection SNR. In a digital receiving system, the predetection point is a key location representing the place where all error-performance analysis is focused. To obtain bit-error probability P_B as a function of E_b/N₀, the important action takes place within the detector block-the more signal energy associated with the sample (vs. N₀), the better will be the error performance.

Thus, the short answer regarding the location of E_b/N₀ is that, by definition, it is located at the predetection point. But the long answer (which we will cover in much detail below) is that the short answer is much too simple, since it doesn't reflect the model that is generally used when specifying such SNRs. As a quick aside, note that E_b/N₀ is defined at a place where there are no bits yet. Bits will appear only after completion of the detection process. Perhaps a better name for E_b/N₀ would be energy per effective bit vs. N₀.

The link budget
In compiling the well-known link budget, the system Pr/N0 is often expressed (with reference to point A of Fig. 3) as shown in Equation 7, where EIRP is the effective isotropic radiated power out of the transmitting antenna, G_r is the gain of the receiving antenna, L_s is the space loss and L_o is a "placeholder" for other losses. In Equation 7, T_S represents system effective temperature, and N₀ = kT_S. We will examine this model later.

EQ7:

A link budget tells us something about the quality of the detected data at the receiving system. That is, the value of P_r/N₀ resulting from Equation 7 can be used to characterize system error performance. To do that, the P_r/N₀ represented here must correspond to the value measured at point C (the predetection point) in Fig. 3. Although the system P_r/N₀ need not be described in terms of a measurement at point C, whatever model (reference point) is used must yield an equivalent value, as if P_r/N₀ were being measured at point C.

History of SNR
In the early days of digital communications, Pr/N0 measurements were made directly at point C in Fig. 3, or else they were made at point A, the output of the receiving antenna, and from there translated to the predetection point by invoking the SNR degradation introduced by the lossy line and receiver. Next, E_b/N₀ was computed via the straightforward relationship described in Equation 8, where R is the data rate in bits per second.

EQ8:

When it came to characterizing a system in terms of the received predetection SNR, it was soon realized that a useful system model of P_r/N₀ would allow the same predetection SNR to be expressed at the output of the receiving antenna (or at any arbitrary reference point in the receiving system), not just at point C.

When referring to textbooks, one sees that P_r/N₀ and E_b/N₀ are often characterized as being located at the output of the receiving antenna. This can be confusing because one might conclude that a simple measurement at the output of the receiving antenna could be used directly as the system SNR that is needed when preparing a link-budget analysis. This is not the case. The system SNR or P_r/N₀ can only be measured in a straightforward way at one place—the predetection point. But, it can be modeled at other places in the receiver. In the sections below, we enumerate the important differences between measurements and models and how to avoid system errors by not confusing the two. A model can conveniently characterize a system so that the system can be represented analytically in a simple way. However, parameters of the model cannot generally be measured. The act of modeling does not modify the underlying process (as measurements usually do), since there are no instruments to interact with the system.

System model of P_r/N₀
Whenever the system is specified or characterized by a predetection SNR with reference to some point in the system, then T is represented by the system temperature (T_S) modeled at that point. We shall designate the system temperature as T_S^x, using a superscript to indicate some reference point x.

Figure 4 illustrates that T_S^x accounts for the system (not just the source) temperature ascribed to that point. For each of the points A', B' and C' shown, T_S^x can be represented as shown in Equation 9, where the parameter T_S^x models the noise temperature of the composite circuitry that follows the point x, and T_S^x is typically different at different reference points in the system.

EQ9:

Fig. 4 illustrates a predetection signal-to-noise ratio (which we can now call the system SNR) being modeled at three different and distinct locations, denoted as reference points A', B' and C'. These points represent the same physical locations as points A, B and C in Fig. 3. The primes are used to emphasize that we are no longer making measurements at these points. Instead, we are now modeling the system SNR (with reference to these points). Such system SNRs can be measured only at point C' (or point C in Fig. 3). For points A' and B' such modeled system signal-to-noise ratios do not physically exist. The primes also ensure that we do not equate the SNR measurements from points A and B in Fig. 3 with the SNR models (having similar labels) from points A' and B' in Fig. 4. For each of the reference points in Fig. 4, we can describe the following TS and Pr/N0 relationships¹:

Point A': See Equations10a and 10b below, where T_S^A' is the system effective temperature with reference to point A', made up of a source noise temperature plus a noise temperature due to the composite circuitry that follows point A'. The concept behind the parameter T_S^A' is that it represents a model which allows us to imagine that all of the noise temperature contributing to the total system noise appears at point A' in Fig. 4. As a result, we can imagine that the total system noise appears at the antenna output.

EQ10a:
EQ10b:

Point B': See Equations 11a and 11b, where T_S^B' is the system effective temperature with reference to point B'. Note that the third term in Equation 11b illustrates how the system effective temperature, when modeled at point B', makes it appear that the noise temperature contributing to the total system noise appears at point B' in Fig. 4.

EQ11a:
EQ11b:

Point C': See Equations 12a and 12b, where T_S^C' is the system effective temperature with reference to point C', and we assume that T_comp^C' = 0.

EQ12a:
EQ12b:

When we compare the Fig. 3 relationships in Equations 4 to 6 with those of Fig. 4 in Equations 10 to 12, we should see that the former, illustrating degraded signal-to-noise ratios in moving from points A to C, reflect real measurements. On the other hand, the latter, which illustrate a fixed-system P_r/N₀ ratio at each of the points A' to C', reflect a model that yields the system (predetection) signal-to-noise ratio at each of the chosen points in the receiving system. Such a modeled predetection SNR can be expressed at any arbitrary reference point in the receiving system. In the model, even though the signal power and the system temperature each differ at diverse locations in the receiving system, the system ratio P_r/N₀ (where N₀ = kT_S) is the same at any chosen reference point in the system.

The real temperature
We look upon system effective temperature, TS, as the parameter that accounts for all the noise power in a receiving system. Therefore, for a given system, isn't there just one fixed value of T_S? Are there several system temperatures? Yes, there are. System effective temperature is a model that accounts for all the noise power ascribed to a chosen reference point within the system. Most engineers become comfortable with just a single reference point, and therefore think in terms of a fixed T_S for a given system. Note that system designers, antenna designers and those working at the transmitter side of a system generally choose the output of the receiving antenna (point A' in Fig. 4) as their reference point for T_S and received power, P_r. But receiver designers generally prefer to describe T_S and P_r at the input to the receiver (point B' in Fig. 4).

Assuming that the antenna and receiver are connected via nothing more elaborate than a lossy line having a loss factor L, then in the context of Fig. 4, T_S^A' = L T_S^B', and P_r^A' = LP_r^B'. And, of course, points A' and B' will exhibit the same system SNR specification. But what about the well-known G_r/T_S figure of merit for a receiving system expressed in Equation 7? How is that affected by different system temperatures? In a similar fashion: At point A', the figure of merit can be expressed as G_r^A'/T_S^A', where G_r^A' is the gain of the receiving antenna. At point B', the same ratio is equivalently expressed as shown in Equation 13.

EQ13:

The reason there is not just a single value for T_S is the same reason there is no single value for P_r or G_r. They are each affected by the gains and losses in the circuit chain, and their individual values depend upon the chosen reference point.

Locating, modeling E_b/N₀
We have already given the short answer as to where E_b/N₀ is located or defined, namely at point C, the predetection point in Fig. 3 (or the same point C' in Fig. 4). We raise the question again in the context of a model. Where exactly in a receiving system can E_b/N₀ be referenced? When dealing with the model of system effective temperature reviewed above, the same value of E_b/N₀ can be gleaned from the system SNR expressed at any arbitrary point in the receiving system. In comparing the SNR in Equation 6b at point C in Fig. 3 with the SNR in Equation 12b at point C' in Fig. 4, we see that the measured SNR at point C is the same as the modeled SNR at point C'. This is so because it is only at this predetection point that the actual noise temperature T_local^C is equal to the system effective temperature T_S^C'. For other locations, system effective temperature represents a convenient model whose manifestations cannot be measured at the point being referenced. In the evolution of specifying SNRs in receiving systems with a model, the output of the receiving antenna (point A' in Fig. 4) is the location that is most often used as a suitable reference point.

Through the above analysis we answered the question regarding where exactly in a receiving system is the proper reference point for establishing or specifying the value of E_b/N₀ and other such SNRs. Assuming an accurate model, the answer is: Any point can be a permissible reference. In the evolution of receiver specifications, however, the output of the receiving antenna has been the most commonly used reference point in such SNR modeling.

Author's Note: A version of this paper was originally presented at the 20th AIAA International Communication Satellite System Conference in Montreal last May.

Related Articles

1. "Improving Accuracy in EDGE-Based Devices"; www.commsdesign.com/story/OEG20010323S0050
2. "Simulating Tradeoffs in W-CDMA/EDGE Receiver Front Ends"; www.commsdesign.com/story/OEG20020103S0048
3. "RF Design of a TDMA Cellular/PCS Handset: Part 1"; www.commsdesign.com/main/2000/02/0002feat1.htm
4. "RF Design of a TDMA Cellular/PCS Handset: Part 2"; www.commsdesign.com/main/2000/03/0003feat2.htm

References

1. Sklar, Bernard, Digital Communications: Fundamentals and Applications, 2nd Edition, Prentice-Hall, Upper Saddle River, N.J., 2001.
2. In private conversations, Raymond Pickholtz, a professor at George Washington University in Washington, D.C., provided me with clarifying ideas and inspired me to write this paper, November 2001.

About the Author
Bernard Sklar (bsklar@ieee.org) is head of advanced systems at Communications Engineering Services, a company he founded in 1984. He has 50 years of electrical-engineering experience at companies that include Hughes Aircraft, Litton Industries and The Aerospace Corp. At Aerospace, he helped develop the Milstar satellite system and was the principal architect for EHF Satellite Data Link Standards. Sklar has taught engineering courses at the University of California, Los Angeles, and the University of Southern California, among others, and has published and presented numerous technical papers. He holds a PhD in engineering from UCLA.