Building Blocks

All those modulation types with all those performance expressions, how do you keep them all straight? Nowadays, anyone who can spell IEEE feels the need to create a new acronym, but fundamental BER performance expressions begin with the same simple idea.

This month, we move back into specific concepts after presenting a few months of broad-based reference material. The funny thing about the acronym list is that about a half-dozen people (a couple of whom are in my workplace) asked if I would also list all the columns over the years. This was before they knew it was coming in January. Voila — break out the champagne (if there’s any left). I actually had a thought process in sync with my audience. I quickly recorded this historical date and called the US Registry. This is a convenient time to thank all of the readers who spent their valuable time reading the “Building Blocks” column last year, and especially those who took the time to provide feedback in 1999. It’s always great to hear from you.

Back to this month’s topic: symbol error calculation mathematics. He doesn’t really mean math does he? Calm your nerves; grab a beer if it helps. It is not so bad. Those funny S-shaped things (known as integrals in academic circles) will make a brief appearance. We will endeavor to keep you from going insane by spreading the material over two columns. This month, I’ll focus on the basics and the simplest example. Next month, I’ll show how this type of mathematics applies to larger signal sets which are of more practical interest. This will let me poke into how other impairments are handled within the boundaries of symbol error calculation math.

First, let’s address the basic scenario. The expressions traditionally presented in textbooks and basic reference articles are always based on the most fundamental and the most unavoidable impairment — additive Gaussian white noise (AWGN). The description says it all: It is a noise signal added to the desired signal, with a Gaussian distribution of the noise amplitudes, and with a white spectrum relative to the signal bandwidth. It is the basic thermal noise associated with real channels. A description of the Gaussian noise process was given in an earlier column (“In the Noise,” Communication Systems Design, April 1997, pp. 14-20). The noise process is defined by a probability density function (PDF) that describes the statistical nature of the process’ randomly varying noise amplitudes. The curve for the Gaussian PDF is described by the expression:

(Equation 1)

The curve is shown in Figure 1 . Its key attributes are the mean (the average), m, and the standard deviation, s, which quantitatively describes the spread of the amplitude values around the mean. For AWGN, the average value is zero. The variable s ² is called the variance (for m = 0 , zero mean) and is defined as the average of the squared noise value. In simpler terms, the variance of zero mean AWGN is simply the power of the noise. The PDF is used to calculate probabilities by noting that the probability that x falls between y and z is the area under the curve between those two points:

(Equation 2)

Now that you are a Gaussian noise expert, let’s look at what this does to a simple constellation — in this case binary phase shift keying (BPSK), also known as antipodal signaling (see Figure 2 ). In its simplest form, BPSK is an RF carrier whose phase is modulated between zero and 180 degrees by a binary data stream, and thus one bit per symbol. In our example constellation, the symbol points are of the same amplitude, b ₁ = -b and b ₂ = b . The distance between these two points is d = 2b. If noise is added to the signal, the constellation points become shifted from their ideal locations. Let’s concentrate on one symbol to see what it would take for this symbol to be incorrectly detected. Without delving into spurious details for the purist, the ideal detector will make a decision based on the y -axis as a decision boundary. For symbol point b (with noise added) the corrupted symbol delivered to the detector at symbol time i is:

(Equation 3)

In this case, the detector merely needs to distinguish along the y -axis, and thus chooses the symbol location of the shorter distance. Assume symbol b ₁ is sent. Then, for the transmission to be in error, symbol b ₂ is chosen if the noisy input is closer to the ideal b ₂ reference than b ₁ . This is mathematically expressed as:

(Equation 4)

Using Equation (3) with b ₁ as the transmission in Equation (4), we get:

(Equation 5)

This simplifies to (trust me):

(Equation 6)

With d = 2b = b ₁ – b ₂ , this becomes:

(Equation 7)

And now it’s time for a deep breath because we are almost there.

Purists in the communication world refer to the symbol detector as the slicer, which is coincidentally what I also call my golf partners on about twelve of eighteen tees each round. The left-hand side has our friend the Gaussian noise sample, which is of zero mean and variance s ² . One of the properties of this statistical type is: If a Gaussian random variable is multiplied by a constant ( d , for example), the new random variable is also Gaussian, with a mean of d m (zero in this case), and variance d ² s ² . This is what sits on the left hand side of Equation (5). The Gaussian random variable is usually written shorthand as N (m,s), where N stands for the normal distribution.

Because it is a random variable, the inequality in Equation (5) may be met sometimes, and not at other times. This is the fun of probability and stochastic systems. When is the inequality met (which indicates that an error has been made)? It occurs as often as a Guassian random variable with zero mean and variance d ² s ² is smaller than - d ² / 2 . In the world of probability distributions, determining this solution is straightforward. We described how to calculate this in Equation (2). The answer is:

(Equation 8)

where f(x) is defined in Equation (1). Unfortunately, the purely analytical effort gets stymied here, as the integration cannot be solved exactly into a closed-form expression. How convenient that this occurs with communications’ most popular noise impairment. However, due precisely to its popularity, the integral has been well characterized numerically, to the point of being well tabularized and part of many standardized numerical recipe toolkits in any decent spreadsheet or mathematical analysis program. As such, it has a well-understood definition similar to Equation (8), and several polynomial series can be used to generate very accurate numerical approximations. The solution, called Q(x) , defines the integration range

(Equation 9)

N (0,1) is a normalized Gaussian random variable. For a Gaussian random variable x with zero mean and variance s ² , it becomes just Prob(x > X) = Q(X/ s ) .

You now have everything you need to complete the BER solution for BPSK with some minor mathematical gymnastics. Figure 1 shows that the BER solution is symmetric. It is easily noted (one of my favorite textbook statements, often meaning “we are too lazy to prove this”) that the area under the curve described by Equation (8) is the same as the area described by integrating on the other side of the (zero) mean from d ² /2 to . In other words, from Equations (8) and (9):

(Equation 10)

Uh-oh, we have now reached double digits in equations. Recall that for a Gaussian random variable x with zero mean and variance (noise power) s ² , Prob(x > X) = Q(X/ s ) . For the noise characteris- tic we are comparing against in Equation (7), N(0,d s ) , we then have Prob(x > X) = Q(X/d s ) . Then, from Equation (10):

(Equation 11)

This may not look helpful yet, but it is truly all you need now. This is because the noise power (s ² ) is commonly written as the product of the noise bandwidth ( B ) and the noise spectral density ( N _o /2, or s ² = N _o B/2 ). The term d can be related to the signal power, and is easy for this case — the average power (P) is d ² . Then, noting B = 1/T for symbol period T , and energy per bit ( E ) is E = PT , Equation (11) can be written as:

(Equation 12)

Poof. In this example, the energy per symbol is the same as the energy per bit since it is one bit per symbol, so

Last equation — congratulations! Thanks for hanging in there. Next month, we will apply this to larger scenarios and see how other expressions unfold from this simple idea.

Rob Howald is the director of systems engineering in the transmission network systems group at General Instrument in Hatboro, PA. He has a BSEE and an MSEE from Villanova University, and received his PhD from Drexel University. He can be reached at rhowald@gi.com .

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