Bit error rate snr

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In digital transmission, the number of bit errors is the number of received bits of a data stream over a communication channel that have been altered due to noise, interference, distortion or bit synchronization errors.

The bit error rate (BER) is the number of bit errors per unit time. The bit error ratio (also BER) is the number of bit errors divided by the total number of transferred bits during a studied time interval. Bit error ratio is a unitless performance measure, often expressed as a percentage.^[1]

The bit error probability p_e is the expected value of the bit error ratio. The bit error ratio can be considered as an approximate estimate of the bit error probability. This estimate is accurate for a long time interval and a high number of bit errors.

Example[edit]

As an example, assume this transmitted bit sequence:

1 1 0 0 0 1 0 1 1

and the following received bit sequence:

0 1 0 1 0 1 0 0 1,

The number of bit errors (the underlined bits) is, in this case, 3. The BER is 3 incorrect bits divided by 9 transferred bits, resulting in a BER of 0.333 or 33.3%.

Packet error ratio[edit]

The packet error ratio (PER) is the number of incorrectly received data packets divided by the total number of received packets. A packet is declared incorrect if at least one bit is erroneous. The expectation value of the PER is denoted packet error probability p_p, which for a data packet length of N bits can be expressed as

,

assuming that the bit errors are independent of each other. For small bit error probabilities and large data packets, this is approximately

Similar measurements can be carried out for the transmission of frames, blocks, or symbols.

The above expression can be rearranged to express the corresponding BER (p_e) as a function of the PER (p_p) and the data packet length N in bits:

Factors affecting the BER[edit]

In a communication system, the receiver side BER may be affected by transmission channel noise, interference, distortion, bit synchronization problems, attenuation, wireless multipath fading, etc.

The BER may be improved by choosing a strong signal strength (unless this causes cross-talk and more bit errors), by choosing a slow and robust modulation scheme or line coding scheme, and by applying channel coding schemes such as redundant forward error correction codes.

The transmission BER is the number of detected bits that are incorrect before error correction, divided by the total number of transferred bits (including redundant error codes). The information BER, approximately equal to the decoding error probability, is the number of decoded bits that remain incorrect after the error correction, divided by the total number of decoded bits (the useful information). Normally the transmission BER is larger than the information BER. The information BER is affected by the strength of the forward error correction code.

Analysis of the BER[edit]

The BER may be evaluated using stochastic (Monte Carlo) computer simulations. If a simple transmission channel model and data source model is assumed, the BER may also be calculated analytically. An example of such a data source model is the Bernoulli source.

Examples of simple channel models used in information theory are:

• Binary symmetric channel (used in analysis of decoding error probability in case of non-bursty bit errors on the transmission channel)
• Additive white Gaussian noise (AWGN) channel without fading.

A worst-case scenario is a completely random channel, where noise totally dominates over the useful signal. This results in a transmission BER of 50% (provided that a Bernoulli binary data source and a binary symmetrical channel are assumed, see below).

In a noisy channel, the BER is often expressed as a function of the normalized carrier-to-noise ratio measure denoted Eb/N0, (energy per bit to noise power spectral density ratio), or Es/N0 (energy per modulation symbol to noise spectral density).

For example, in the case of QPSK modulation and AWGN channel, the BER as function of the Eb/N0 is given by:
.^[2]

People usually plot the BER curves to describe the performance of a digital communication system. In optical communication, BER(dB) vs. Received Power(dBm) is usually used; while in wireless communication, BER(dB) vs. SNR(dB) is used.

Measuring the bit error ratio helps people choose the appropriate forward error correction codes. Since most such codes correct only bit-flips, but not bit-insertions or bit-deletions, the Hamming distance metric is the appropriate way to measure the number of bit errors. Many FEC coders also continuously measure the current BER.

A more general way of measuring the number of bit errors is the Levenshtein distance.
The Levenshtein distance measurement is more appropriate for measuring raw channel performance before frame synchronization, and when using error correction codes designed to correct bit-insertions and bit-deletions, such as Marker Codes and Watermark Codes.^[3]

Mathematical draft[edit]

The BER is the likelihood of a bit misinterpretation due to electrical noise . Considering a bipolar NRZ transmission, we have

for a «1» and for a «0». Each of and has a period of .

Knowing that the noise has a bilateral spectral density ,

is

and is .

Returning to BER, we have the likelihood of a bit misinterpretation .

and

where is the threshold of decision, set to 0 when .

We can use the average energy of the signal to find the final expression :

±§

Bit error rate test[edit]

BERT or bit error rate test is a testing method for digital communication circuits that uses predetermined stress patterns consisting of a sequence of logical ones and zeros generated by a test pattern generator.

A BERT typically consists of a test pattern generator and a receiver that can be set to the same pattern. They can be used in pairs, with one at either end of a transmission link, or singularly at one end with a loopback at the remote end. BERTs are typically stand-alone specialised instruments, but can be personal computer–based. In use, the number of errors, if any, are counted and presented as a ratio such as 1 in 1,000,000, or 1 in 1e06.

Common types of BERT stress patterns[edit]

• PRBS (pseudorandom binary sequence) – A pseudorandom binary sequencer of N Bits. These pattern sequences are used to measure jitter and eye mask of TX-Data in electrical and optical data links.
• QRSS (quasi random signal source) – A pseudorandom binary sequencer which generates every combination of a 20-bit word, repeats every 1,048,575 words, and suppresses consecutive zeros to no more than 14. It contains high-density sequences, low-density sequences, and sequences that change from low to high and vice versa. This pattern is also the standard pattern used to measure jitter.
• 3 in 24 – Pattern contains the longest string of consecutive zeros (15) with the lowest ones density (12.5%). This pattern simultaneously stresses minimum ones density and the maximum number of consecutive zeros. The D4 frame format of 3 in 24 may cause a D4 yellow alarm for frame circuits depending on the alignment of one bits to a frame.
• 1:7 – Also referred to as 1 in 8. It has only a single one in an eight-bit repeating sequence. This pattern stresses the minimum ones density of 12.5% and should be used when testing facilities set for B8ZS coding as the 3 in 24 pattern increases to 29.5% when converted to B8ZS.
• Min/max – Pattern rapid sequence changes from low density to high density. Most useful when stressing the repeater’s ALBO feature.
• All ones (or mark) – A pattern composed of ones only. This pattern causes the repeater to consume the maximum amount of power. If DC to the repeater is regulated properly, the repeater will have no trouble transmitting the long ones sequence. This pattern should be used when measuring span power regulation. An unframed all ones pattern is used to indicate an AIS (also known as a blue alarm).
• All zeros – A pattern composed of zeros only. It is effective in finding equipment misoptioned for AMI, such as fiber/radio multiplex low-speed inputs.
• Alternating 0s and 1s — A pattern composed of alternating ones and zeroes.
• 2 in 8 – Pattern contains a maximum of four consecutive zeros. It will not invoke a B8ZS sequence because eight consecutive zeros are required to cause a B8ZS substitution. The pattern is effective in finding equipment misoptioned for B8ZS.
• Bridgetap — Bridge taps within a span can be detected by employing a number of test patterns with a variety of ones and zeros densities. This test generates 21 test patterns and runs for 15 minutes. If a signal error occurs, the span may have one or more bridge taps. This pattern is only effective for T1 spans that transmit the signal raw. Modulation used in HDSL spans negates the bridgetap patterns’ ability to uncover bridge taps.
• Multipat — This test generates five commonly used test patterns to allow DS1 span testing without having to select each test pattern individually. Patterns are: all ones, 1:7, 2 in 8, 3 in 24, and QRSS.
• T1-DALY and 55 OCTET — Each of these patterns contain fifty-five (55), eight bit octets of data in a sequence that changes rapidly between low and high density. These patterns are used primarily to stress the ALBO and equalizer circuitry but they will also stress timing recovery. 55 OCTET has fifteen (15) consecutive zeroes and can only be used unframed without violating one’s density requirements. For framed signals, the T1-DALY pattern should be used. Both patterns will force a B8ZS code in circuits optioned for B8ZS.

Bit error rate tester[edit]

A bit error rate tester (BERT), also known as a «bit error ratio tester»^[4] or bit error rate test solution (BERTs) is electronic test equipment used to test the quality of signal transmission of single components or complete systems.

The main building blocks of a BERT are:

• Pattern generator, which transmits a defined test pattern to the DUT or test system
• Error detector connected to the DUT or test system, to count the errors generated by the DUT or test system
• Clock signal generator to synchronize the pattern generator and the error detector
• Digital communication analyser is optional to display the transmitted or received signal
• Electrical-optical converter and optical-electrical converter for testing optical communication signals

See also[edit]

• Burst error
• Error correction code
• Errored second
• Pseudo bit error ratio
• Viterbi Error Rate

References[edit]

 This article incorporates public domain material from Federal Standard 1037C. General Services Administration. (in support of MIL-STD-188).

External links[edit]

• QPSK BER for AWGN channel – online experiment

The signal-to-noise ratio (SNR) and the resulting BER for a specific modulation format depend on the demodulation scheme employed. This is so because the noise added to the signal is different for different demodulation schemes. In this tutorial we consider the shot-noise limit and discuss BER for the three demodulation schemes discussed in the demodulation schemes tutorial. In the next tutorial we will focus on a more realistic situation in which system performance is limited by other noise sources introduced by lasers and optical amplifiers employed along the fiber link.

1. Synchronous Heterodyne Receivers

Consider first the case of the binary ASK format. The signal used by the decision circuit is given

with φ = 0. The phase difference φ = φ_s — φ_LO generally varies randomly because of phase fluctuations associated with the transmitter laser and the local oscillator. We consider such fluctuations later in the next tutorial but neglect them here as our objective is to discuss the shot-noise limit. The decision signal for the ASK format then becomes

where I_p ≡ 2R_d(P_sP_LO)^1/2 takes values I₁ and I₀ depending on whether a 1 or 0 bit is being detected. We assume no power is transmitted during the 0 bits and set I₀ = 0.

Except for the factor of 1/2 in this equation, the situation is analogous to the case of direct detection discussed in the optical receiver sensitivity tutorial. The factor of 1/2 does not affect the BER since both the signal and the noise are reduced by the same factor, leaving the SNR unchanged. In fact, one can use the same result

where the Q factor can be written as

In relating Q to SNR, we used I₀ = 0 and set σ₀ ≈ σ₁. The latter approximation is justified for coherent receivers whose noise is dominated by shot noise induced by the local-oscillator and remains the same irrespective of the received signal power. As shown in the coherent detection tutorial, the SNR can be related to the number of photons N_p received during each 1 bit by the simple relation SNR = 2ηN_p, where η is the quantum efficiency of the photodetectors employed.

The use of the two equations above with SNR = 2ηN_p provides the following expression for the BER:

One can use the same method to calculate the BER in the case of ASK homodyne receivers. The last BER equation and the Q equation above still remain applicable. However, the SNR is improved by 3 dB in the homodyne case.

The last BER equation can be used to calculate the receiver sensitivity at a specific BER. Similar to the direct-detection case discussed in the optical receiver sensitivity tutorial, we define the receiver sensitivity  as the average received power required for realizing a BER of 10^-9 or less. From the two equations above, BER = 10^-9 when Q ≈ 6, or when SNR = 144 (21.6 dB). We can use the following equation to relate SNR to 

if we note that

simply because signal power is zero during the 0 bits. The result is

In the ASK homodyne case,  is smaller by a factor of 2 because of the 3-dB homodyne-detection advantage. As an example, for a 1.55-μm ASK heterodyne receiver with η = 0.8 and Δf = 1 GHz, the receiver sensitivity is about 12 nW and reduces to 6 nW if homodyne detection is used.

The receiver sensitivity is often quoted in terms of the number of photons N_p using the BER expression above because such a choice makes it independent of the receiver bandwidth and the operating wavelength. Furthermore, η is also set to 1 so that the sensitivity corresponds to an ideal photodetector. It is easy to verify that for realizing a BER of 10^-9, N_p should be 72 and 36 in the heterodyne and homodyne cases, respectively. It is important to remember that N_p corresponds to the number of photons within a single 1 bit. The average number of photons per bit, , is reduced by a factor of 2 in the case of binary ASK format.

Consider now the case of the BPSK format. The signal at the decision circuit is given by

The main difference from the ASK case is that I_p is  constant, but the phase φ takes values 0 or π depending on whether a 1 or 0 is being transmitted. In both cases, I_d is a Gaussian random variable but its average value is either I_p/2 or — I_p/2, depending on the received bit. The situation is analogous to the ASK case with the difference that I₀ = — I₁ in place of being zero. In fact, we can use
for the BER, but Q is now given by

where I₀ = — I₁ and σ₀ = σ₁ was used. By using SNR = 2ηN_p, the BER is given by

As before, the SNR is improved by 3 dB, or by a factor of 2, in the case of PSK homodyne detection.

The receiver sensitivity at a BER of 10^-9 can be obtained by using Q = 6. For the purpose of comparison, it is useful to express the receiver sensitivity in terms of the number of photons N_p. It is easy to verify that N_p = 18 and 9 for heterodyne and homodyne BPSK detection, respectively. The average number of photons/bit, , equals N_p for the PSK format because the same power is transmitted during 1 and 0 bits. A PSK homodyne receiver is the most sensitive receiver, requiring only 9 photons/bit.

For completeness, consider the case of binary FSK format for which heterodyne receivers employ a dual-filter scheme, each filter passing only 1 or 0 bits. The scheme is equivalent to two complementary ASK heterodyne receivers operating in parallel. This feature allows us to use 

for the FSK as well. However, the SNR is improved by a factor of 2 compared with the ASK case because the same amount of power is received even during 0 bits. By using SNR = 4ηN_p, the BER is given by

In terms of the number of photons, the sensitivity is given by

The figure below shows the BER as a function of ηN_p for the ASK, PSK, and FSK formats, demodulated by using a synchronous heterodyne receiver.

It is interesting to compare the sensitivity of coherent receivers with that of a direct-detection receiver. The table below shows such a comparison.

As discussed in the optical receiver sensitivity tutorial, an ideal direct-detection receiver requires 10 photons/bit to operate at a BER of ≤ 10^-9. This value is considerably superior to that of heterodyne schemes. However, it is never achieved in practice because of thermal noise, dark current, and many other factors, which degrade the sensitivity to the extent that

is usually required. In the case of coherent receivers,  below 100 can be realized because shot noise can be made dominant by increasing the local-oscillator power.

2. Asynchronous Heterodyne Receivers

The BER calculation for asynchronous receivers is more complicated because the noise does not remain Gaussian when an envelop detector is used. The reason can be understood from this equation showing the signal processed by the decision circuit.

In the case of an ideal ASK heterodyne receiver, φ can be set to zero so that (subscript d is dropped for simplicity)

Even though both i_c and i_s are Gaussian random variable with zero mean and the same standard deviation σ, where σ is the root mean square (RMS) noise current, the probability density function (PDF) of I is not Gaussian. It can be calculated by using a standard technique and is found to be given by

where I₀(x) represents a modified Bessel function of the first kind and I varies in the range 0 to ∞ because the output of an envelop detector can have only positive values. This PDF is known as the Rice distribution. When I_p = 0, the Rice distribution reduces to a Rayleigh distribution, well known in statistical optics.

The BER calculation follows the analysis of the optical receiver sensitivity tutorial with the only difference that the Rice distribution needs to be used in place of the Gaussian distribution. The BER is given by

with

where I_D is the decision level and I₁ and I₀ are values of I_p for 1 and 0 bits. The noise is the same for all bits (σ₀ = σ₁ = σ) because it is dominated by the local oscillator power. The integrals in the equations can be expressed in terms of Marcum’s Q function, defined as

The result for the BER is

The decision level I_D is chosen such that the BER is minimum for given values of I₁, I₀, and σ. It is difficult to obtain an exact analytic expression of I_D. However, under typical operating conditions, I₀ ≈ 0, I₁/σ ≫ 1, and I_D is well approximated by I₁/2. The BER then becomes

Using SNR = 2ηN_p, we obtain the final result,

A comparison with the following equation,

obtained in the case of synchronous ASK receivers shows that the BER is larger in the asynchronous case for the same value of ηN_p. However, the difference is so small that the receiver sensitivity at a BER of 10^-9 is degraded by only 0.5 dB. If we assume η = 1,

shows that BER = 10^-9 for

while

in the synchronous case.

Consider next the PSK format. As mentioned earlier, asynchronous demodulation cannot be used for it. However, DBPSK signals can be demodulated by implementing the delay-demodulation scheme in the microwave regime. The filtered current in this following equation:

is divided into two parts, and one part is delayed by exactly one symbol period T_s. The product of two currents depends on the phase difference between any two neighboring bits and is used by the decision current to determine the bit pattern.

To find the PDF of the decision variable, we write this equation in the form

where

Here, n = i_c + ii_s is a complex Gaussian random process. The current used by the decision circuit can now be written as

If ω_IFT_s is chosen to be a multiple of 2π and we can approximate ψ with φ, then I_d = ±r(t)r(t — T_s) as the phase difference takes its two values of 0 and π. The BER is thus determined by the PDF of the random variable r(t)r(t — T_s).

It is helpful to write this product in the form

where

Consider the error probability when φ = 0 for which I_d > 0 in the absence of noise. An error will occur if r₊ < r_— because of noise. Thus, the conditional probability is given by

This probability can be calculated because the PDFs of r_±² can be obtained by noting that n(t) and n(t — T_s) are uncorrelated Gaussian random variables. The other conditional probability, P(0|π), can be found in the same manner. The final result is quite simple and is given by

A BER of 10^-9 is obtained for ηN_p = 20. As a reminder, the quantity ηN_p is just the SNR per bit in the shot noise limit.

3. Receivers with Delay Demodulation

In the delay-demodulation scheme shown in the demodulation schemes tutorial, one or more MZ interferometers with one-symbol delay are used at the receiver end. In the DBPSK case, a single MZ interferometer is employed. The outputs of the two detectors in this case have average currents given by

The decision variable is formed by subtracting the two currents such that

The average currents for 0 and 1 bits are R_dP₀ and — R_dP₀ for Δφ = 0 and π, respectively.

To see how the noise affects the two currents, note that from this equation

that I_d can be written in the form

where

is the optical field entering the receiver. Here, n(t) represent the noise induced by vacuum fluctuations that lead to the shot noise at the receiver. A comparison of this equation with

obtained in the case of heterodyne detector with delay implemented in the microwave domain, shows the similarity between the two cases. Following the discussion presented here, one can conclude that the BER in the DBPSK case is again given by

As before, the SNR per bit, ηN_p, sets the BER, and a BER of 10^-9 is obtained for ηN_p = 20.

The analysis is much more involved in the case of the DQPSK format. Proakis has developed a systematic approach for calculating error probabilities for a variety of modulation formats that includes the DQPSK format. Although his analysis is for a heterodyne receiver with delay implemented in the microwave domain, the results apply as well to the case of optical delay demodulation. In particular, when the DQPSK format is implemented with the Gray coding, the BER is given by

where I₀ is the modified Bessel function of order zero and Q₁(a, b) is Marcum’s Q function introduced earlier.

The following figure shows the BER curves for the DBPSK and DQPSK formats and compares them with the BER curve obtained in the case in which a heterodyne receiver is employed to detect the BPSK or the QPSK format (without differential encoding).

When DBPSK is used in place of BPSK, the receiver sensitivity at a BER of 10^-9 changes from 18 to 20 photons/bit, indicating a power penalty of less than 0.5 dB. In view of a such a small penalty, DBPSK is often used in place of BPSK because its use avoids the need of a local oscillator and simplifies the receiver design considerably. However, a penalty of close to 2.4 dB occurs in the case of DQPSK format for which receiver sensitivity changes from 18 to 31 photons/bit.

Because of the complexity of the BER expression above, it is useful to find its approximate analytic form. Using the upper and lower bounds on Marcum’s Q function, this BER expression can be written in the following simple form

This expression is accurate to within 1% for BER values below 3 x 10^-2. If we now employ asymptotic expansions

valid for large values of x, and use a and b from

we obtain

This expression is accurate to within a few percent for values of ηN_p > 3.

Having built a simple digital communication system, it is necessary to know how to measure its performance. As the names say, Symbol Error Rate (SER) and Bit Error Rate (BER) are the probabilities of receiving a symbol and bit in error, respectively. SER and BER can be approximated through simulating a complete digital communication system involving a large number of bits and comparing the ratio of symbols or bits received in error to the total number of bits. Hence,
begin{equation}label{eqCommSystemSER}
text{SER} = frac{text{No. of symbols in error}}{text{Total no. of transmitted symbols}}
end{equation}
and
begin{equation}label{eqCommSystemBER}
text{BER} = frac{text{No. of bits in error}}{text{Total no. of transmitted bits}}
end{equation}

Till now, the only imperfection between Tx and Rx entities we have covered is AWGN. It then makes sense that more distortions should corrupt the signal further and result in more errors. Hence, SER or BER plotted against some measure of signal-to-noise ratio (SNR) is the benchmark against which the performance of a system can be measured. Another useful parameter that combines the effects of all impairments into a single value is Error Vector Magnitude (EVM).

Next, we discuss how we choose a specific measure of SNR against which SER and BER should be plotted.

Figure of Merit: $E_b/N_0$

---

The SNR in a digital communication system is defined as a ratio of signal power in a symbol $P_M$ to noise power $P_w$ as
begin{equation}label{eqCommSystemSNR}
SNR = frac{P_M}{P_w}
end{equation}
where the signal power $P_M$l is modulation average symbol energy per unit symbol time
begin{equation}label{eqCommSystemPowerEnergy}
P_M = frac{E_M}{T_M}
end{equation}

For PAM and QAM modulation schemes, average $E_M$ was defined in their respective articles, see Pulse Amplitude Modulation (PAM) and Quadrature Amplitude Modulation (QAM). Moreover, how to determine the noise power $P_w$ was detailed in the article on AWGN.

When specified in dB units and using a factor of 10 for power conversions, it is
begin{align}
SNR_{text{dB}} &= P_{M,text{dB}} – P_{w,text{dB}} nonumber \
&= 10 log_{10} P_M – 10log_{10} P_w label{eqCommSystemSNRdB}
end{align}
A visual measure of finding $mathrm{SNR_{dB}}$ is shown in Figure below.

Using Eq eqref{eqCommSystemPowerEnergy} and Eq eqref{eqCommSystemNoisePower1}, notice that the SNR can also be written as
begin{align*}
P_M cdot frac{1}{P_w} &= frac{E_M}{T_M} cdot frac{1}{N_0 B} \
&= E_M R_M cdot frac{1}{N_0B}
end{align*}
The above equation can be rearranged as
begin{equation*}
frac{E_M}{N_0} = frac{P_M}{P_w} cdot frac{B}{R_M}
end{equation*}
Recall that there are $log_2 M$ bits in one symbol,
begin{equation}label{eqCommSystemEMEb}
E_b = frac{1}{log_2M} cdot E_M
end{equation}
where $E_b$ is energy per bit. Similarly, the bit rate is
begin{equation}label{eqCommSystemRMRb}
R_b = log_2 M cdot R_M
end{equation}
Plugging the values for $E_b$ and $R_b$,
begin{equation}label{eqCommSystemEbNo}
frac{E_b}{N_0} = frac{P_M}{P_w} cdot frac{B}{R_b} = text{SNR} cdot frac{B}{R_b}
end{equation}
So we can say that $E_b/N_0$ is equal to SNR scaled by ratio of bandwidth to bit rate.

Now the question is: why is $E_b/N_0$ a natural figure of merit in digital communication systems instead of SNR.

• Intuitively, the ratio of signal power $P_M$ to noise power $P_w$ in the system (i.e., SNR) should be a good criterion for measuring performance of different communication systems. And indeed it is true for analog communication systems where the signal continuity transcends the need to partition the signal into some units of time. On the other hand, a symbol is the basic building block of a digital communication system and signal energy within each symbol turns out to be a more useful parameter.
• As we saw in the article on packing more bits in one symbol, many bits ($log_2M$ to be precise) can be packed into every symbol for an $M$-symbol modulation. Consequently, within every $T_M$ seconds, either 1 bit is travelling through the air (BPSK), or 2 bits (QPSK, 4-PAM), or 4 bits (16-QAM, 16-PSK), and so on. As shown in Eq eqref{eqCommSystemEbNo}, using $E_b/N_0$ instead of SNR automatically incorporates this bits per symbol factor through $R_b$.
• Again from Eq eqref{eqCommSystemEbNo}, bandwidth is also taken into account. A system using less bandwidth than the other will have this factor reflected in its $E_b/N_0$ values.

In summary, $E_b/N_0$ is a normalized SNR measure, which proves really useful when comparing the SER/BER performance of digital modulation schemes with different bits per symbol and bandwidths.

The ratio $E_b/N_0$ is dimensionless as
begin{equation*}
frac{E_b}{N_0} = frac{text{Joules}}{text{Watts/Hz}} = frac{text{Watts} cdot text{seconds}}{text{Watts} cdot text{seconds}}
end{equation*}
It is frequently expressed in decibels as $10log_{10} E_b/N_0$.

Scaling Factors

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In most of the figures in this text, you would have noticed a very clear labeling on the x-axis. On the y-axis though, the emphasis has been on the shape rather than the exact values. Now when the SER/BER measurements are to be taken, scaling factors on y-axis become equally important.

Constellation

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Starting with the symbol constellations, we scale the values such that the average energy per symbol is 1. This can be achieved by dividing the constellation values by the square root of average symbol energy (sum of squares of the constellation points). For example, for a PAM modulation with $A=1$, $E_M$ was shown to be equal to
begin{equation*}
E_{M-text{PAM}} = frac{M^2-1}{3}
end{equation*}
which yields
begin{align*}
E_{2-text{PAM}} &= 1 \
E_{4-text{PAM}} &= 5 \
E_{8-text{PAM}} &= 21 \
vdots
end{align*}

When the constellation points $pm 1$, $pm 3$, $pm 5$, $cdots$, are divided by respective $sqrt{E_M}$, various $M$-PAM constellations become normalized to have average symbol energy equal to $1$, as illustrated in Figure below.

As a verification, consider from Figure above,
begin{align*}
E_{2-text{PAM,norm}} &= frac{1}{2}left(1^2 + 1^2right) = 1 \
E_{4-text{PAM,norm}} &= frac{1}{4}cdot frac{1}{5}left(-3^2 + -1^2 + 1^2 + 3^2 right) = frac{1}{20}cdot 20 = 1 \
E_{8-text{PAM,norm}} &= 1
end{align*}
where the subscript `norm’ stands for normalized.

On a similar note, the scaling factor for QAM with $A=1$ was shown as
begin{equation*}
E_{M-text{QAM}} = frac{2}{3} left(M-1right)
end{equation*}
which yields
begin{align*}
E_{4-text{QAM}} &= 2 \
E_{16-text{QAM}} &= 10 \
E_{64-text{QAM}} &= 42 \
vdots
end{align*}
For example, when scaled with square-root of this normalization factor, a normalized $4$-QAM constellation is shown in Figure below. Here, using Pythagoras theorem and realizing that all $4$ symbols have equal energies,
begin{equation*}
E_{4-text{QAM,norm}} = frac{1}{4}cdot 4cdot left(frac{1}{2} + frac{1}{2} right) = 1
end{equation*}

Tx Pulse

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To understand the scaling factor for Tx pulse shape, also known as Tx filter, consider the block diagram of a PAM modulator or a QAM modulator. The next step after converting bits to symbols is upsampling the symbol train before it is filtered by a square-root Nyquist filter. This square-root Nyquist filter performs two operations:

1. It lowpass filters the spectral replicas arising as a result of increasing the sample rate, see the post on sample rate conversion for details.
2. It shapes the symbol sequence so that it is bandlimited in the frequency domain according to given specifications.

We also discussed in sample rate conversion that to maintain the same peak time domain amplitude in the upsampled signal as before, the filter should be designed with a gain equal to the upsampling factor, $L$ in this case. The three available choices, time domain amplitudes and energies were presented there. When we choose this option, the symbol energy gets scaled by $L$ as well as derived in that post. Therefore, denoting energy/symbol in the upsampled and filtered case with $E’_M$,
begin{equation}label{eqCommSystemScalingEnergy}
E_M’ = L cdot E_{Mtext{,norm}} = L
end{equation}

Remember from the post on pulse shaping filter that the peak value in a Square-Root Raised Cosine filter is $1 – alpha + 4alpha/pi$. In an actual implementation of the system, the pulse coefficients should be scaled by this value as well so that the peak value becomes unity and the precision with which the coefficients are represented in a fixed-point arithmetic processor can be maintained. However, here the purpose is only to simulate the BER due to which we ignore this factor.

Noise Power

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In the article on AWGN, the noise power $P_w$ within a bandwidth $B$ was derived as
begin{equation}label{eqCommSystemNoisePower1}
P_w = N_0B
end{equation}
For a discrete signal with sampling rate $F_S$, it is understood that the bandwidth has already been constrained by an analog filter within the range $pm 0.5F_S$ to avoid aliasing. Plugging $B=0.5F_S$ in the above equation, the noise power in a sampled bandlimited system is given as
begin{equation*}
P_w = N_0cdot frac{F_S}{2}
end{equation*}

A digital modulation signal is sampled at rate $F_S = L cdot R_M$ where $L$ is samples/symbol. Bit rate $R_b$ can change with different modulation schemes and sample rate $F_S$ can be increased or decreased during the processing chain. On the other hand, the symbol rate $R_M$ is a standard parameter which is directly proportional to the signal bandwidth as shown in the modulation bandwidths, and hence it remains constant throughout the system implementation. For this reason, $R_M$ in most simulation systems is chosen to be equal to $1$ and the other parameters can be scaled accordingly. Then,
begin{equation}
P_w = N_0cdot frac{L}{2} label{eqCommSystemNoisePower2}
end{equation}

Finally, the relationship between noise power $P_w$ and $E_b/N_0$ can be written as
begin{equation*}
P_w = frac{E_b}{E_b/N_0} cdot frac{L}{2}
end{equation*}
which can be modified using $log_2 M$ bits per symbol,

begin{equation}label{eqCommSystemNoisePower3}
P_w = frac{E_M}{log_2 M cdot E_b/N_0} cdot frac{L}{2}
end{equation}

where $E_M$ is the symbol energy. In case normalized energy is used for symbol constellations, it can be replaced with $1$.

Notice that noise power increases by a factor of $L$ due to oversampling the signal. On the other hand, the signal energy increases by a factor of $L$ due to scaling shown in Eq eqref{eqCommSystemScalingEnergy}. The overall result is that both factors cancel out in the numerator and denominator of SNR expression in Eq eqref{eqCommSystemSNR}. This is illustrated in Figure below.

Rx Pulse

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Both the signal and noise enter the system before filtered by the Rx pulse. Therefore, any scaling has no effect on the SNR and it only changes the time domain amplitudes. Since the Raised Cosine pulse is a convolution of Tx and Rx Square-Root Raised Cosine pulses defined above, it has a peak amplitude of $1$ and hence maps the symbols exactly on the constellation.

Finally, remember that the group delay defined in pulse shaping filter is now given by the contribution from both Tx and Rx square-root Nyquist filters. For that reason, we remove $LG/2+LG/2=LG$ samples at the start and $LG$ samples at the end of the received sequence.

BER Plots

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Now we are at a stage where we can plot the symbol and bit error rates of all three modulation schemes discussed above, namely PAM, QAM and PSK. To avoid any confusion, we leave SER plots and just concentrate on BER plots. SER plots are an obvious byproduct of the simulation setup discussed above.

The BER plots for 2, 4 and 8-PAM modulation are shown in Figure below.

Note that for a fixed $E_b/N_0$, the BER increases with $M$ because the same symbol energy has to be divided among $log_2M$ bits. Similarly, for a fixed BER, more energy is required as $M$ increases to maintain similar distances among constellation points. As an example, for a target BER of $10^{-4}$,

• $E_b/N_0$ needed by $2$-PAM is $8.4$ dB,
• $E_b/N_0$ needed by $4$-PAM is $12.2$ dB,
• $E_b/N_0$ needed by $8$-PAM is $16.5$ dB,

A gray code is a method of assigning bits to symbols in a manner that adjacent symbols differ in only one bit. For example, symbols in a $4$-PAM constellation can be assigned bits 00, 01, 11 and 10 (as opposed to 00,01,10 and 11) such that all symbols have neighbours with only one bit difference. Since Gaussian noise has a higher probability around the mean value, the most likely symbol errors end up in adjacent symbols. As a result, these most likely symbol errors produce one erroneous bit and remaining $M-1$ correct bits. Hence, the relation between BER and SER can be defined as
begin{equation*}
text{BER} = frac {1}{log_2 M} text{SER}
end{equation*}

Moving on to QAM, the BER plots for $4$, $16$ and $64$-QAM modulation are shown in Figure beow. Similar conclusions hold for BER vs $E_b/N_0$ as in the case of PAM.

Finally, the BER plots for 2, 4 ,8 and 16-PSK modulation are shown in Figure below. Notice that the plots for BPSK and QPSK overlap with each other because QPSK can be regarded as a pair of orthogonal BPSK systems. We will have more to say about this point in the next section.

Spectral Efficiency

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In BER vs $E_b/N_0$ plots above, we noticed that for a target BER of $10^{-4}$, $E_b/N_0$ required by $2$-PAM, $4$-PAM and $8$-PAM are $8.4$ dB, $12.2$ dB and $16.5$ dB, respectively. Does this mean that $8$-PAM is $8$ dB worse than $2$-PAM modulation scheme? The answer is no, because the bandwidth has not been taken into consideration in this comparison. For the same symbol rate (which means same bandwidth), $8$-PAM transmits $3$ bits of information as compared to just $1$ bit of information by $2$-PAM.

For that reason, bandwidth must also be considered to get a complete picture of how different modulations compare to one another. Thus, spectral efficiency is defined as the ratio of bit rate to the bandwidth and its units are bits/s/Hz.
begin{equation}label{eqCommSystemSpectralEfficiency}
text{SE} = frac{R_b}{B} quad text{bits/s/Hz}
end{equation}
In essence, spectral efficiency is a way to figure out how fast the system is per unit of bandwidth. For QAM and PSK constellations with a square-root Nyquist filter of excess bandwidth $alpha$, we saw in the post on modulation bandwidths that the required bandwidth is $(1+alpha)R_M$. In terms of bit rate,
begin{equation*}
B_{text{QAM}} = left(1+alpharight) frac{R_b}{log_2 M}
end{equation*}
from which spectral efficiency can be determined as
begin{equation}label{eqCommSystemSEQAM}
SE_{text{QAM}} = frac{log_2 M}{1+alpha} quad text{bits/s/Hz}
end{equation}

Turning towards PAM, although the bandwidth required is half as compared to QAM, it transmits half the number of bits as well due to using $I$ channel only. Therefore, its spectral efficiency is similar to QAM.

Different constellations with different number of points can be compared by taking all the parameters into account, namely bits/symbol, bandwidth and SNR performance. This can be achieved by plotting the spectral efficiency against the value of $E_b/N_0$ required to obtain a specific target BER. Figure below shows spectral efficiency versus $E_b/N_0$ for QAM and PSK constellations for BER = $10^{-5}$ and excess bandwidth $alpha = 0.5$, where $E_b/N_0$ values can be taken from the x-axis of BER curves in the previous section. Note that QAM provides better spectral efficiency than PSK because it packs the constellation points in a more efficient manner. This is why it is the modulation of choice in most of the high-speed wireless networks today.

In general, better modulations operate at high spectral efficiency and low $E_b/N_0$, i.e., to the top left of this curve.

On a final note, we rewrite Eq eqref{eqCommSystemEbNo} here.
begin{equation*}
frac{E_b}{N_0} = frac{P_M}{P_w} cdot frac{B}{R_b} = text{SNR} cdot frac{B}{R_b}
end{equation*}
The above equation can be written as
begin{equation*}
text{SE} = frac{R_b}{B} = frac{text{SNR}}{E_b/N_0}
end{equation*}
As a result, SE can also be seen as the ratio of SNR to $E_b/N_0$. Thus, the lesser $E_b/N_0$ a modulation scheme requires for a given error rate, the more spectrally efficient it is.