SIGNAL & NOISE SCALING IN VSS SIGNAL & NOISE SCALING IN VSS By: Dr. Kurt R. Matis Director of Systems Research SIGNAL AND NOISE SCALING FOR LINK ERROR ANALYSIS This document describes

. . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . .SIGNAL & NOISE SCALING IN VSS

By: Dr. Kurt R. Matis
Director of Systems Research
SIGNAL AND NOISE SCALING FOR LINK ERROR ANALYSIS

This document describes the signal power and noise scaling conventions used in the Visual System Simulator (VSS). In this note, we discuss the fundamental sig-nal representation used in VSS. In addition, the signal and noise power relation-ships in an end-to-end communication link simulation are described. Both concepts are comprehensively illustrated with two simulation examples. In order to make this note somewhat self-contained, some elementary background from statistics is presented.

Basic Measurements on Noisy SignalsWe begin this note with some basic background information on signal measure-ments made in noisy environments. A communication receiver is actually a mea-surement device, much like a spectrum analyzer or oscilloscope. It must measure the parameters of the incoming signal to detect data bits and to per-form ancilliary functions like synchronization. The major difference between communications receivers and most test equipment is that communications receivers generally operate in a much noisier environment. Signal-to-noise power ratios are typically in the range of 0-30 dB and can be significantly lower for spread-spectrum signals. It is beneficial to examine how noise occurs and how it is processed in a digital communication link.

The are many sources and types of noise that can degrade the signal on its path from transmitter over a communication channel, through a receiver�s front end to the point where it is finally detected and the bit or symbol value is declared. We generally consider noise to be an additive disturbance, thus excluding other phenomena such as multiplicative fading and multipath. Additive noise can occur both from environmental sources and from thermal sources within equip-

V i s u a l S y s t e m S i m u l a t o r 1

S I G N A L & N O I S E S C A L I N G I N V S S

Signal and Noise Scaling for Link Error Analysis

ment. Receiver front-end noise is normally modeled as a white Gaussian process with intensity proportional to receiver temperature and resistance [1]. Other sources of noise that can exhibit a Gaussian characteristic are galactic noise and impulse noise. Impulse noise usually occurs as a mixture of a high-level noise excursions superimposed on a background of WGN. In any case, Gaussian noise is ever-present in communication systems, and receiver SNR performance is usually calibrated against an assumed background component of AWGN. Even in fading channels, BER performance is calibrated with respect to the aver-age received signal power divided by the noise power.

A zero mean white Gaussian noise process, w(t), is characterized by its flat power spectral density, . Here, represents the double-sided power spectral density. White noise does not exist in practice, since such a process would have infinite power. It is a mathematical abstraction for noise that possesses a power spectrum which is flat over a range of frequencies that is greater than the effective receiver front-end bandwidth, B. In this case, the receiver sees a band-limited Gaussian noise process with spectrum confined essentially to . An isolated sample (measurement) of this band-limited Gaussian noise process is a Gaussian random variable with variance equal to:

(1)

When the power of a Gaussian noise process is referenced, the implication is that the process is observed through a measurement filter of some finite bandwidth, B.

Simulation Model

Samples of band-limited Gaussian noise taken at intervals of 1/B are approxi-mately statistically independent. In what follows, we assume that the receiver front-end filter is a �brickwall� filter with a rectangular frequency response. In this case, the auto-correlation function of noise samples at the filter output has zero crossings at intervals of 1/B, making noise samples taken at these intervals uncorrelated. Since the process is assumed Gaussian, the samples are precisely statistically independent. This assumption of a brickwall front-end filter does not result in any loss of generality, since this filter can conceptually precede other frequency-dependent elements that model more general frequency charac-teristics. The fact that the noise samples are independent has some convenient implications for the implementation of a sampled-data simulation. We assume that our simulated signal corresponds to an ideally sampled band-limited analog

Sww f( ) N0 2⁄= N0 2⁄

B 2⁄ B 2⁄,�{ }

σ2 B N0 2⁄⋅=

2 V i s u a l S y s t e m S i m u l a t o r

S I G N A L & N O I S E S C A L I N G I N V S SSignal and Noise Scaling for Link Error Analysis

waveform with bandwidth B = . Here, represents the number of samples taken per unit time. The specific time unit can vary, depending on the applica-tion. In VSS, it can be set automatically to be the default system unit (seconds, microseconds, etc.). It can also be set to a normalized digital symbol time in the case of digital communications applications. Here, the user of the simulation would be specifying the number of samples used to represent the analog wave-form during one digital signalling interval, or �samples per symbol�. If we assume that we are generating a signal that is equivalent to an ideally sampled analog waveform, we can generate a sequence of noise samples, , which are independent with variance

(2)

This is the assumption made in most sampled-data simulation models and it is the assumption made in VSS. Figure 1 shows a conceptual diagram illustrating this model.

Figure 1. Band-limited Noise and Simulation Model

FILTERING OF BAND-LIMITED NOISE

Filtering is a signal-processing operation that is often employed to minimize the effects of random noise. Averaging is a special case of filtering. Averaging of a signal, , is simply a weighted integration and is mathematically represented by:

fs fs

wi

σ2 Ns N⋅ 0 2⁄=

Rectangular(brickwall)

FilterB!"#

t = 0, T$ ,2T$,�.

wi = wB(iT$) wi ~ GaussianMean=0Variance = BN 0/2

w(t)

B!"#

-B/2 B/2

wB(t)

N0/2

RwBwB!%#

B%0 2 3-2-3 1-1

1.0

"

T$&'()

Spectrum of Bandlimited White Noise Autocorrelation Function of Bandlimited White Noise

r t( )

T e c h n i c a l N o t e s 3



, (3)

where T represents the integration interval. This is equivalent to passing the process, , through a low-pass filter, , with frequency response:

. (4)

This is the familiar sin(x)/x response associated with a finite-time integrator and the dual of the transform pair corresponding to the brickwall filter shown in Figure 1, with the sinc function now being the frequency response. The term �finite-time� derives from the integration taking place over a sliding window of finite length, not over the infinite past. If white noise is passed through this fil-ter, the magnitude of the high-frequency spectral components is reduced by a

factor of , with the corresponding power reduced by a factor of . If we pass band-limited white noise through such a filter, the spectral quantities are as shown in Figure 2.

Figure 2. Finite-Time Integration Operation with Spectral Quantities

As Figure 2 indicates, the WGN process is passed first through the brickwall fil-ter with bandwidth B, then through an integrator, with integration length T. We assume that here, although this assumption is not necessary. Quantiz-

y t( ) 1 T⁄ r t( ) tdt T�( )

t

∫⋅=

r t( ) h t( )

H f( ) jπfT�( )exp πfTsinfT

---------------- ⋅=

1 f⁄ 1 f2⁄

B!"#

t = 0, T$ ,2T$,�.

w�i ~ GaussianMean=0Variance = N0/(2T)

A+w(t)

B!"#

-B/2 B/2

A+wB(t)

0"

T$&'()

H!"#

H!"#

BrickwallFilter

Integrator B0w ' A+w ( )

nT

i t dt∆= ∫

1/T-1/T

Overlaid Spectra of Brickwall and Integrating Filters

T nT∆=



ing time to the sampling interval simplifies the simulation, and it produces suffi-ciently accurate results in the majority of applications. Such quantization is not always appropriate, however, and VSS provides the capability to interpolate between samples when necessary.

The integrator actually has infinite bandwidth, since the sinc function extends to infinity. However, its bandwidth is essentially limited to 2/T. Here, we assume that the integrator bandwidth is significantly smaller than the double-sided rect-angular noise bandwidth, B, as the figure indicates. A sample of the averaged noise process, taken at the end of the integration interval now has variance approximately equal to:

. (5)

The approximate equality follows from the assumption that the integrator had a much narrower bandwidth than the band-limited noise. This assumption does not limit the generality of the result, it only simplifies the calculation. Our result indicates that the variance of the noise sample is reduced by a factor of , when the integration interval is much longer than . This implies that we can reduce the noise variance to whatever level we want simply by increasing the integration interval. A simulation example helps to illustrate this point by dem-onstrating the averaging of a constant (D.C.) signal buried in AWGN. Figure 3 shows a constant real value of 1.0 that is corrupted with Gaussian noise. The noise is bandlimited on B=fs=1.0 Hz, with power spectral density Watts/Hz. If the band-limited Gaussian noise is simply sampled at any time, t, the variance of the noise sample is just equal to . If, instead, the signal is averaged over T seconds before sampling, the variance of the noise sample is reduced by a factor of T. Figure 3 shows two waveforms generated within VSS overlaid on the same set of axes. The first waveform consists of just samples of the constant 1.0 with band-limited Gaussian noise added, without integration. Samples are taken at intervals of The second waveform is the normalized integral of this original waveform. As the integration interval becomes longer and longer, the noise level becomes lower and lower, allowing the constant level to be seen more easily.

σy2 N0

2------ H f )( ) 2 fd

B 2⁄�

B 2⁄

∫⋅=N02

------ H f )( ) 2 fd

∞�

∞

∫⋅≈N02

------ 1

T---⋅=

BT1 B⁄

N0 2⁄ 1.0=

1.0

T∆ 1 B⁄ 1 sec.= =




Figure 3. VSS Waveforms Demonstrating the Effects of Signal Averaging

In the VSS sampled-data simulation model, (3) is implemented as a normalized sum of samples:

. (6)

Here, the samples are taken at the rate per second, corresponding to Figure 1. Figure 4 illustrates how the data samples are actually generated. A sream of discrete numerical values are generated to represent a constant plus sampled WGN. Independent Gaussian variates are generated and added to con-stant samples. These samples are processed by a �running average� model block to create the statistic . We use the subscript i to indicate that this could be a sequence of filtered outputs, although our current experiment displays only a single (the first) integrated noise output. Recall that the variance of a sum of

independent random variables with variance is equal to and that the variance of a random variable defined as a constant C times a random variable

r 1Ns------ r′j

j 0=

Ns 1�

∑=

r'j B fs=

ri

Ns

σ2 Nsσ2



with variance is . With these facts in mind, it can be easily verified that the variance of the output sample possesses a variance that decreases with the number of samples averaged.

Figure 4. Sampled-Data Implementation of Analog Signal Plus Noise Model in VSS.

Link Error Rate EvaluationThe goal of an end-to-end link analysis is to measure how often a transmitted bit is received in error. This event is called a bit error. It�s sometimes preferable to deal with symbols, the actual entity that is emitted over the channel. A symbol can be a bit or a group of bits that is encoded in some way. For example, in Quadrature-Phase-Shift-Keyed (QPSK) transmission, bits are grouped in pairs to form 4-ary symbols. In either case, an error analysis is designed to predict how often bits or symbols are received in error. Where no confusion results, we simply refer generically to this as Bit Error Rate (BER). Where Symbol Error Rates are specifically under consideration, we use �SER�.

It is usually desireable to tabulate estimates of error rate versus one or more sys-tem parameters, such as signal-to-noise ratio, a filter�s bandwidth, amplifier bias point, etc.. In the latter two cases, the goal is often to optimize system perfor-mance with respect to certain equipment settings. In a signal-to-noise ratio anal-ysis, the goal is usually to determine how much energy must be expended by the transmitter to drive the bit error rate down to an acceptable level. At some point, the signal level overcomes any deleterious effects that most types of ran-dom noise can produce. The output of such an analysis is presented in the famil-iar BER vs SNR curve.

Link error rate evaluations are normally performed as a function of the signal-to-noise ratio at the output of the digital receiver. In this note, we assume a lin-

σ2 C2σ2

BrickwallFilterB!"#

t = 0, T$ ,2T$,�.

~ GaussianMean=0Variance = BN0/2

T$&'()

( )j Bw w jT ∆′ =

( )( )

r tA w t

=+

jjr A w′ ′= +

jw′

( )1

*

0

1 Ns

jjNs

−

=∑

i ir A w= +

~ GaussianMean=0Variance = BN0/(2Ns)

iw

VSS Simulation




ear �matched� filter demodulator operating in the presence of AWGN. The matched filter sampled at the optimum time instant is shown in Figure 5.

We have assumed antipodal signalling for clarity, so 1�s and 0�s correspond to negative and positive outputs from the matched filter.

Figure 5. Matched Filter Receiver for Antipodal Signals in AWGN

Define , where the latter equality holds since we assume that the noise has zero mean value.

For rectangular signaling pulses, the matched filter is simply an integrator. The mathematical operation performed by the integrator receiver is:

. (7)

This is precisely the integration operation discussed in the previous section, with T now equal to , the channel signalling interval. Rectangular signaling corre-sponds to the transmission of a constant value, A, during the signalling interval. The demodulated output statistic then becomes:

. (8)

Matched Filterh(t0-t)

t = 0, Ts ,2Ts,�.r(t)

ri = si + wi

wi ~ GaussianMean=0Variance = N0/2

si = +- Es

w(t)

s(t)

Es ri2 si

2==

ri1Ts----- r t( ) td

t Ts�( )

t

∫=

Ts

ri1Ts----- A w t( )+ td

t Ts�( )

t

∫= A wi+=



Here, is a Gaussian random variable with zero mean and variance

. represents the double-sided power spectral density of the (white) noise process, as usual. We saw in the previous section that the noise component could be made smaller and smaller by simply increasing the integra-tion interval. Since the signal is constant for only seconds, integrating over this interval is the best we can do. In this case, we are employing a demodulation filter at the receiver that is �matched� to the transmission filter. In Gaussian noise environments, it can be shown that this is the best we can ever do to reduce the effects of the noise. Specifically, signal-to-noise ratio is defined as the mean-square value of the signal component divided by the variance of the noise component, or:

. (9)

Here, represents the received energy during each channel signalling interval, measured at the demodulator output. Note that symbol error rates are normally graphed as a function of , which is actually half of the received signal-to-noise ratio.

SNR SCALING OPTIONS IN VSS

When simulating end-to-end link operation, scaling is somewhat arbitrary, pro-vided the required value of at the receiver output is obtained. In typical receivers, there are automatic gain control circuits which adjust the level of the received signal (desired signal plus noise) for various purposes. This is necessary, for example, to avoid overloading analog devices or to avoid exceeding the dynamic range of digital circuitry that might use small integers for arithmetic. Furthermore, we do not generally wish to model the physical details of the transmission channel, which includes transmission loss, loss/gain due to anten-nas, and a variety of other factors. As a result, we are free to scale the signal sam-ples in a variety of ways. The option chosen in VSS is to set the numerical value of the transmitter output waveform samples to represent the actual voltage across a resistance equal in value to the system impedance, . This presenta-tion is intuitive to hardware engineers. The voltage value, A, is then calculated as:

. (10)

wi

σw2 N0 2Ts( )⁄= N0 2⁄

Ts

SNR ri2 σi

2⁄= A2 N0 2Ts⁄( )⁄ 2A2Ts( ) N0⁄ 2Es( ) N0⁄= = =

Es

Es N0⁄

Es N0⁄

Z0

A PT Z0⋅=




Here, is the average transmitted power, expressed in Watts. Note that arbi-trary pulse shaping may be employed to achieve and/or emulate desired fre-quency characteristics. Under these conditions, (10) holds on an average power basis. To reconcile received SNR goals with the signal scaling chosen, additive

noise samples are generated with a variance given by ,

which differs in form from (2) only by the factor . The sampling rate, , is expressed in samples per unit time and the power spectral density is expressed in Watts/(cycle per unit time).

Our digital receiver performs the correlation operation against incoming sam-ples, :

. (11)

Here, represents the number of samples employed to represent the analog

waveform during one symbol period. The samples represent an energy-normalized stored digital pulse shape that the demodulator correlates the input samples against during each symbol interval. Note that the transmit pulse can-not be energy-normalized, since in (10) we are explicitly transmitting constant sample values. The length, N, of the pulse may extend over more than one sym-bol. In this case, pulses will overlap, and the correlation is performed over more than the symbol time. We assume in this note that the pulse shape is matched (except for a scale factor) to the digital pulse employed at the transmitter. Mis-matches would cause available received energy to be reduced, but the develop-ments presented here are still applicable. Depending on how the user wants to interpret unit time, three scaling options are available as parameter choices in the Additive White Gaussian Noise CHannel (AWGNCH) model. The first option is to specify the power spectral density in units of Watts per (cycle/sec-ond), or Watts per Hertz. In this case, noise samples are generated with variance:

(12)

Note that the user specifies , in Watts/Hertz, whereas represents the

PT

σ2 Z0 fs⋅ N⋅ 0 2⁄=

Z0 fs

r'k{ }

ri 1

Ns Ts---------------- hj r• ′i Ns j+⋅ i

j 0=

N 1�

∑= 0 1 2 3...∞, , ,=

Nshj{ }

σj B(Hertz)N02

------• Watts/Hertz)

Ns(samples/symbol) RT(symbol/second)N02

------•• (Watts/Hertz)

=

=

N0 2⁄ RT



symbol rate of the digital communication system being simulated. is a global system parameter associated with the simulation. This scaling produces a received SNR that is dependent on the data rate of the digital signal transmis-sion:

(13)

is the mean-square value of the demodulator decision statistic. The variance of the noise component at the demodulator output is:

, (14)

which is independent of the symbol rate, with the normalization in (11). If the transmitter power (in Watts) is held constant, BER curves will shift to the left or right as the data rate is varied. This is due to the fact that the matched filter is integrating a signal with a fixed power level over a shorter period of time. This scaling would be appropriate if the user wanted to correlate physical noise mea-surements taken from equipment with the simulation results. Physical measure-ments are usually taken in absolute time/frequency units and BER results would actually depend on data rate in physical hardware. Since the transmitter models emit a fixed number of samples at a fixed average level, the noise component is scaled to acheive desired demodultor output SNR behavior under these condi-tions. Thus, the varaince of each noise sample added to the sampled-data wave-form will change with data rate. On the other hand, it is sometimes convenient to normalize power quantities to the digital data rate, specifically to avoid the changing of BER with varying data rate. This is due to the fact that the sample rate in digital communications simulation is normally specified relative to the data rate - i.e., number of samples per symbol. For this purpose, two other scal-ing options are provided in the AWGNCH model. The user may elect to specify power spectral density in units of Watts per (cycle/symbol) or Watts per (cycle/bit). (Recall in modulation schemes with n bits per symbol, energy per symbol is higher than energy per bit by a factor of n. For example, symbol energy is approximately 3 dB higher than bit energy in QPSK.) Clearly the two quantities can be derived from Watts/Hz simply by dividing by the symbol or bit rate respectively. The advantage is simply one of convenience to the user who wishes to parametize a BER simulation by either or , without worrying

about the data rate tag, , associated with the digital data stream. The normal-ization is performed internally within the AWGNCH model. This additional set of normalization options allows the simulation to employ physical transmitted

Ns

si2 PT(Watts) Ts(seconds)⁄=

σi2 RT

N02

------ Ts⁄•N02

------= =

Es N0⁄ Eb N0⁄

RT




analog waveform samples corresponding to real-world voltage levels at interme-diate points within the link, while simultaneously allowing digital link analysis to be performed conveniently, with power quantities that are normalized to the data rate. We refer to these additional options for noise power normalization as �Energy-Per-Symbol� and �Energy-Per-Bit� normalization, respectively.

We must be careful to understand that in (10) we are now normalizing to power that is generated at the output of the transmitter rather than the energy of the demodulator decision statistic. In a real system, the power levels at the receiver front-end would be tiny - on the order of picowatts or lower, in some cases. However, we don�t necessarily want to model the actual gain or loss on the channel that would produce these physical received voltage levels. This would involve scaling the time samples in the sampled-data waveform up or down. If there are no models incorporated in the simulation that depend on the ABSO-LUTE power levels at the receiver front end, modeling the physical received voltage levels serves no useful purpose. Instead, we adjust the noise power to produce the desired SNR at the receiver output, based on the somewhat arbi-trary choice of scaling in (10). Should the user desire to model actual power loss on the channel, the AWGNCH model offers a loss parameter for this purpose. The loss is applied to the signal before the noise is added.

If there are additional gains or losses in any simulation models between the transmitter and the point at which Gaussian noise is added, these gains/(losses) must be offset by raising/(lowering) the noise power to ensure that the target demodulator output SNR is not changed. A physical device like an amplifier will (hopefully) produce a power gain. In simulating such a device, the nominal gain of the amplifier is often normalized out of a link simulation so that SNR scaling at the demodulator output is preserved. On the other hand, it�s sometimes nec-essary to model the absolute voltage levels at various points in a simulation topology. This needs to be done within models that depend on these absolute levels. In order to increase the efficiency of the simulation and avoid unneeded complexity, this should be done only when necessary.

Quadrature Signals

The above development is true whether the quantities r(t) and ri are real or com-plex. To emphasize that r is a complex quantity, we follow standard convention and use a tilde �~�.

The most important use of complex quantities in the communications field is associated with coherent signaling and reception of such signals. In this case, we assume that the receiver has knowledge of the phase of the incoming signal and



that orthogonal cosine and sine channels are identifiable. These channels are often referred to as �inphase� and �quadrature� (I/Q) channels. Figure 6 shows a simple conceptual model of an ideal quadrature receiver.

Figure 6. Ideal Quadrature Receiver Indicating Use of Complex Signal and Noise Quantities

In this model, we assume that I and Q envelopes are ideally extracted from an RF signal with known center frequency. There are many models within VSS that simulate non-ideal behaviors within the down-conversion process, but these are beyond the scope of this note. We make the assumption that the I/Q channels are independent, being ideally down-converted from orthogonal waveforms. In this case, the I/Q demodulator noise components are independent, with each posessing exactly the same statistics as described previously. Figure 6 illustrates the analog processing of analog waveforms. The demodulator which operates on the I/Q envelopes output from this downconverter could also be imple-mented in analog hardware. In recent years it has become more common to employ digital techniques to implement demodulators, however. In this case, the signal is sampled at a rate that may be four times the symbol rate or higher, allowing digital filtering and other DSP techniques to be used in the demodula-tion process. In this scenario, we can often select the simulation parameters so that the digital simulation imitates the behavior of the target digital system more or less exactly. When the implementation is digital, the configuration in Figure 1 represents a typical implementation as well as the simulation model. The sam-pler would represent the sampling and A/D conversion process and the brick-wall filter would represent an anti-aliasing filter. Again, models are available within VSS to simulate the behavior of practical A/D conversion circuitry, but we do not discuss these in this note.

cos(" ct)

sin(" ct)

LPF

LPF

r(t) r( ) r ( ) r ( )c st t j t= +%

Received RF SignalI/Q Components

Of Complex Envelope

To Demodulator




We employ the usual notation for complex-envelope signals (See Jeruchim, Bal-aban and Shanmugan[2], for example) and write:

, (15)

where

(16)

represents the complex envelope of the signal component and

(17)

represents the complex envelope of the noise component. As described, the noise component is jointly Gaussian with equal component variances.

ExamplesWe present two simple examples in this section that are intended to illustrate the procedure for generating BER and SER curves and plotting these curves in con-junction with theoretical curves.

QPSK Example

Figure 7 illustrates a simulation of QPSK link. In addition to the specific model names, the block diagram is annotated with generic names corresponding to the function of each individual block by choosing Schematic > Add Text. Addition-ally, on-screen display of the model parameters was disabled for clarity on the diagram by choosing Options > Project Options, clicking the Schematic/Dia-grams tab, and selecting Hide parameters.

Finally, the grid was turned off by choosing Diagram > Show Grid Snap to tog-gle this option off.

r� i s�i w� i+=

s�i sc i, jss i,+=

w� i wc i, jws i,+=



Figure 7. Illustration of QPSK Link - Block Diagram

In this example, a QPSK1_TX transmitter model is fed by a RND_D model. The RND_D model is a general-purpose source of pseudo-random digits. The alphabet size may be chosen as any power of 2. In this case, we have set the alphabet size to 2, so the digits are binary. The QPSK1_TX model is designed to accept bits and group them into pairs in order to select between the four signal quadrants. AWGN is added at the output of the transmitter model to corrupt the signal with noise. A QPSK1_RX model accepts the modulated, noise-cor-rupted waveform and attempts to estimate the original transmittted bit stream. The QPSK1_RX model has both real-valued I/Q outputs and detected outputs. The detection process is performed by thresholding the received I/Q samples against 0.0 and making bit declarations based on the result of these compari-sons. For equiprobable antipodal signals in AWGN, this procedure is optimum. A BER COUNTER model compares the output of the QPSK1_RX with the originally transmitted bit stream and keeps a tally of errors. It performs this operation by regenerating the psuedo-random bit stream internally and compar-ing each regenerated bit to the receiver estimates.

In the simulation of Figure 7, bits are generated by the random source at a rate of 2 per channel signalling interval. The transmitter maps these bits into seg-ments of a sampled-data waveform. This waveform represents the envelope of the analog RF signal that would actually be transmitted. In this case, the sam-pling rate has been chosen as 4 samples per symbol. Complex-envelope wave-forms and a scatter plot are illustrated in Figure 8 and Figure 9. The graphical plots used in this note have been annotated and customized by using the many plot customization functions available within VSS. These are accessed by right-clicking on a plot. The complex envelope of a signal can be viewed as either




�inphase�, �quadrature�, �magnitude�, or �phase�. The four views of the sam-pled-data QPSK transmitter output plus noise are shown in Figure 8.

Figure 8. QPSK Complex Envelope Waveforms

Rectangular signalling is selected at the transmitter, so the magnitude of the envelope is constant. QPSK is often considered to be a �constant-envelope� transmission scheme, but more general pulse shaping renders the envelope non-constant. The positive and negative modulating pulses are clearly seen in the I and Q waveforms. Also, the four samples per symbol are clearly seen. Energy-Per-Symbol noise power normalization is employed in the AWGNCH model. This set of waveforms was collected at a signal-to-noise ratio of 34 dB, so the pulse-like modulated waveforms are relatively noise-free. For QPSK, this is a SNR value much higher than any needed to provide reasonable error rate goals. It was selected to allow the waveforms to have a clear and observable shape. A �scatter� diagram is constructed by overlaying 1000 I/Q receiver outputs. This diagram is shown in Figure 9.



Figure 9. QPSK Scatter Plot

Note that the complex data number at the receiver I/Q output port represents the end-symbol decision statistic, produced at the symbol rate. This is in con-trast to the sampled-data waveform, which contains four data numbers (in this case of four samples per symbol) per channel symbol. It is the demodulated decision statistic on which the detector makes its bit decisions. The �scatter� diagram derives its name from the fact that the amount of �scatter� in the dia-gram can be used as a diagnostic for receiver performance. For example, this is the dataset that is used to derive the so-called �Error Vector Magnitude� (EVM) measure of performance. This measure of performance is called for in many communication standards.

From the perspective of implementing the signal-processing model in Figure 7, this type of simulation is often referred to as a �multi-rate� simulation because the transmitter turns two bits into four complex samples at its output. This con-version effectively doubles the rate of data values passed from model block to




model block in the simulation. Conversely, the receiver turns four complex sam-ples back into two bits, with an analogous rate reduction.

Figure 10 shows the graphed results of a BER simulation taken on the QPSK system.

Figure 10. QPSK Link Bit and Symbol Error Rates

Obviously, in order to generate a BER curve, some method of varying the SNR must be available. For this purpose, a stepped global variable may be established by simply typing it in to the Global Definitions window as shown in Figure 11.



Figure 11. Global Definitions Window

For this experiment, a stepped variable �Es_N0� has been established to sweep out a range of values from 0 to 12, in increments of 1. This stepped variable is used as the value for the transmitter output power parameter, thus raising the signal level as the variable is stepped. In this case, the AWGN power parameter is set to 0.0 dB. Note that the variation in received could have been effected by varying the noise power while keeping the signal level constant. Which approach is taken depends on the goals of the user. In this case, the user can verify that the value of the I/Q components of the complex envelope wave-forms in Figure 8 correspond to a value of 34 dB at which the waveforms were captured. To capture waveform or other data at a fixed power level, the user may fix the step range to one step or temporarily replace the transmitter power parameter with a fixed value. Otherwise, the waveform data would start at 0 dB and proceed in steps to the final value. Viewing the waveforms in this fashion is also sometimes desirable, since it allows the user to see noisy waveforms �clean up� as SNR increases.

Note that two distinct curves are present in Figure 10. The curve with markers actually represents two curves overlaid. The first curve is plotted with markers and represents the experimental BER tabulated by the symbol error rate counter. The solid curve is the theoretical curve representing BER as a function of energy per bit, or . This presentation of the QPSK error rate is often preferred, since it allows QPSK to be compared directly to BPSK on an energy efficiency per bit basis. QPSK can allow the same bit error probability on a per I/Q channel basis as does BPSK, although the energy transmitted per symbol is twice as high. When I and Q channels are driven by two independent bit streams, bit error rates for QPSK are the same as for BPSK, and can be calcu-

Es N0⁄

Eb N0⁄




lated exactly through the use of the Cumulative Distribution Function (CDF) for a Gaussian variable. The solid curve is derived in this way. Theoretical curves are available for a variety of coherent modulation schemes and can be added to any BER plot. These theoretical curves may be added to a BER plot by choos-ing Add Measurement, and then under the Meas. Type �System� category, selecting the �BER� sub-category. The curve on the right of the graph illustrates symbol error rate as a function of . For some modulation schemes, it is easier to analytically calculate symbol error rate than bit error rate. The inclusion of the additional curve demonstrates the flexibility to present either statistic. Customized error rate plots and other plots may be constructed by overlaying data files from multiple simulation runs. Plots can be customized as needed for reports with the extensive plot editing capabilities within VSS.

Results from the final experiment in this note are shown in Figure 12, Figure 13, and Figure 14. This simulation shows error rates and other data from a 64-QAM system. The block-diagram topology is the same as in Figure 7. The QPSK transmitter and receiver are simply replaced with the corresponding QAM mod-els.

In Figure 12, we exhibit the scatter plot for the QAM system operating at an of 34 dB. 64-QAM is a high-order modulation scheme and requires a

much higher received SNR to have reasonable error probability.

Es N0⁄

Es N0⁄



Figure 12. 64-QAM Scatter Plot

The 64-QAM I/Q trajectory plot is shown in Figure 13.




Figure 13. 64-QAM I/Q Trajectory Plot

The components of the complex waveform are the I and Q envelopes that are associated with the cosine and sine channels of the transmitted RF signal.The simulation is again taken at a sample rate of four samples per digital symbol. In this case, the symbols are 64-ary symbols. Note the difference between the I/Q trajectory and the scatter plot. The two are sometimes confused, since both are plotted as parametric X/Y graphs. The term parametric applies since the I and Q values are not plotted with respect to an independent variable, for example, time or frequency, but are both considered parameters plotted versus each other on X/Y (I/Q) axes. In the case of the I/Q trajectory plot, the curve is designed to show continuous motion between sampled-data points representing a com-plex analog waveform. For this reason, an I/Q trajectory plot is normally drawn with connecting lines between the markers to show transitions between sample points. In this simulation, four samples are employed to represent the analog signal during each symbol interval. We have elected to overlay 200 sym-bols on the I/Q trajectory plot. This means that a total of 800 samples from the



sampled-data waveform have been overlaid. Note that not all constellation points are visited during the 200 symbols shown here because the transmitted data is generated by a psuedo-random source and it is improbable that all sym-bols from 0 to 63 will appear in this short simulation run. This is in contrast to the QPSK case, where it is highly likely that all four constellation points will be visted in a run containing 200 symbols. In this simulation, there is no filtering or any other process that causes the I/Q trajectory to deviate from the ideal. Thus, points are clustered about the nominal signal constellation points. The AWGN noise perturbs their position only slightly. As in the previous QPSK experiment, the scatter plot of Figure 12 represents the overlaid end-symbol demodulated I/Q decision statistics. These numbers are not samples of an analog waveform, but are the actual real-valued complex decision statistics measured by the detec-tor in the process of making estimates of the original transmitted symbols. The optimum detector for reception in AWGN partitions the I/Q plane into �deci-sion regions� in an attempt to map received symbols into the closest point in the signal constellation. The symbol corresponding to this point is declared as the transmitted symbol. In this scatter plot, the grid lines are adjusted so that they correspond to the decision regions that are employed in the detector. QAM detectors in VSS assume a received I/Q statistic that is normalized to have unit energy. Receivers have an ideal front-end AGC that scales the received signal by the inverse of the transmitter power gain so that the decision statistic has unity average energy. For this reason, it is important not to introduce systematic gains or losses into the signal transmission path when using an amplitude-sensitive modulation type. Because QPSK is normally detected by thresholding against zero on the I and Q channels, QPSK is not sensitive to received amplitude lev-els. This not the case for 64-QAM or any other modulation scheme where the detection process depends on the received amplitude of the signal. If, for exam-ple, an amplifier model is incorporated in the transmit end, the amplifier charac-teristic should either be scaled for zero quiescent (DC) gain or the DC gain must be incorporated into the receiver AGC scaling parameter. This last adjustment is simple and is accomplished by simply setting a parameter �AGC_LEVEL� in the receiver.

The final figure in this note, Figure 14, shows BER and SER plots for the 64-QAM system. For simplicity, error rates are tabulated as a function of

only. The same statistics could be calculated vs , with the resulting BER

and SER curves translated to the right by . In contrast to QPSK, BER is not simple to calculate analytically for 64-QAM. BER is a strong function of the mapping from bits to constellation points in the 64-ary constel-

Eb N0⁄

Es N0⁄

10log10 64( ) 18dB≅




lation. It is much easier to calculate symbol error probability, since the probabil-ity of each individual constellation point crossing a detection boundary by virtue of noise can be calculated. These probabilities can be averaged and an exact SER can be analytically calculated. Both analytical and experimental SER�s are shown on the left of the graph. It is also popular to bound the SER by overesti-mating the probability of crossing a detection boundary. It is often assumed that the probability of crossing a detection boundary is bounded by the probability of an interior point crossing a detection boundary. It is easy to show that this provides an upper bound on symbol error probability.

Figure 14. BER and SER with Analytical Results for the 64-QAM Link



It involves calculations that are only a bit more tedious to predict exact symbol error rate for square QAM. This is the approach taken to generate the theoreti-cal curves in VSS as displayed in figure 13.

BER curves are shown on the left of the plot. As stated previously, BER depends on the mapping from symbols to constellation points. When a �pseudo-Gray� 1 mapping is used to map symbols onto constellation points, only one bit is in error when a constellation point is confused with one of its nearest neighbors by the receiver. In this case, one symbol error produces one bit error and bit error probability is proportional to symbol error probability. This is only a lower bound on bit error probability, however, accurate at high signal-to-noise ratios. We see that the bound does indeed diverge from the experimental results at low values of SNR, with the value saturating at the sym-bol error rate times the reciprocal of the number of bits per symbol, or at a value of , as opposed to the correct asymtotic value of 0.5.

The small squares indicating �active simulation� were being collected from a running simulation as they were overlaying previously stored results. The plot was captured as the simulation was running.

ConclusionWe have illustrated the signal scaling and modeling philosophy for communica-tion link simulation within VSS. We hope that this information is helpful to those users employing VSS simulations for end-to-end link performance analy-sis.

References[1] Carlson , A. B.; �Communication Systems�, second edition; McGraw-Hill, 1975.[2] 2)Jeruchim, Michel C., Balaban, Philip and Shanmugan, K. Sam; �Simulation of Communica-

tion Systems�; Plenum, 1992.[3] 3)Korn, I.; �Digital Communications�; Van Nostrand, 1985.

1. Strictly speaking, Gray coding is applicable only to circular constellations. Gray�s original work[3] de-scribed a procedure for digitally encoding the rotation of a shaft. Nevertheless, this term is often appliedto mapping schemes which attempt to cause �nearby� points in arbitrary constellations to differ in onlyone bit. We prefer the term �pseudo-Gray� mapping for these schemes.

1 1 64⁄( )�( ) 6⁄


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SIGNAL & NOISE SCALING IN VSS SIGNAL & NOISE SCALING IN VSS By: Dr. Kurt R. Matis Director of Systems Research SIGNAL AND NOISE SCALING FOR LINK ERROR ANALYSIS This document describes