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In this module, we first introduce the eye diagram and constellation diagram as qualitative ways of evaluating the symbol error probability of a digital communication system. We discuss various symbol alphabets, such as QAM, PAM, and PSK, and their associated decision regions. Finally, we derive the symbol error probability for PAM and QAM in additive white Gaussian noise, using the Q and erfc functions, and discuss Gray coding.

Recall the figure below, from the module Discrete-Time Implementation of Digital Communication . When the channel is trivial and noiseless and thepulses satisfy the Nyquist criterion (i.e., g [ k ] * q [ k ] = δ [ k ] ), the digital comm system will work perfectly, yielding y [ n ] = a [ n ] .

This flowchart shows movement from a[m] to a box containing an up arrow and the variable P, then with an arrow labeled a-up-arrow[k] shows movement to the right to a box labeled g[k]. To the right of this is an arrow labeled m-tilde[k] that points to the right at a box labeled h-tilde[k]. To the right of this is an arrow that points to the right at a plus-circle. Above the plus-circle is the expression w-tilde[k], which points down at the circle. To the right of the plus-circle is an arrow labeled v-tilde[k] that points to the right at a box labeled q[k]. To the right of this is an arrow labeled y-up-arrow[k] that points to the right at a box containing a down arrow and the variable P. To the right of this is an arrow that points to the right at a final expression, y[n]. This flowchart shows movement from a[m] to a box containing an up arrow and the variable P, then with an arrow labeled a-up-arrow[k] shows movement to the right to a box labeled g[k]. To the right of this is an arrow labeled m-tilde[k] that points to the right at a box labeled h-tilde[k]. To the right of this is an arrow that points to the right at a plus-circle. Above the plus-circle is the expression w-tilde[k], which points down at the circle. To the right of the plus-circle is an arrow labeled v-tilde[k] that points to the right at a box labeled q[k]. To the right of this is an arrow labeled y-up-arrow[k] that points to the right at a box containing a down arrow and the variable P. To the right of this is an arrow that points to the right at a final expression, y[n].

In practice, however,

  • the pulses g [ k ] and q [ k ] will be truncated to finite length,
  • the channel will not be trivial (i.e., h ˜ [ k ] δ [ k ] ), and
  • the channel will not be noiseless (i.e., w ˜ [ k ] 0 ),

leading to y [ n ] a [ n ] , in which case we must infer the value of a [ n ] from the received samples { y [ m ] } m = - . For now, we consider using only the single sample y [ n ] to infer a [ n ] .

Key question: What are the mechanisms by which errors are made?

To better understand error behavior, we can plot the “eye diagram” or the “constellation diagram” and calculate the symbol error rate (SER).

Eye diagrams

Usually used when a [ n ] R , the eye diagram is a plot which superimposes T -second segments of Re { y ( t ) } over the time intervals t [ n T - T 2 , n T + T 2 ) for many integers n .

This graph shows a series of waves on a graph of t and Re[y(t)], and specific points of these waves are drawn to correspond to a prism-like graph of width T called eye diagram. This graph shows a series of waves on a graph of t and Re[y(t)], and specific points of these waves are drawn to correspond to a prism-like graph of width T called eye diagram.

In MATLAB, the eye diagram can be made by superimposing P -sample segments of Re { y [ k ] } corresponding to intervals k { n P - P 2 , , n P + P - 1 2 } for many n . (Usually P 8 .)

This figure illustrates two prism-like graphs. The first has the caption, when the eye is 'open,' decisions will be reliable. The graph is in four segments and has various intersecting lines of rather uniform shape. In comparison, the second graph is labeled, as the eye 'closes', decisions get more unrelaible, and the graph displayed has the same four sections but with many more lines in the graph drawn and a chaotic amount of intersections across the different sections and curves. This figure illustrates two prism-like graphs. The first has the caption, when the eye is 'open,' decisions will be reliable. The graph is in four segments and has various intersecting lines of rather uniform shape. In comparison, the second graph is labeled, as the eye 'closes', decisions get more unrelaible, and the graph displayed has the same four sections but with many more lines in the graph drawn and a chaotic amount of intersections across the different sections and curves.

If a [ n ] C , eye diagrams can be plotted for both the “I” and “Q” channels using Re { y ( t ) } and Im { y ( t ) } , respectively.MATLAB for digital mod/demod with eye diagram:

"% design SRRC P = 16; % oversampling factoralpha = 0.5; % SRRC rolloff param D = 2; % truncation to [-DT,DT]g = srrc(D,alpha,P); % SRRC pulse Ng = length(g);% generate symbols N = 100; % # symbolsM = 4; % alphabet size sig2a = 1; % symbol variancea = pam(N,M,sig2a); % symbol sequence % pulse-amplitude modulatea_up = zeros(1,N*P); a_up(1:P:end) = a; % upsampled symbolsm = conv(a_up,g); % PAM % matched-filter demodulatey_up = conv(m,g); % use SRRC again % remove causal filtering delayleft = -P/2; % left edge of eye diagram k = left+[1:P*N]; % desired time indices dly = (Ng-1)/2+(Ng-1)/2;% delay due to pulsesy_up = y_up(k+dly); % remove delay y = y_up([1-left:P:P*N]);% downsample % plot received signalfigure(1) plot(left/P+[0:P*N-1]/P,y_up,’r’,[0:N-1],y,’.’);axis(‘tight’) title([‘SRRC (\alpha=‘,num2str(alpha),...‘) truncated to \pm’,num2str(D),’T’])xlabel([‘symbol index’])% plot eye diagram figure(2)Y_up = reshape(y_up,P,N); % extract segments plot(left/P+[0:P-1]/P,Y_up) % superimpose them title([‘SRRC (\alpha=‘,num2str(alpha),...‘) truncated to \pm’,num2str(D),’T’])xlabel([‘relative symbol index’]) This figure contains two graphs. The first is titled SRRC (α=0.5) truncated to ±2T The horizontal axis is labeled as symbol axis, and ranges from 0 to 100. The graph is a seres of nonuniform waves with peaks in several spots labeled by blue dots. The second graph has the same title, but the horizontal axis is labeled relative symbol index, and it shows a large number of multi-colored curves all reaching from different spots from either side of the figure but meeting at four intersection points along the vertical axis, at vertical values of -1.5, -0.5, 0.5, and 1.5. This figure contains two graphs. The first is titled SRRC (α=0.5) truncated to ±2T The horizontal axis is labeled as symbol axis, and ranges from 0 to 100. The graph is a seres of nonuniform waves with peaks in several spots labeled by blue dots. The second graph has the same title, but the horizontal axis is labeled relative symbol index, and it shows a large number of multi-colored curves all reaching from different spots from either side of the figure but meeting at four intersection points along the vertical axis, at vertical values of -1.5, -0.5, 0.5, and 1.5.

Constellation diagrams

The constellation diagram is a plot of Im { y [ n ] } vs. Re { y [ n ] } for many integers n . When the comm system is working well, the points cluster around thesymbol alphabet values:

This figure consists of two graphs. The first has sixteen splat-marks that follow a general grid, occurring from -1 to 1 horizontally, -1 to 1 vertically, and everything evenly-spaced in between. The second is similar except that it only contains four splats, all occurring at a vertical value of 0, and evenly spaced across the graph from -1.5 to 1.5. This figure consists of two graphs. The first has sixteen splat-marks that follow a general grid, occurring from -1 to 1 horizontally, -1 to 1 vertically, and everything evenly-spaced in between. The second is similar except that it only contains four splats, all occurring at a vertical value of 0, and evenly spaced across the graph from -1.5 to 1.5.

Recall that y [ n ] C due to the complex-baseband channel model, regardless of whether a [ n ] R or a [ n ] C .

Sometimes it is instructive to superimpose a plot of Im { y [ n ] } vs. Re { y [ n ] } , which approximates the trajectory of y ( t ) in the complex plane:

This is a chaotic graph of hundreds of random-looking yellow lines that connect to a sixteen-dot grid across the graph from -1 to 1 vertically and horizontally. This is a chaotic graph of hundreds of random-looking yellow lines that connect to a sixteen-dot grid across the graph from -1 to 1 vertically and horizontally.

MATLAB for digital mod/demod with constellation diagram:

% design SRRC P = 8; % oversampling factoralpha = 0.5; % SRRC rolloff param D = 2; % truncation to [-DT,DT]g = srrc(D,alpha,P); % SRRC pulse Ng = length(g);% generate symbols N = 100; % # symbolsM = 4; % (sqrt) alphabet size sig2a = 1; % symbol variancea = qam(N,M,sig2a); % symbol sequence % pulse-amplitude modulatea_up = zeros(1,N*P); a_up(1:P:end) = a; % upsampled symbolsm = conv(a_up,g); % PAM % matched-filter demodulatey_up = conv(m,g); % use SRRC again % remove causal filtering delayk = [1:P*N]; % desired time indicesdly = (Ng-1)/2+(Ng-1)/2;% delay due to pulses y_up = y_up(k+dly); % remove delayy = y_up(1:P:end); % downsample % plot received signalfigure(1) plot3([0:P*N-1]/P,real(y_up),imag(y_up),... ‘r’,[0:N-1],real(y),imag(y),’.’); xlabel(‘symbol index’);ylabel(‘I’); zlabel(‘Q’); view(20,30); % to see full trajectory%also try view(0,90), view(0,0), view(90,0) % plot constellation diagramfigure(2) plot(real(y_up),imag(y_up),’y’,real(y),imag(y),’.’);xlabel(‘I’); ylabel(‘Q’); title([‘SRRC (\alpha=‘,num2str(alpha),...‘) truncated to \pm’,num2str(D),’T’])axis(‘equal’); This figure contains two graphs. The second is a chaotic graph of hundreds of random-looking yellow lines that connect to a sixteen-dot grid across the graph from -1 to 1 vertically and horizontally. The first appears to be a three-dimensional  version of the same thing. This figure contains two graphs. The second is a chaotic graph of hundreds of random-looking yellow lines that connect to a sixteen-dot grid across the graph from -1 to 1 vertically and horizontally. The first appears to be a three-dimensional  version of the same thing.

QAM : “quadrature amplitude modulation”
PAM : “pulse amplitude modulation”
PSK : "phase shift keying"

This figure contains three graphs. The first is labeled 16-QAM, and shows on a graph of Re against Im a grid of sixteen evenly-spaced dots, centered at the origin, and the lateral distances between the dots is measured as variable delta. The second is labeled 8-PAM, and plots eight evenly-spaced dots along the horizontal axis on a graph of Re against Im. The width between the dots is measured as variable delta. The third graph is titled 8-PSK, and shows eight dots arranged in a circle around the origin, with distance between them measured as variable delta. This figure contains three graphs. The first is labeled 16-QAM, and shows on a graph of Re against Im a grid of sixteen evenly-spaced dots, centered at the origin, and the lateral distances between the dots is measured as variable delta. The second is labeled 8-PAM, and plots eight evenly-spaced dots along the horizontal axis on a graph of Re against Im. The width between the dots is measured as variable delta. The third graph is titled 8-PSK, and shows eight dots arranged in a circle around the origin, with distance between them measured as variable delta.

Note that: "QPSK" = 4-QAM = 4-PSK and "BPSK" = 2-PAM = 2-PSK

When the alphabet entries are spaced by Δ and picked with equal probability, the symbol variance σ a 2 = E { | a [ n ] | 2 } obeys:

alphabet M 2 -QAM M -PAM M -PSK
σ a 2 Δ 2 6 ( M 2 - 1 ) Δ 2 12 ( M 2 - 1 ) Δ 2 4 sin 2 ( π / M )

Decision regions

A reasonable way to infer the transmitted symbol a [ n ] from the received sample y [ n ] is to decide that a [ n ] was the alphabet element nearest to y [ n ] .

Nearest-element decision making is equivalent to using decision regions whose boundaries are equidistant from the two nearest alphabet elements:

This figure contains four graphs. The first is labeled 16-QAM, and shows on a graph of Re against Im a grid of sixteen evenly-spaced dots, centered at the origin. In between the grid of dots are dashed lines that make their own grid on the graph. The second plots Re against Im on a graph labeled 4-PAM, with four evenly-spaced dots along the horizontal axis, and two vertical dashed lines in between dots 1 and 2, and 3 and 4, separating the dots a column-like grid. The third graph is labeled 8-PSK and shows eight dots in the shape of a circle centered at the origin. In between each dot is a line that extends from the origin out to the quadrants of the graph, dividing the dots up into individual sections. The fourth graph is labeled with three question marks, and shows a grid of nine dots centered at the origin, with the top-right dot missing, and a series of dashed lines in between the dots both horizontally and vertically dividing them. In the absence of the top-right dot, there is a diagonal dashed line extending from the grid of dashed lines further into the first quadrant. This figure contains four graphs. The first is labeled 16-QAM, and shows on a graph of Re against Im a grid of sixteen evenly-spaced dots, centered at the origin. In between the grid of dots are dashed lines that make their own grid on the graph. The second plots Re against Im on a graph labeled 4-PAM, with four evenly-spaced dots along the horizontal axis, and two vertical dashed lines in between dots 1 and 2, and 3 and 4, separating the dots a column-like grid. The third graph is labeled 8-PSK and shows eight dots in the shape of a circle centered at the origin. In between each dot is a line that extends from the origin out to the quadrants of the graph, dividing the dots up into individual sections. The fourth graph is labeled with three question marks, and shows a grid of nine dots centered at the origin, with the top-right dot missing, and a series of dashed lines in between the dots both horizontally and vertically dividing them. In the absence of the top-right dot, there is a diagonal dashed line extending from the grid of dashed lines further into the first quadrant.

When a [ n ] = a , the symbol error rate (SER) equals the probability that y [ n ] lies outside the decision region corresponding to alphabet member a . Writing y [ n ] = a [ n ] + e [ n ] , we represent the cumulative effect of noise and ISI by the error e [ n ] . Usually we model e [ n ] as a Gaussian random variable with mean 0 and variance σ e 2 .

Symbol error rate (ser) for M -pam

Let's first consider an M -PAM alphabet, where a [ n ] R . Since the decision regions show that Im { y [ n ] } is not useful, we'll consider only the real parts of y [ n ] and e [ n ] .

When a [ n ] = a , we have y [ n ] = a + e [ n ] , implying that y [ n ] is Gaussian with mean a and variance σ e 2 , abbreviated as “ N ( a , σ e 2 ) ”. This is illustrated below for the case of 4-PAM:

Formally, we say that p y [ n ] | a [ n ] ( y | a ) , the probability density function (pdf) of y [ n ] conditioned on a [ n ] = a , obeys

p y [ n ] | a [ n ] ( y | a ) = 1 2 π σ e 2 exp - ( y - a ) 2 2 σ e 2 N ( a , σ e 2 ) .

Basically, p y [ n ] | a [ n ] ( y | a ) tells us how likely it is that y [ n ] = y given that a [ n ] = a .

Consider first the case where a is an “interior” (not an “edge”) element of the symbol alphabet.Given that a [ n ] = a , we make an error when y [ n ] < a - Δ 2 or when y [ n ] > a + Δ 2 . To find the probability of the latter error event, i.e.,

Pr { y [ n ] > a + Δ 2 a [ n ] = a ,

we integrate p y [ n ] | a [ n ] ( y | a ) over y ( a + Δ 2 , ) :

a + Δ 2 p y [ n ] | a [ n ] ( y | a ) N ( a , σ e 2 ) d y = a + Δ 2 1 2 π σ e 2 exp - ( y - a ) 2 2 σ e 2 d y .

The integral represents the shaded area below:

This figure is the same bell curve as in the previous figure, except that the area to the right of the rightmost vertical dashed line is shaded yellow, and the bell curve itself is labeled p_y[n]a[n](y|a). This figure is the same bell curve as in the previous figure, except that the area to the right of the rightmost vertical dashed line is shaded yellow, and the bell curve itself is labeled p_y[n]a[n](y|a).

This integral is often solved via

x 1 2 π σ 2 exp - ( y - μ ) 2 2 σ 2 N ( μ , σ 2 ) d y = Q x - μ σ ,

using the “Q function”:

This figure plots a curve on a graph of x against Q(x) the curve begins at (-3, 1) and decreases at an increasing rate until (0, 0.5), where the curve continues decreasing, but at a decreasing rate, where it meets the bottom-right corner of the graph at (3, 0). This figure plots a curve on a graph of x against Q(x) the curve begins at (-3, 1) and decreases at an increasing rate until (0, 0.5), where the curve continues decreasing, but at a decreasing rate, where it meets the bottom-right corner of the graph at (3, 0).

While the Q function is not represented in MATLAB, it can be calculated using the “complementary error function” erfc :

Q ( x ) = 1 2 erfc x 2

In any case, the latter error event occurs with probability

Pr { y [ n ] > a + Δ 2 a [ n ] = a = Q ( a + Δ 2 ) - a σ e = Q Δ 2 σ e .

By symmetry, the former error event probability is also

Pr { y [ n ] < a - Δ 2 a [ n ] = a = Q Δ 2 σ e .

Since these two events are disjoint, the probability of making a decision error on an interior symbol equals their sum:

Q Δ 2 σ e + Q Δ 2 σ e = 2 Q Δ 2 σ e .

For edge symbols, we experience half the decision error probability, since there is only one decision boundary to cross.

Finally, we average over the conditional error probabilities:

Pr { error } = a alphabet Pr { error | a [ n ] = a } Pr { a [ n ] = a } = 1 / M for all a = Q Δ 2 σ e · 2 M + 2 Q Δ 2 σ e · M - 2 M = 2 M - 1 M Q Δ 2 σ e ,

Using σ a 2 = Δ 2 12 ( M 2 - 1 ) , we can finally write

SER M -PAM = 2 M - 1 M Q 3 ( M 2 - 1 ) σ a 2 σ e 2 .

Symbol error rate (ser) for M 2 -qam

With QAM, we have complex-valued y [ n ] , a [ n ] , e [ n ] . We'll assume that Re { e [ n ] } and Im { e [ n ] } are uncorrelated and equal variance.To calculate SER, we can re-use the PAM approach with a few modifications:

  1. integration is done on the complex plane,
  2. σ e 2 -variance e [ n ] σ e 2 2 -variance Re { e [ n ] } & Im { e [ n ] } ,
  3. M 2 -QAM has 4 corner points, 4 ( M - 2 ) edge points, and M 2 - 4 M + 4 interior points,
  4. calculate Pr { error | a [ n ] = a } via 1 - Pr { correct | a [ n ] = a } , since the regions of integration are simpler:
    This figure contains two graphs. The first plots Re against Im, and is titled Pr[error|a[n] = a]. It displays an array of sixteen evenly-spaced dots and a grid of dashed lines around the dots. The square containing the dot in the second row and the third column of the grid is labeled a, and is the only portion of the graph not shaded yellow. Conversely, the second graph has the same grid, same array, but this second row, third column dot is the only shaded box of the graph. The second graph is titled Pr[correct|a[n] = a]. This figure contains two graphs. The first plots Re against Im, and is titled Pr[error|a[n] = a]. It displays an array of sixteen evenly-spaced dots and a grid of dashed lines around the dots. The square containing the dot in the second row and the third column of the grid is labeled a, and is the only portion of the graph not shaded yellow. Conversely, the second graph has the same grid, same array, but this second row, third column dot is the only shaded box of the graph. The second graph is titled Pr[correct|a[n] = a].

After a bit of algebra, we find

SER M 2 -QAM = 1 - 1 - 2 M - 1 M Q 3 ( M 2 - 1 ) σ a 2 σ e 2 2 .

Bit error rate (ber) and gray coding

With an M -ary alphabet, there are log 2 M bits per symbol, so 1 symbol error could cause up to log 2 M bit errors.

Gray coding is a clever way of mapping bits to symbols so that neighboring symbols differ by only a single bit.Since the vast majority of errors occur when y [ n ] falls into a neighboring decision region, Gray coding yields BER SER.

This figure contains two graphs. The first is titled 4-PAM, and on a graph plotting Re against Im, there are four evenly-spaced dots on the horizontal axis, labeled from left to right 00, 01, 11, and 10. There are dashed vertical lines in between the dots. The second graph is titled 16-QAM, and shows a grid of 16 evenly-spaced dots centered at the origin, with dashed lines creating a grid around the dots. The dots are labeled across in the first row 0000, 0100, 1100, 1000. In the second row, they are labeled 0001, 0101, 1101, and 1001. In the third row they are labeled 0011, 0111, 1111, and 1011. In the fourth and final row they are labeled 0010, 0110, 1110, and 1010. This figure contains two graphs. The first is titled 4-PAM, and on a graph plotting Re against Im, there are four evenly-spaced dots on the horizontal axis, labeled from left to right 00, 01, 11, and 10. There are dashed vertical lines in between the dots. The second graph is titled 16-QAM, and shows a grid of 16 evenly-spaced dots centered at the origin, with dashed lines creating a grid around the dots. The dots are labeled across in the first row 0000, 0100, 1100, 1000. In the second row, they are labeled 0001, 0101, 1101, and 1001. In the third row they are labeled 0011, 0111, 1111, and 1011. In the fourth and final row they are labeled 0010, 0110, 1110, and 1010.

Questions & Answers

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Source:  OpenStax, Introduction to analog and digital communications. OpenStax CNX. Sep 14, 2009 Download for free at http://cnx.org/content/col10968/1.2
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