Distribution of focal lengths used during three weeks in Hawai'i. Blue bars include shots that were stitched into panoramas, green bars represent only single-frame images.

Only the shots that I kept out of the 3,180 total were used to make the figure.

Interestingly, the short focal lengths see the most use, with the minimum possible length dominating the statistics, even more so when including the shots comprising the 205 panoramas. I guess that’s not unusual, since most of these are pictures of landscape (after all, it’s Hawai’i). However, I was obviously to lazy (or too hurried) to even once switch to the 14mm pancake lens that I lugged around with me.
There were a couple of situations (<1.5% of the pictures) where I likely could have used a longer lens, but had to settle for the 55mm long end of my 18-55mm zoom.

OTDM Recap

Some basics of OTDM / Nyquist WDM are explained in The Art of Nyquist WDM which I recommend reading if you’re new to the concept. Recently, I find the name OTDM much more fitting due to its duality with OFDM. However, what’s commonly termed OTDM in optics (multiplexing really short pulse trains) I assume stands for optical TDM, since it doesn’t make any use of an orthogonality relation (though technically, the tributaries are orthogonal).

Anyway, the basic premise of OTDM is that $K$ pulses are overlapping in time. Only by their special shape (the truncated sinc function) can they be detected without crosstalk. The sinc function has the fortunate property of having zeros at regular intervals. By making it so that these zeros occur at the centers of all neighboring symbol slots, there will only be one symbol not equal to zero at each slot center (this being the symbol that is centered on that particular slot). At all other times within the symbol slot, all symbols can have values other than zero and interfere. Hence, we immediately expect quite different signal statistics depending on the position within the symbol slot we are interested in.

The spectrum of the sinc function is rectangular (or almost rectangular for the truncated sinc) so that many such channels could be put very close to each other without spectral crosstalk.

Importance Sampling

In contrast to OFDM which was (in a special case) the superposition of randomly chosen letters from the same alphabet (the QAM constellation diagram) with equal weights, the weights of the different superposed symbols in OTDM are defined by the sinc function. At the symbol slot center, for instance, the weight for the symbol centered in that slot is unity while for all other symbols it is zero. In this case the PMF for a quadrature, the signal or its power can be simply determined from the constellation diagram of the modulation format.

At all other positions the weights are different and non-zero. We cannot therefore use combinatorical means to tackle this problem. Due to the low probabilities involved (which are expected to be similar to OFDM), an exhaustive numerical simulation is not feasible and a Monte-Carlo simulation will not yield the relevant low-probability details. We can, however, apply the concept of importance sampling to our Monte Carlo simulations [1]. Here, we bias the random drawing (of points from the constellation diagram for each symbol) to favor the occurrence of signals with high powers, i.e. we prefer to stay in the corners of the constellation diagram and switch to the opposite corner whenever the sign of the particular sinc-weight for a symbol becomes negative. In this way we get samples for which all the constituents are quite large and in phase. We have to retain some randomness, however, since we are interested in the PMF/CDF and not just the maximum possible value (more on that below). The biased probability for the symbols is compensated when later calculating the PMF.

Biasing toward higher sample powers results in a smaller number of samples with low power, which in turn leads to larger variance in that part of the PMF. We would like to have an accurate representation of the whole PMF, though. Hence, we will generally simulate the same modulation format with different bias values and later combine their results using the balance heuristic which is basically just a pretty optimal way of combining such data [2].

You will find the MATLAB code (not optimized for speed or anything) at the end of the article.

Signal Power

Since the code is available to play around with, we will look at just a few results to highlight the differences between OFDM and OTDM. In particular, the symbol length will be $16T$, i.e. 16 times the time between consecutive symbols (or the inverse symbol rate), and thus $K=16$. The alphabets used will be 16-QAM ($N=4$) and 64-QAM ($N=8$). As can be seen in Fig. 5 in the OFDM article, this results in probabilities in the vicinity of $10^{-28}$ for the largest powers in OFDM signals, something not feasible to be figured out by “regular” Monte Carlo techniques.

Figure 1: Cumulative distribution functions of sample power for OTDM signals with K = 16 overlapping symbols. Also shown are comparable distributions of OFDM signals (K = 16) and the χ-squared distribution corresponding to complex Gaussian signals. Samples are taken at the symbol slot boundaries.

The figure shows the worst case regarding high signal powers, with the sampling instants chosen at the symbol slot boundaries. See also the next section for more details. While the probabilities of the maximum values are the same (they depend only on $K$ and $N$ as shown in the OFDM post), the maximum observable power is less than half in OTDM. At high probabilities like $10^{-6}$ the difference is smaller. This is a result of the heterogeneous weights of the contributions from the different symbols to each sample, as opposed to equal (magnitudes of) weigths in OFDM.

Sometimes there will be “kinks” in the graphs so obtained by importance sampling. These are a result of outliers whose effect is amplified by the algorithm (in the code, data with very low dataweights contribute significantly more). This is a known problem of importance sampling. As these occur randomly, they might or might not be there. Their position is also random, so that they can be identified by multiple runs of the code.

Other Sample Times

In OFDM, the sample power distribution varied only little with sample timing within the symbol slot (cf. Figs. 6 and 7 in the OFDM post). As the constellation diagrams of consecutive OTDM symbols do not rotate relative to each other, it is quite straightforward to determine the maximum possible sample power at various instants within the symbol slot by simply adding the weight magnitudes (plus squaring and scaling). They can even be given analytically.$^1$

Figure 2: Maximum possible sample power versus sampling instant within the symbol slot. Colored markers correspond to curves in Fig. 3.

Unlike OFDM, the maximum power does not have multiple local maximums within the symbol slot, but varies by about an order of magnitude between symbol slot center and boundaries. The sample power CDFs vary accordingly:

Figure 3: Cumulative distribution functions of sample power for OTDM signals with K = 16 overlapping symbols and 16-QAM modulation for various sampling instances within the symbol slot.

For $t_S=0.5$ (the symbol slot center), the CDF corresponds to the constellation diagram, as at this instant only one of the overlapping symbols is non-zero.

Oversampling

For (always necessary) oversampling, the distributions for the appropriate sampling instances can simply be combined. Interestingly, for simple two-fold oversampling, half the samples will be taken in the symbol slot center and the other half at its boundaries. Thus, the worst case regarding symbol power is included and the high-power samples resulting thereof need to be accounted for by leaving appropriate headroom in the digital-analog converters, amplifiers and whatnot.

MATLAB Code

clear variables

LoS = 16; % length of symbol in units of symbol slots
QAM = 64; % number of possible values in each quadrature for 16-QAM - only powers of 2 allowed
NoS = 2^20; % number of Monte Carlo samples

offset = 0.5; % relative sampling position within symbol slot - 0.5 is center

% list of biasing coefficients used; need not be integer
explist = [0,1,2,3]; % works well for 16-QAM, LoS = 16
% explist = [0,2,4,6,8]; % works well for 64-QAM, LoS = 16

bins = linspace(0,16,1001); % set histogram bins for graphing

%%% set-up of some variables

normalization = 1/sqrt(2/3*(QAM-1)); % alphabet normalization
normalization = normalization / 0.987345; % unit energy per symbol for LoS = 16
% normalization = normalization / 0.993669; % unit energy per symbol for LoS = 32
normweight = (1/QAM)^LoS; % probability of any particular symbol w/o biasing

% sinc coefficients of each sample within the symbol
coeffs = sinc((-LoS/2:LoS/2-1) + offset);

% constellation diagram
NoV = sqrt(QAM); % number of possible values in I and Q
values = 0:2:2*(NoV-1); values = values - NoV + 1; % possible values in I and Q for each symbol
values = values * normalization;
symbols = repmat(values, [NoV,1]) + 1i * repmat(values.', [1,NoV]);
symbols = symbols(:); % complex QAM alphabet

bins = [bins, Inf];
PMF = zeros(length(bins), length(explist));
bincounts = PMF;

%%% importance sampling Monte Carlo simulation

for jj = 1:length(explist)

    % biasing is based on distance to opposite corner symbols (depending on coeff sign)
    distplus = abs(symbols - symbols(1));
    wplus = 1 ./ (1 + distplus).^explist(jj); % weights avoid division by zero
    wplus = wplus / sum(wplus); % normalization

    distminus = abs(symbols - symbols(QAM)); % opposite corner
    wminus = 1 ./ (1 + distminus).^explist(jj);
    wminus = wminus / sum(wminus);

    origdata = rand(NoS,LoS); % generate random data (symbol plus neighbors)

    data = zeros(NoS,LoS);
    dataweights = zeros(NoS,LoS);

    for kk = 1:LoS
        if coeffs(kk) >= 0
            for ii = (QAM):-1:1; % draw from biased distribution
                data(origdata(:,kk) < sum(wplus(1:ii)),kk) = symbols(ii);
                dataweights(origdata(:,kk) < sum(wplus(1:ii)),kk) = wplus(ii);
            end
        else % for negative coeffs bias towards opposite corner
            for ii = (QAM):-1:1;
                data(origdata(:,kk) < sum(wminus(1:ii)),kk) = symbols(ii);
                dataweights(origdata(:,kk) < sum(wminus(1:ii)),kk) = wminus(ii);
            end
        end
    end

    data = data .* repmat(coeffs, [NoS, 1]); % each row corresponds to one sample

    powers = abs(sum(data, 2)).^2;
    dataweights = prod(dataweights, 2);

    if explist(jj) ~= 0 % remove IS outliers
        index = find(dataweights >= normweight);
        powers = powers(index);
        dataweights = dataweights(index) * NoS / 4;
            % division by 4 is necessary to compensate for biasing towards only 1 corner
    else
        dataweights = dataweights * NoS;
    end

    % build PMF histograms for each biasing value
    for ii = 1:length(bins)-1
        index = find(powers >= bins(ii) & powers < bins(ii+1));
        PMF(ii,jj) = sum(normweight ./ dataweights(index));
        bincounts(ii,jj) = length(index);
    end

end

% average histograms using balance heuristic
weights = PMF .* bincounts ./ repmat(sum(PMF .* bincounts, 2), [1, length(explist)]);
weights(isnan(weights)) = 0; % bincount of zero causes weights to become NaN

avgPMF = sum(PMF .* weights, 2);

% complementary CDF
for ii=1:length(avgPMF)
    CDF(ii) = sum(avgPMF(ii:end));
end

Note the normalization of the symbol amplitude in order to obtain unit mean energy per symbol. This accounts for the distribution of the constellation points for the particular QAM format as well as the truncated sinc pulse shape. Since

$$\intop_{-K/2}^{K/2} \mathrm{sinc}^2 x \, dx < 1 \quad \text{for} \quad K < \infty$$

the energy reduction resulting from truncation is compensated by appropriate scaling.

1 The main problem is to find

$$S_\mathrm{max}^2 = \left[\sum_{k=0}^{K-1} \left| \mathrm{sinc} \left(k - K/2 + t_S\right) \right| \right]^2 = \left[\sum_{k=0}^{K-1} \left| \frac{\sin \bigl[\pi \left(k - K/2 + t_S\right)\bigr]}{\pi \left(k - K/2 + t_S\right)} \right| \right]^2$$

which is then multiplied by the highest power occuring in the constellation diagram of the particular modulation format (which also depends on $K$ when properly normalized). For simplicity, we'll assume $K$ to be even and we will only try and find $S_\mathrm{max}$, and leave the squaring to the interested reader. The sine in the numerator will then have a constant value, determined by $t_S$ that simply alternates its sign with $k$ and thus can be written

$$S_\mathrm{max} = \frac{\sin \pi t_S}{\pi}\sum_{k=0}^{K-1} \left| \frac{c_S \left(-1\right)^k}{k - K/2 + t_S} \right|$$

with

$$c_S = \begin{cases} 1 & K/2 \text{ even}\\ -1 & K/2 \text{ odd}\end{cases}$$

We split the sum to get rid of the negative denominator and with it the absolute value function and the sign coefficients:

$$S_\mathrm{max} = \frac{\sin \pi t_S}{\pi}\sum_{k=0}^{K/2-1} \frac{1}{k + t_S} - \frac{1}{k - K/2 + t_S}$$

These are then just two general harmonic series. Their solution can be given by generalized harmonic numbers or in terms of the Digamma function $\psi(\cdot)$ [3]

$$S_\mathrm{max} = \frac{\sin \pi t_S}{\pi} \left[ \psi\left(t_S - \frac{K}{2}\right) + \psi\left(t_S + \frac{K}{2}\right) - 2 \psi\left(t_S \right) \right]$$

[1] Importance Sampling [Wikipedia]
[2] E. Veach, Robust Monte Carlo methods for light transport simulation, dissertation, Stanford University, 1998.
[3] T. M. Rassias and H. M. Srivastava, "Some classes of infinite series associated with the Riemann zeta and polygamma functions and generalized harmonic numbers," Applied Mathematics and Computation, vol. 131, pp. 593–605, 2002. ]]> http://www.marcuswinter.de/archives/1612/feed 0 http://www.marcuswinter.de/archives/1610 http://www.marcuswinter.de/archives/1610#comments Mon, 05 Dec 2011 13:19:46 +0000 Marcus http://www.marcuswinter.de/?p=1610 I am still catching up to getting some order into the pictures I took during our last vacation on Hawai’i. And that was back in September! Anyway, here are some of the better panoramas from our first stop, Big Island. Click through for the large version at flickr, with a short description.

scenic point on Hwy 19, Big Island, Hawai'i (panorama)

Pololu Valley lookout, Big Island, Hawai'i (panorama)

halfway down Pololu Valley, Big Island, Hawai'i (panorama)

Pololu Valley, Big Island, Hawai'i (panorama)

Pololu Valley, Big Island, Hawai'i (vertorama)

Rainbow Falls, Big Island, Hawai'i (panorama)

cove in Onomea Bay, Big Island, Hawai'i (panorama)

off Hawai'i Belt Road, Kau, Big Island, Hawai'i (panorama)

Halema'uma'u Crater in Kilauea Caldera, Volcanoes National Park, Big Island, Hawai'i (panorama)

cinder cone off Saddle Road, Big Island, Hawai'i (panorama)

clouds rolling in, Mauna Kea Access Road, Big Island, Hawai'i (panorama)

central Big Island seen from Mauna Kea Visitor Center, Hawai'i (panorama)

seeing double at Mauna Kea Visitor Center, Big Island, Hawai'i (panorama)

W.M. Keck Observatory, Mauna Kea summit, Big Island, Hawai'i

cinder cones and observatories, Mauna Kea summit, Big Island, Hawai'i

Lake Waiau, Mauna Kea summit, Big Island, Hawai'i (panorama)

Kealakekua Bay, Big Island, Hawai'i (panorama)

Kealakekua Bay, Big Island, Hawai'i (panorama)

First of all, for a finite number of subchannels, the probability distribution will be a discrete one, described by a probability mass function (PMF). Second, unlike the Gaussian, the supports of the involved distributions are bounded and there exists a maximum value that occurs with finite probability. So, what could be more interesting than to figure out the exact PMFs involved? Ultimately, we are interested in the statistics of the sample powers. Often, these are given as a peak-to-average-power ratio (PAPR) [1]. We will discuss why this has almost no meaning in just a little while.

We’ll start with OFDM in this article since it allows us to use some results from combinatorics to save some work. OTDM, on the other hand, will require a brute force – or comprehensive – approach, and we’ll save that for later.

Caveat emptor: As with almost all articles on this blog, the content presented is not just reproduced from some book, but derived with the help of the cited sources only. I cannot guarantee that it is correct or that it is not treated elsewhere in a more complete way. If you find it discussed somewhere not cited here, or if you find an error, please let me know in the comments.

OFDM Recap

In the article on OFDM Basics an OFDM symbol $C$ was described as

$$C(t) = \sum_{k=0}^{K-1} C_k(t) = \sum_{k=0}^{K-1} c_k \cdot \exp\bigl(i\omega_k t\bigr)\tag{1}$$

where the $c_k$ are the coefficients containing the data. Often, these are taken from a complex M-QAM alphabet which resembles a square in phase space when $M = 2^{2m}$ with integer $m$, e.g. $M=4$, $M=16$ or $M=64$. If we (initially) restrict our analysis to the case $t=0$, i.e. a particular sampling point in the duration of the symbol, then the exponentials vanish for all $k$ and all subchannel symbols $C_k$ would use the same alphabet – without this restriction, the alphabet for each symbol would be rotated by a $t$- and $k$-dependent amount.

We start by looking at the signal statistics for a single quadrature (I or Q), since both are equal with square alphabets / constellation diagrams, and then combine these statistics into a PMF for the symbol power.

Single Quadrature

The signal in a single quadrature of an M-QAM symbol can take $N = \sqrt{M}$ different values. The exact possible values depend on our normalization. If we normalize to average power $\bigl\langle P \bigr\rangle = 1$ then we have the following alphabets

$$\begin{gather}
A_\mathrm{4-QAM} = \frac{1}{\sqrt{K} \sqrt{2}} \left\lbrace -1, 1 \right\rbrace \\
A_\mathrm{16-QAM} = \frac{1}{\sqrt{K} \sqrt{10}} \left\lbrace -3, -1, 1, 3 \right\rbrace \tag{2}\\
A_\mathrm{64-QAM} = \frac{1}{\sqrt{K} \sqrt{42}} \left\lbrace -7, -5, -3, -1, 1, 3, 5, 7 \right\rbrace
\end{gather}$$

Since all coefficients $C_k(t=0)$ are chosen from the same alphabet, the order in which they appear in the sum $C(t=0)$ does not matter, so we are reduced to the problem of figuring out the possible combinations of symbols in $A$ in a sequence of length $K$ and the probability of each possibility occurring. The problem is equivalent to the combinatorical problem of choosing $K$ elements from the alphabet $A$ with repetitions, but without order (since the order of the occurrence of the various symbols is not relevant for their sum), and the number of combinations is given by the binomial coefficient [2]

$$\begin{pmatrix} N + K – 1 \\ K \end{pmatrix} \tag{3}$$

The probability of each such combination is then determined by the multinomial distribution [3].

$$p = \begin{cases}\frac{K!}{\prod x_n!}\left(\frac{1}{N}\right)^K & \text{when} \; \sum_{n=1}^N x_n = K \\ 0\vphantom{\frac{1}{1}} & \text{otherwise}\end{cases} \tag{4}$$

where $x_n$ is the number of occurrences of symbol $n$ from alphabet $A$. In (4) we assumed that all symbols appear with equal probability $N^{-1}$. Figure 1 compares the number of combinations for various $N$ as described by (3) and with a comprehensive approach, which has $N^K$ combinations. The comprehensive approach produces combination counts which are orders of magnitude larger for even small values of the sequence length $K$, and can only be analyzed using Monte Carlo techniques except for small $N$ and $K$.

Figure 1: Number of possible combinations of choosing K times from a set of size N (with repetitions); regarding order (solid lines, comprehensive) and disregarding order (dashed lines).

The MATLAB code to compute the PMF of a single quadrature signal is quite straightforward$^2$:

clear variables
NoS = 8; % number of subchannels
NoV = 4; % number of possible values in each quadrature for 16-QAM - only powers of 2 allowed
normalization = 1 / sqrt(NoS) / sqrt(2/3*(NoV^2 - 1)); % alphabet normalization according to (2)
    % see http://www.dsplog.com/2007/09/23/scaling-factor-in-qam/
    % for the origin of the factor (2/3*(NoV^2 - 1)
probability = 1 / NoV; % (uniform) probability of each value to occur

values = 0:2:2*(NoV-1); values = values - NoV + 1; % generate integer part of the alphabet

support = min(values)*NoS:max(values)*NoS; % range of integers that the signal can take

[count, tuples] = nsumk(NoV, NoS);
    % listing of (ordered) NoV-tuples of non-negative integers adding up to NoS
    % see http://www.mathworks.com/matlabcentral/fileexchange/28340-nsumk for details on nsumk()

p = factorial(NoS) ./ prod(factorial(tuples), 2) * probability^NoS; % multinomial PDF of each tuple row
sums = sum(tuples .* repmat(values, [size(tuples, 1), 1]), 2); % sum of all symbols

% add PMFs of equal sum values but different addends (if any)
PMF = zeros(size(support));
for ii = 1:size(p)
    index = find(support == sums(ii));
    PMF(index) = PMF(index) + p(ii);
end

support = normalization * support;

and a typical PMF plot will look like this:

Figure 2: Probability mass functions for two exemplary combinations of signal levels (N) and number of subchannels (K). More signal levels will lead to more combination possibilities and thus a finer-grained PMF.

The PMF is unfortunately not a very clear tool due to the colored dots all over the place – often it is easier to compare distributions with discrete values by their cumulative density function which is – like for continuous probability densities – the integral of the $\mathrm{PDF}_C(C)$.

$$\mathrm{CDF}_C(C) = \intop_{-\infty}^C \mathrm{PDF}_C(C’) \, dC’ \tag{5}$$

which in MATLAB can be done via

CDF = (sum(tril(repmat(PMF, [size(PMF, 2), 1])),2);

The CDFs are step functions:

Figure 3: Cumulative distribution functions of single-quadrature signal levels for various QAM formats and K = 8 (top), K = 16 (middle) and K = 32 (bottom) subchannels, together with the CDF for a Gaussian distribution with equal power.

We can see that each QAM format and subchannel count will result in a slightly different distribution of values. As $N$ and $K$ increase, the CDFs (and thus PDFs) approach the Gaussian distribution. However, each $(N,K)$ pair has a distinctive minimum value $C_\mathrm{min} = K \cdot \min(A)$ and associated probability $\mathrm{PDF}_C(C_\mathrm{min})$ for which we will give a closed expression later. The CDF can be used to determine clipping probabilities and the CDF of the clipped portion of the signal (where we also have to pay attention to the positive side which does, however, have the same statistics due to symmetry).

Signal Power

For each in-phase amplitude $C_i$ with associated probability $\mathrm{PMF}_C(C_i)$ and quadrature amplitude $C_q$ with associated probability $\mathrm{PMF}_C(C_q)$ we can assign a sample power

$$P = C_i^2 + C_q^2\tag{6a}$$

with joint probability

$$\mathrm{PMF}_P(P) = \mathrm{PMF}_C(C_i) \cdot \mathrm{PMF}_C(C_q) \tag{6b}$$

since the data in both quadratures is independent. We simply calculate all power values and associated probabilities, clean the list by combining the probabilities of equal power values and sort the values$^2$:

ll = length(support);
power = repmat(support,[ll, 1]).^2 + repmat(support.',[1, ll]).^2; % all possible power values
p = repmat(PMF_C,[ll, 1]) .* repmat(PMF_C.',[1, ll]); % joint probability

support_P = unique(power(:)); % find unique power values and sort them
PMF_P = zeros(size(support_P));

for ii = 1:length(p(:))
    index = find(support_P == power(ii));
    PMF_P(index) = PMF_P(index) + p(ii);
end

The result looks like this:

Figure 4: Probability mass function of the sample power for N = 8 (64-QAM), K = 8 subchannels and unit mean power.

Again, it’s more useful to plot the CDF. In particular, the complimentary CDF can be used to determine the probability of the sample power being larger than some value of interest, which can be related to the PAPR.

Figure 5: Cumulative distribution functions of sample power for various QAM formats and K = 8 (top), K = 16 (middle) and K = 32 (bottom) subchannels, together with the CDF for a χ-squared distribution with two degrees of freedom and equal power.

While there are significant deviations from the $\chi^2$ distribution$^3$ visible in Figure 5, these occur mostly at low probabilities.

Other Sample Times

Generally, a single OFDM symbol comprising $K$ subchannels is made up of at least $K$ samples, more if there is oversampling and/or cyclic prefix. Since sampling times other than multiples of $\Delta\omega \cdot t = \pi/2$ result in different coefficient alphabets$^1$ for different subchannels, we cannot apply our previous simplifications and have to resort to comprehensive methods. We’ll discuss these in more detail in a separate post on Nyquist WDM signal statistics.

As an example, we’ll look at the highest sample power at various sampling instances of OFDM symbols with eight 16-QAM subchannels obtained by exhaustively searching all combinations (which takes about half an hour PER POINT to compute on my machine and which could probably be done much faster with a little more thinking). We would expect the maximum to occur at $\Delta\omega \cdot t = m \cdot \pi/2$ (quarter rotations of the constellation alphabet between neighboring subchannels) because only then do the constellation points with the highest powers in each subchannel potentially add in phase for all subchannels. Figure 6 confirms our assumption:

Figure 6: Maximum possible single-sample power depending on the position of the sample within the OFDM symbol for N = 4 and K = 8. To reduce calculation times, only the non-repeating part (solid) was actually calculated and then repeated (dashed). The symbols correspond to the colors in Figure 7.

The CDFs will of course also differ with sample timing, but not by much as Figure 7 shows.

Figure 7: Cumulative distribution functions of sample power for 16-QAM format and K = 8 for different sample times within the OFDM symbol, together with the CDF for a χ-squared distribution with two degrees of freedom and equal power.

Limit Probabilities and PAPR

Usually, it’s computationally not feasible to calculate the CDFs when there are either a lot of QAM levels $M$ or subchannels $K$. We can, however, quite easily determine the probability of the highest possible sample power to occur. We know that it appears for $t=0$ and we can thus use the derivations made above. We also know that it is just the square of $K$ times the largest-amplitude signal in a single subchannel, or with (6),

$$\max(P) = \max\left(C_i^2 + C_q^2\right) = 2 K^2 \cdot \bigl[\max(A)\bigr]^2 \tag{7}$$

with the alphabet $A$ as in (2). Since or normalization is such that $\left\langle P \right\rangle=1$, $\mathrm{max}(P)$ is also the “true” PAPR of an OFDM signal.

Let us check how probable an occurrence of $\max(P)$ actually is. We start with the probability for all coefficients $C_k$ being equal in a single quadrature, which corresponds to all $x_n$ in (4) being zero except for the one denoting that particular coefficient which is equal to $K$ (it does not matter which one that is when all coefficients are equally likely).

$$p_\text{equal quadrature} = \left(\frac{1}{N}\right)^K\tag{8}$$

The probability that the other quadrature also has the same coefficient in each subchannel (but possibly a different one than the first quadrature) is

$$p_\text{equal symbol} = p^2_\text{equal quadrature} = \left(\frac{1}{N}\right)^{2K} = \left(\frac{1}{M}\right)^{K}\tag{9}$$

where $M$ is the number QAM constellation points, since we still assume the data in both quadratures to be independent. This is the probability that any single constellation point appears simultaneously in all subchannels. There are four constellation points which cause the symbol to have the maximum possible power – the corner points. Hence the probability for the maximum power to occur is

$$p_{\max(P)} = 4 \left(\frac{1}{M}\right)^{K} \tag{10}$$

This can be very, very small, as we can also see in Figure 5, easily down to the $10^{-50}$s. Furthermore, (10) is valid only for the sample times within the OFDM symbol for which $\Delta\omega \cdot t = m \cdot \pi/2$, as discussed above. For all other sample times, the probability for $\max(P)$ to occur is zero.

Such probabilities are utterly irrelevant for communication even at Terabit/s. If just observing an OFDM signal for a while and giving the PAPR as the ratio involving the highest observed sample power, the result would be somewhat random and should only be given with proper confidence bounds. It would make more sense to define a power value (or rather, ratio) which is only exceeded with a probability lower than some value like $10^{-3}$ or $10^{-6}$ maybe. At these probabilities, the differences between the different sample timing (cf. Figure 7) seem to be negligible. As far as I know, such a definition does not exist (yet).

Anyway, if PAPR is a serious issue in a system, one should maybe look at Nyquist WDM which has nicer overall sample statistics (non-Gaussian).

1 Here, $\Delta\omega$ is the subchannel frequency separation. When $\Delta\omega \cdot t = \pi/2$, then all subchannel alphabets are rotated by multiples of a quarter circle (neighboring subchannels are rotated a quarter circle against each other, the next subchannels a half circle, and so forth) which results in identical alphabets for square QAM. The length of the “core” OFDM symbol is given by $\Delta\omega \cdot t = 2\pi$.

2 This is the fastest code I could hack together in a short time. Take it as a starting point.

3 The (central) $\chi^2$ distribution with 2 degrees of freedom describes the distribution of power in a signal with two-dimensional Gaussian amplitude distribution [4].

[1] PAPR [Wikipedia]
[2] Combinations [Wikipedia]
[3] Multinomial Distribution [Wikipedia]
[4] $\chi^2$ distribution [Wikipedia]

Series Expansions Reference

This is something that I use so often that I should have done it much sooner. Series expansions are a very useful tool that they don’t talk about nearly enough in college classes. Most of the time, for e.g. small signals, the first non-constant term is sufficient for an approximation – and instead of using inverse trigonometric functions or worse to solve your equations, you have a linear or maybe quadratic set of equations that is so much easier to handle. I can never remember the correct coefficients, though, and frequently look them up in Wikipedia. With this nice sheet tacked to the old office wall, these times will be gone.

One needs to be careful, however, not to let the function argument deviate too much for a series approximation to become invalid. That’s why I added graphs of the function itself (dashed black), the linear term (red), the quadratic term (blue) and the cubic term (green) in the vicinity of $x=0$. This way it should be much easier to estimate the validity of the various approximations.

Download the Series Expansions Cheat Sheet.

Today’s (or this week’s) ampersand is from Rooney Pro by Jan Fromm, a very nice-looking serif family that will set you back a mere $546 if you want them all. Unfortunately, it does not contain Greek letters, so it cannot really be used for scientific work requiring mathematics. I really like the small caps and indices, though.

It’s a nice variation on the very common & shape that consists of not one but two strokes. So purrty.