6.10  Exercises

6.1. One can aim at increasing the number of bits per symbol by using an increasing number of basis functions. The tradeoff is that increasing the number of functions also increases the required bandwidth. Consider discrete-time processing and complex exponentials as basis functions. Find the maximum number $N$ of orthogonal complex exponentials of $D$ samples.
6.2. For all questions, assume an ADSL deployment with a total bandwidth of $1.104$ MHz and estimated background noise at the receiver with a flat (white) unilateral PSD level of $N_0=\; -140$ dBm/Hz. a) What is the total background noise power in Watts a receiver modem has to deal with? b) Given that the tone width is $\mathrm{\Delta }f\; =\; 4.3125$ kHz, what is the background noise power in a single tone? c) To obtain an $\mathrm{\text{SNR}}=\; 20$ dB for a given tone $k$, what should be the received power $P_k^r$ and PSD at this tone assuming it is flat (“white”)? d) To obtain the same $\mathrm{\text{SNR}}=\; 20$ dB for tone $k$, what should be the transmitted power $P_k^t$ and (flat) PSD level if, before adding the background noise, the LTI channel at this tone $k$ is assumed to have a flat magnitude of $|H(k)|\; =\; 0.3$?
6.3. The ADSL standard adopts a tone spacing $\mathrm{\Delta }f\; =\; 4.3125$ kHz and a maximum total power of 20.4 dBm for downstream transmission. Without further restrictions such as a “PSD mask”, what would be the maximum value the PSD could assume in a given tone assuming it is flat over this tone? Assuming the noise is AWGN with unilateral PSD $N_0=\; -140$ dBm/Hz, what is the maximum SNR of a tone?
6.4. Margin. A DSL system was designed to achieve a probability of symbol error $P_e$ with uncoded QAM and a corresponding gap $\Gamma$. During operation, an ${\mathrm{\text{SNR}}}_k=\; 15$ dB was estimated at tone $k$. Without any margin, the number of bits $b_k={log}_2\left(1\; +\frac{{\mathrm{\text{SNR}}}_k}{\Gamma }\right)$ that can be loaded to cope with the required $P_e$ is $b_k=\; 6$. a) What is the margin $\gamma$ if the actual number of loaded bits is 4? b) If ${\mathrm{\text{SNR}}}_k$ (in linear scale, not dB) increases to $8{\mathrm{\text{SNR}}}_k$, what is the increase in margin?
6.5. A DSL system uses a symbol rate of $R_{\mathrm{\text{dmt}}}=\; 1000$ bauds, a DFT of $N\; =\; 128$ points and a prefix-length of $L_{}$

cp=20$samples.a)Assumingthatthereisnooversampling,whatisthesamplingratefortheADCandDACchips?b)Whatisthespacing$Δ_f$betweenFFTfrequencies(tonespacing)?$
6.6. Design a DMT system assuming bitloading is given. Assume that a channel probing procedure estimated the channel impulse response as $h(n)\; =\; \delta (n)\; +\; 2\delta (n\; -\; 1)\; +\; \delta (n\; -\; 3)$. The DFT has $N\; =\; 8$ points the symbol rate is $R_{\mathrm{\text{dmt}}}=\; 4$ kHz. Assume PAM for the DC and Nyquist (real) tones and QAM constellations for all other tones. The bitloading $[b_0,b_1,\dots ,b_5]$, where $b_0$ corresponds to the DC tone $k\; =\; 0$, is [1, 2, 2, 3, 2] bits per symbol. The intention is to transmit the bit stream $B\; =\; \{0110110010\}$. Do not worry about the used power (simply use constellations without minding about their normalization). Inform: a) the length of the cyclic prefix that must be used, b) the constellations you adopted, c) the FEQ project, d) the total bit rate in bps, e) the $F_s$ the DAC must work with, f) the frequency interval between neighboring tones and g) using Figure 6.1 as reference, specify the values of the signals Xk, xn, xp, r, yn, Yk and Zk for the transmission of B assuming the channel is initially at rest, that is, the initial conditions are zero.
6.7. Normalization of transmit constellations. A DMT system with a unitary DFT of $N\; =\; 4$ points has bitloading [2, 3, 2] bits per tone and power per tone given by $P_k$=[4, 10, 20] Watts. Find a) the constellations assuming all they are normalized to have average energy equal to 1 and b) their scaling factors to produce $P_k$. Using these scaled constellations, c) convert the bitstream B=[1011001] into a DMT symbol. d) Show the input and output of the IFFT modulator. e) Using a cyclic prefix with $L_{}$

cp=2$samples,calculatethepowerinWattsoftheIFFToutput\mathbf{\text{corresponding to this symbol}}(youcanconsiderthesymbolperiod$T=2$millisecondsandthesamplingperiod$T_s = T/6$milliseconds,butyoushouldnotneedeither).f)Ifyoudonotusecyclicprefix,whatisthepower?g)DoestheParsevaltheoremholdinthiscase?h)Inpractice,whenaveragingthesignaloveralongtime,doesthelengthofthecyclicprefixincreaseordecreasethepower?Thesolutionisdiscussedinscript\mathsf{\text{MatlabOctaveCodeSnippets/snip\_multicarrier\_constellations.m}}.$
6.8. Possible problems and their solutions. Take the script Listing 6.3 and artificially create problems to observe the consequences. a) Use $L_{}$

cp=2$asthelengthofthecyclicprefix.Doyouobserveerrorsinthereceivedbitstream?HowaboutthereceivedQAMsymbols?b)Repeatfor$L_cp=1$.c)Usethechannel$h(n)=2$\delta (n)\; +\; \delta (n\; -\; 1)\; +\; \delta (n\; -\; 3)$. Is it possible to use the DC tone with this channel?
6.9. Find power and bitloading using waterfilling (version B: with PAMs). This one is similar to previous question but shows the required attention when dealing simultaneously with PAM and QAM tones. Assume, as in the previous question, $N\; =\; 8$, a total transmit power of 20.4 dBm, the rate-adaptive version of waterfilling, 4000 bauds, ${\Gamma }_{dB}=\; 6$ dB and the noise at the receiver is AWGN with $N_0=\; -140$ dBm/Hz. Change only the estimated channel to $h(n)\; =\; 10^{-7.7}(2\delta (n)\; +\; \delta (n\; -\; 1)\; +\; 3\delta (n\; -\; 3))$ to have a smaller received power. Provide: a) a plot of the channel frequency response, b) the signal-to-noise ratio ${\mathrm{\text{SNR}}}_n$ per tone, c) the number $b_n$ of bits per tone assuming this number can be non-integer and d) the margin in dB associated with item c). Note the different treatment of PAM and QAM tones.³
6.10. Dealing with constellations. The previous solution (ak_loading_dmt_b.m) was a DMT with $N\; =\; 8$ and powerPerTone = [0.0150 0.0115 0.0105 0.0253 0 0.0253 0.0105 0.0115] Watts for a bitloading bits = [1.3378 0.5663 0.5057 1.8027 0]. Note that while powerPerTone shows the power for the negative frequencies, bits shows only for positive frequencies. To have an integer bitloading, use bits=[1 1 1 2 0]. Find the constellations and their scaling factors, assuming all constellations are normalized to have average energy equal to 1. What is the new margin ${\gamma }_n$ per tone?
6.11. Error probability. A previous solution (script ak_loading_dmt_b.m) was a DMT with $N\; =\; 8$ and powerPerTone = [0.0150 0.0115 0.0105 0.0253 0 0.0253 0.0105 0.0115] Watts for a bitloading bits = [1.3378 0.5663 0.5057 1.8027 0]. The noise at the receiver is AWGN with $N_0=\; -140$ dBm/Hz. To have an integer bitloading, use bits=[1 1 1 2 0]. The channel was $h(n)\; =\; 10^{-7.7}(2\delta (n)\; +\; \delta (n\; -\; 1)\; +\; 3\delta (n\; -\; 3))$. a) Calculate the symbol error probability for each tone. b) Does this bitloading allow to operate with a symbol error probability less than $10^{-5}$? c) Change to bits=[1 0 0 1 0] and check whether or not the symbol error probabilities are less than $10^{-5}$. d) What is the problem when rounding a non-integer bitloading to a larger integer value?⁴
6.12. SVD. Assume a channel $h(n)\; =\; 2\delta (n)\; +\; \delta (n\; -\; 3)\; +\; \delta (n\; -\; 4)$ and partition it in $N\; =\; 6$ tones using SVD. Find the number of samples for the guard band and the FEQ. Provide the transmitter matrix $M$ and the receiver matrix $F$. Send the 6 QAM symbols X=[1+j 1-j 2 0 1 2+j2] over the channel and reconstruct it. What is the SNR per tone? Use the code Listing 6.7 as guideline. Does the guard band need to be a cyclic prefix in SVD partitioning? Do the same partitioning but now use DFT with cyclic prefix. Compare SVD and DMT under similar circumstances (e.g., maximum transmit power and same channel).⁵
6.13. ICI and ISI. Assuming a DMT system of your choice, with channel dispersion $M$ and cyclic prefix ($L_{}$

cp<M$),estimateISIandICIonthetimedomain.ToestimateISIyoucanusethescheme\; :\; assumethatthechannelfinishedtransferringthesecondsymbol(thechannelmemoryisnotzero)andtransmitasequenceofzeros(allsymbolsarezeroonbothtimeandfrequencydomains,beforeandaftertheIFFT).ThechanneloutputwillbetheISI.ToestimateICI,sendafirstsymbol(channelwasatrest)withacyclicprefixof$L_cp=M$samplesandthensendthesamesymbolwithcyclicprefixof$L_cp$samples.ThedifferencebetweenthetwosignalswillbetheICI.$
6.14. Do the DMT basis functions (sinusoids) remain orthogonal after passing through the channel?
6.15. Prove that the multi-channel SNR for a set of parallel channels is

You may confirm your answer by reading the chapter on multicarrier modulation at [ url5cln].
6.16. Study the article [vWNDM10] and the references therein to describe an expression for the PSD of an IEEE 802.11a WLAN signal using Eq. (13) of that paper.