A COMBINED STOCHASTIC/DECISION-AIDED SNR ESTIMATOR. Witthaya Panyaworayan and Rolf Matzner

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1 A COMBINED STOCHASTIC/DECISION-AIDED SNR ESTIMATOR Witthaya Payaworaya ad Rolf Matzer Istitute for Commuicatios Egieerig Federal Armed Forces Uiversity Muich, Germay ABSTRACT A problem of a blid stochastic SNR estimator ad a priciple of SNR estimatio combiig a blid estimator ad decisio-aided estimator are preseted. Sigal ad oise ca be detected from the received sigal by a decisio depedig o a decisio threshold that is made available by usig the secod ad fourth-order momets of the received sigal. Subsequetly, the SNR is computed from the detected sigal ad the detected oise. The bias of SNR estimatio (based o detectio errors) is removed by applyig aalytically bias correctio. source x(t) (theory) source chael (t) oise chael (t) oise y(t) y(t) detector estimatio algorithm detector estimatio algorithm sik (practice) estimate sik estimate. INTRODUCTION The sigal-to-oise ratio (SNR) at the receiver s side of a commuicatio lik is a importat measure for the quality of the commuicatio chael. The SNR is obviously iflueced by two ukow power measures, the received sigal power S of the sigal emitted by the seder ad atteuated by the chael, ad the received oise power N due to thermal oise, crosstalk from eighboured chaels, etc. I geeral, both are ukow ad added to the total received power S N of the received superpositio of sigal ad oise, which ca actually be observed. The task of a SNR estimator is to aalyze the superimposed received sigal ad to determie which part of the power is due to the trasmitted sigal ad which part is due to oise. We wat to distiguish two basically differet approaches: Data-aided Estimatio If digital data are trasmitted, the receiver tries to recostruct the trasmitted sigal by meas of the decoded data, ad iterpret the differece betwee (a appropriately scaled versio of) the received sigal ad the recostructed sigal as oise. The major drawback of this method is, of course, that the decoded data become less reliable with decreasig SNR, ad thus the estimate exhibits a radom error due to decodig errors. This effect is obviously especially serious i a low SNR eviromet. Figure : Data-aided SNR estimatio (top) ad blid SNR estimatio (bottom). The data-aided SNR estimatio relies o the data set (theory), but actually uses the data detected (practice). Blid Estimatio methods do ot use decoded data. Istead, they use data-idepedet properties of the received sigal that bear iformatio about the data ad the oise compoets. Due to the abstractio from the actual data beig trasmitted these properties usually are of statistical ature, such as higher order momets or cumulats. Revertig the above argumet, oe advatage of blid estimators is the robustess agaist erroeously decoded data, makig them especially well-suited for low-snr chaels. Aother useful property of blid estimators is that they work for digital messages (data) ad aalog messages (e.g. aalogously trasmitted audio), whereas there s o way to obtai a decoded versio of a aalog sigal for a data-aided estimator. Figure illustrates the differeces betwee both approaches. 2. STATEMENT OF THE PROBLEM Sigal xètè ad oise ètè are modelled as stochastic processes with probability desity fuctios (p.d.f.) f x èxè ad f èè, resp. The received sigal yètè is a superpositio of

2 sigal ad oise: yètè =xètè ètè () We assume that xètè ad ètè are i.i.d. stochastic processes, ad mutually idepedet. This is valid e.g. for Nyquist systems with samplig after a matched filter ad additive white Gaussia oise. We further assume that sigal s ad oise s p.d.f.s fulfill the symmetry coditios f x èxè =f x è,xè; f èè =f è,è; (2) ad that ot both, sigal ad oise, are Gaussia. If xètè ad yètè are complex valued stochastic processes, they are described by their joit p.d.f.s of real ad imagiary part. For the rest of the paper, we restrict our aalysis to M-PAM sigallig schemes. However, extesio of the proposed algorithms to complex AM/PM schemes is straight forward. For sake of simplicity, oise is assumed to be Gaussia. A discrete-time AWGN chael is cosidered. 3. THE CRAMR-RAO BOUND Give a stochastic process with p.d.f. f x èxjè depedet o a parameter, the Cramr-Rao Boud (CRB) [] gives us a lower boud for the variace varfg of a estimatio for : varfg æ l 2 (3) where Efæg deotes the expected value ad is the umber of observed samples. Give a M-PAM trasmitter with trasmitted power S ad p.d.f. f x èxjsè = M M, k= æ è x, è2k, M è r 3S M 2, ad Gaussia oise with variace N we obtai for the received p.d.f. f y èyjs; N è= M p 2N M, k= y,è2k,mèr 3S 2 e, M 2, Substitutig y = p S y ad the SNR = N S obtai: f y èy jè = p M p 2 M, k= e, 2! (4) 2N (5) i (5) we y,è2k,mèq 3 M 2, 2 (6) Figure 2 ad 6 also depict umerical results for the equatios (3) ad (6) for M 2f2; 4; 8g ad a observatio size of = 52. For larger SNR the CRB is almost proportioal to 2. For this reaso, variaces of estimates will be ormalized to 2 i the sequel. The ormalized CRB shows a promiet maximum at low SNR. 4. THE BLIND STOCHASTIC SNR ESTIMATION 4.. Algorithm Usig the secod ad fourth-order momets the SNR ca be estimated by observig the oisy sigal if oly the shapes of sigal s ad oise s p.d.f.s are kow [2]. The estimate of the received SNR is computed by [3]: q G y =G x =, = q (7) G y =G x where, æ deotes the estimated sigal-to-oise ratio S=N, æ is the estimated sigal-to-total ratio S=èS N è, æ G x deotes the trasmitted Gauss-ulikeess æ G x = m x;4 =m 2 x;2, 3; G y deotes the estimated received Gauss-ulikeess. Additioally, this algorithm ca be see i [4] Aalysis This blid stochastic SNR estimatio was simulated with the AWGN chael model, ad a measure of efficiecy of the SNR estimator with the CRB was used. The CRB for the blid SNR estimatio was computed umerically with (3) ad (6). Figure 2 shows the results of the simulated blid stochastical SNR estimator for 2-PAM (usig simulatio rus with 5 packets, each cotaiig 52 samples) compared to the this CRB. The result approachs the theoretical boud for SNR estimatio. It turs out that for 2-PAM the blid stochastic estimator is efficiet with respect to the CRB Problem of Algorithm The blid stochastic estimator has severe disadvatages whe it is used for badwidth-efficiet modulatio schemes. The estimatio remais stable as log as is equal or greater tha but smaller tha. Estimatio for M-PAM, M é 2, ca obtai values for close to the pole i (7),

3 4 x 3 mea var{ρ 2PAM }/ρ 2 CRB 2PAM /ρ 2 I fig. 3 the mea stadard deviatio of simulated estimated sigal-to-total ratio usig blid stochastical SNR estimator (usig simulatio rus with 5 packets, each cotaiig 52 ad 24 samples) for 4-PAM are compared to the correspodig CRBs. The CRB of 24 samples is lower tha the CRB of 52 samples. If the actual SNR ad the actual sigal-to-total ratio icrease, the stadard deviatio of estimated will approach the CRB. For 8-PAM the problem is similar. This SNR estimator is ot well-suited for 4 ad 8-PAM, but it ca be used to estimate the sigal-to-total ratio. Variace ormalized to actual SNR liear actual SNR Figure 2: Mea ormalized variace varfg= 2 of the simulated blid stochastical SNR estimator for 2-PAM ad CRB of varfg= 2 causig istability ad explodig variace of the estimated. However, it has to be oted that while varfg explodes due to the pole, varfg remais stable..4.2 mea std(κ ) 52 mea std(κ ) 24 CRB std52 5. A COMBINED STOCHASTIC/DECISION-AIDED ESTIMATOR 5.. Priciple of Operatio Revestig the last sectio, this leads to a estimator usig blid stochastical estimatio of, ad the computig a decisio-aided estimatio for. The sigal-to-total ratio is used to adjust the decisio threshold d of the decisioaided estimator, for 4-PAM: r æ my;2 d = (8) 5 where m y;2 deotes secod-order momet of the received sigal, ad 5 is simply the ormalized power of 4-PAM. I the sequel, we use x æ to deote the detected sigal, ad æ for the detected oise (i.e. æ = y, x æ ). We ca fid the sigal power S x æ from the detected sigal x æ,adthe oise power N æ from the detected oise æ. The estimated SNR æ is provided by relatio of S x æ ad N æ. Due to the detectio errors, the estimated æ is biased. However, this bias ca easily be removed by applyig a aalytically computed oliear characteristic i the sectio 5.2. Fig. 4 illustrates the priciple of this operatio. Stadard Deviatio CRB std liear actual sigal to total ratio κ Figure 3: Mea stadard deviatio of the simulated estimated sigal-to-total ratio usig blid stochastical SNR estimator with 52 ad 24 samples for 4-PAM ad their correspodig CRBs of stdfg Gy my;4 m my;266 2, 3 my;4 y 6 i 2 yi 4 i= y :::y r æm y;2 6 r Gy Gx 6 d i= 6 6? - s - Decisio 4PAM =5 8PAM =2 G x;4pam =,:36 G x;8pam,:24 x æ i-s i= -? h æ i- S x æ i 2 x æ - i= S x æ N æ æ i 2 N æ 6 æ - Bias Correctio Figure 4: Combied Stochastic/Decisio-aided SNR Estimator for 4- ad 8-PAM ~ æ -

4 5.2. Bias Removal The characteristic of the estimated SNR æ is aalytically computed i [5]. The received p.d.f. for 4-PAM ca be give by (5), therefore the sigal power S x æ for 4-PAM is obtaied by usig the received p.d.f. ad the decisio threshold d.,2d S x æès; N è = è,3dè 2 f y èyè dy, liear actual SNR ρ Bias correctio for 4 PAM Bias correctio for 8 PAM è,dè 2,2d 2d èdè 2 è3dè 2 2d f y èyè dy f y èyè dy f y èyè dy (9) Additioally, the oise power N æ for 4-PAM is obtaied by usig the received pdf ad the decisio threshold d. N æ è æ js; N è = 2ë d d d æ2 f y è æ 3dè d æ æ2 f y è æ dè d æ æ2 f y è æ, dè d æ æ2 f y è æ, 3dè d æ ë () The aalytically estimated SNR æ for 4-PAM is ow computed as the relatio of S x æ i (9) ad N æ i (). The aalytically estimated SNR æ for 8-PAM ca be computed i the same way as for 4-PAM. Figure 5 depicts curves of actual SNR for4-ad8-pam depedig o the aalytically estimated SNR æ. The aalytically estimated SNR æ for 8-PAM is oly greater tha the actual SNR, whereas the aalytically estimated SNR æ for 4-PAM is equal to the actual SNR, whe the actual SNR is greater tha 6. These curves are used to remove the bias of the estimated SNR æ liear aalytically estimated SNR ρ * Figure 5: Bias correctio to correctig the estimated SNR æ for4-ad8-pam 5.3. Results Figure 6 shows the results of the simulated combied stochastic/decisio-aided SNR estimator for 4- ad 8-PAM (usig simulatio rus with 5 packets, each cotaiig 52 samples) with bias correctio compared to the CRB. Both variaces varf~ æ g= 2 of simulatio remai stable, ad they have similar characteristics to the CRBs. The varf~ æ g= 2 of 4-PAM is closer its CRB tha the varf~ æ g= 2 of 8-PAM. Figure 6 also clarifies that the combied stochastic/decisio-aided SNR estimator has early the same efficiecy for 4-PAM as for 8-PAM with respect to the CRB. Variace ormalized to actual SNR ~* mea var{ρ 8PAM } /ρ 2 CRB 8PAM /ρ 2 ~* mea var{ρ } /ρ 2 4PAM CRB /ρ 2 4PAM liear actual SNR Figure 6: Mea ormalized variaces varf~ æ g= 2 of the simulated combied stochastic/decisio-aided SNR estimator for 4- ad 8-PAM ad their correspodig CRBs of varfg= 2

5 6. CONCLUSION The blid stochastic SNR estimator has a good efficiecy i the sese of the CRB for 2-PAM. However, it is ot very well-suited for 4-PAM ad higher PAM or QAM modulatio schemes, because estimatio for M-PAM, M é 2, at high actual SNR ca obtai values for close to the pole, causig istability ad explodig variace of the estimated. If detected data are available or ca be made available, the combied stochastic/decisio-aided SNR estimator solves this problem i a ear-optimum maer. 7. REFERENCES [] Paul G. Hoel, Sidey C. Port, Charles J. Stoe, Itroductio to Statistical Theory, Houghto Miffli Compay, Bosto, 97. [2] R. Matzer, A SNR estimatio algorithm for complex basebad sigals usig higher order statistics, Facta Uiversitatis, Series: Electroics ad Eergetics, vol. 6, pp. 4 52, 993. [3] R. Matzer ad F. Eglberger, A SNR estimatio algorithm usig fourth-order momets, i It. Symp. If. Theory, (Trodheim), 994. [4] R. Matzer, F. Eglberger, ad R. Siewert, Aalysis ad desig of a blid statistical SNR estimator, AES 2d Covetio, Muich, Germay, March 997. [5] Witthaya Payaworaya, Ei kombiiert stochastisches/etscheidugsgestuetztes Verfahre zur SNR- Schaetzug, Diploma thesis, Federal Armed Forces Uiversity Muich, 997. (i Germa).

Problem Set 4 Due Oct, 12

EE226: Radom Processes i Systems Lecturer: Jea C. Walrad Problem Set 4 Due Oct, 12 Fall 06 GSI: Assae Gueye This problem set essetially reviews detectio theory ad hypothesis testig ad some basic otios