Single-Pole IIR Channel Power Prediction with Variable Delays
PPaper presented at IEEE GLOBECOM 2015, San Diego, California.
Single-Pole IIR Channel Power Prediction withVariable Delays
Jes´us Arnau
Mathematical and Algorithmic Sciences LabFrance Research Center, Huawei Technologies Co. Ltd.20 Quai du Point du Jour, 92100 Boulogne-Billancourt, France.Email: [email protected]
Abstract —Exploiting outdated channel quality indicators iscrucial in most adaptive wireless communication systems. This isoften done through channel prediction based on previous receivedindicators. In this paper, we analyze the case where the feedbackdelay experienced by the quality indicators is not constant, butrandom. Focusing on a single-pole IIR predictor, we obtainanalytical expressions for the MSE and the filter parameters, andstudy the throughput behavior through Monte Carlo simulations.Results show that prediction provides a performance advantagefor average delays smaller than ms for low terminal speeds. Index Terms —Channel prediction; variable delays; time-varying correlation; LTE uplink; M2M.
I. I
NTRODUCTION
In wireless communication systems using adaptive codingand modulation (ACM), dealing with outdated channel qualityindicators (CQI) poses a relevant challenge. Consider thefollowing uplink example: in order to select the appropriatemodulation and coding scheme (MCS), the user equipment(UE) receives an indicator from the base station (BS) statingthe quality of the channel. Because of the delays involved,such indicator will be outdated at the moment of using it, aphenomenon sometimes called
CQI aging [1] .Apart from permanent delays given by BS processing andpropagation, a crucial delay contribution is given by the timedifference with the last channel estimation. For example, theLTE uplink uses periodic sounding reference signals (SRS) toestimate the channel, and so their period will determine howoutdated the channel estimates are at the moment of usingthem.Another possibility is to extract the CQI only from previ-ous data transmissions. This uses fewer signaling resources,but introduces a new problem: the delays involved are now random , as a consequence of the packet arrival instants beingrandom themselves. It shall be remarked that this is speciallyrelevant for machine to machine (M2M) communications. Insuch systems, terminals use the channel sparingly and totransmit short messages, and superfluous signaling is kept toa minimum.In this work, we compare the performance of channelprediction under fixed and random feedback delays. For itssimplicity, and competitive performance, we illustrate the caseof the single-pole IIR predictor in [2]. We obtain closed-formexpressions for the prediction mean-squared error (MSE), andfor the optimum filter parameters. These will be complemented by Monte Carlo simulations that showcase throughput underrealistic conditions. Results will show the different trendspresent with both types of delays. Moreover, they will illus-trate the tipping points above which prediction offers littleperformance advantage.The need for channel prediction to compensate for CQIaging has been longly recognized in the literature (see e.g. [3],[4], and references therein). Because of the numerous trade-offs involved, the topic has kept attracting attention in the lastyears [1], [5]–[7]. To the best of our knowledge, existing worksdo not specifically address random feedback delays.The remainder of the document is structured as follows:Section II explains the system model and assumptions, Sec-tion III collects the analytical derivations of the MSE forfixed and variable delays, Section IV reports some simulationresults, and finally conclusions are summarized in Section V.II. S
YSTEM MODEL
A. Signal model
The signal model considered is given by y (cid:96) = √ snr · h (cid:96) s (cid:96) + w (cid:96) (1)where s (cid:96) denotes the (cid:96) -th block of B independent, unit-powertransmitted symbols, s (cid:96) = [ s (cid:96) , s (cid:96) +1 , . . . , s (cid:96) + B ] T ; y (cid:96) denotesthe received block, and w (cid:96) contains B zero-mean, unit-powersamples accounting for Gaussian noise plus interference.We assume that each transmitted block undergoes a sin-gle channel realization ( block fading ), and that blocks aretransmitted using a finite set of M modulation and codingschemes, C = { C , C , · · · , C M } , characterized by theirassociated signal-to-interference-plus noise (SINR) thresholds S and spectral efficiencies R . We will further assume that the (cid:96) -th block, using the j -th MCS, is transmitted successfully ifand only if snr | h (cid:96) | > S j .The channel represents a Rayleigh fading scenario, so that h (cid:96) ∼ CN (0 , ; time variations in the fading process are ruledby the Doppler spectrum, whose bandwidth is given by f d = v/cf c ; v is the terminal speed, c is the speed of light ( · m/s) and f c is the carrier frequency. We will go deeperinto the details of the channel’s time correlation in the nextsection.For future reference, we should note that each power sam-ple | h (cid:96) | follows a scaled Chi-squared distribution with two a r X i v : . [ c s . I T ] D ec egrees of freedom, | h | ∼ χ (2) ; therefore, the followingidentities hold: E (cid:2) | h | (cid:3) = 1 , E (cid:2) | h | (cid:3) = 2 . B. Traffic pattern
As explained, we will allow the transmission time instants tobe random. This means that the set h (cid:96) will not be a periodicsampling of a continuous-time channel process: instead, theunderlying channel will be sampled at random time instants,and thus the correlation between adjacent samples will vary.Let us assume that arrivals follow a Poisson process. Then,the time interval between two consecutive channel samples, h (cid:96) and h (cid:96) +1 , denoted as τ (cid:96) , will be exponentially distributed τ (cid:96) ∼ Exp(1 /T ) (2)where T denotes the mean time between blocks. We shallremark that such an exponential process is memoryless, andthat, as suggested by the notation in (2), the random variables τ (cid:96) are i.i.d; we will often drop the subindex when unnecessary. C. Time correlation
The classical Jakes-Clarke model states that, if h (cid:96) is a Rayleigh fading process, then E (cid:2) h (cid:96) h ∗ (cid:96) + L (cid:3) = J (2 πf d τ | L − (cid:96) | ) . However, the presence of the Besselfunction makes it difficult to analyze this model, andspecially in our case where τ is not a constant.For the rest of the paper, we will assume a simplifiedcorrelation model, where h (cid:96) is a first-order autoregressiveprocess: h (cid:96) +1 = ρh (cid:96) + (cid:112) − ρ · e (cid:96) +1 (3)where the e (cid:96) are independent complex standard normal randomvariables, and ρ is the correlation factor between adjacentsamples: ρ · = E (cid:2) h (cid:96) h ∗ (cid:96) +1 (cid:3) . (4)
1) Evenly spaced arrivals:
If, as traditionally, channelsamples are evenly spaced over time, then the correlationcoefficient between them will be constant. It trivially followsthat E [ h (cid:96) + L h ∗ (cid:96) ] = ρ L . (5)
2) Randomly spaced arrivals:
The case where arrivals arerandomly spaced is more difficult to handle (even in the caseof i.i.d intervals from Section II-B). To tackle it, we proposethe following model with random correlation coefficients: h (cid:96) +1 = ρ (cid:96) h (cid:96) + (cid:113) − ρ (cid:96) e (cid:96) +1 . (6)Here, each ρ (cid:96) is random, and given by ρ (cid:96) = J (2 πf d τ (cid:96) ) . (7)We should recall that τ (cid:96) are exponentially-distributed randomvariables.Correlation between any two channel samples is now givenby E h [ h (cid:96) + L h ∗ (cid:96) ] = L (cid:89) i =1 ρ (cid:96) + i , (8) which again is a random variable and depends on the arrivalsprocess. Thus, it will be useful to define the average correla-tion coefficient : ρ · = E τ [ ρ ] = E τ (cid:20) E h (cid:2) h (cid:96) h ∗ (cid:96) +1 (cid:3)(cid:21) . (9) D. Method under study
Let γ (cid:96) denote the (cid:96) -th realization of the process we want topredict, then the single-pole IIR predictor proposed in [2] isgiven by ˆ γ (cid:96) +1 = (1 − α )ˆ γ (cid:96) + αγ (cid:96) . (10)For future usage, note that the mean squared error of theprediction is given by [2, Eq. 13] mse · = E (cid:2) | ˆ γ (cid:96) +1 − γ (cid:96) +1 | (cid:3) = 22 − α (cid:32) E (cid:2) | γ | (cid:3) − α ∞ (cid:88) i =1 (1 − α ) i − E (cid:2) γ (cid:96) γ ∗ (cid:96) − i (cid:3)(cid:33) . (11)Before going further, we should note that (10) admits thefollowing expression in terms of the infinite impulse responseof the prediction filter, denoted below by g : ˆ γ (cid:96) +1 = ∞ (cid:88) i =0 g i γ (cid:96) − i = α ∞ (cid:88) i =0 (1 − α ) i γ (cid:96) − i . (12)III. A NALYSIS OF SINGLE - POLE
IIR
PREDICTION
In this section, we analyze the performance of the predictorabove in terms of MSE. We will start by comparing twodifferent approaches to power prediction, and choose one ofthem for the rest of the paper. We will analytically obtain theMSE as a function of α , both with fixed and random blockspacing, and also the expression of the optimum α . A. Power vs. amplitude estimation
As explained in Section II-A, each block is transmittedusing an MCS whose transmission success will depend onthe instantaneous SINR it experiences. Consequently, we willbe interested in predicting the instantaneous channel power,given by snr | h (cid:96) +1 | . We will briefly analyze the behavior oftwo different power prediction alternatives.Assume that we have a prediction of the future channelsample h (cid:96) +1 , and that we want to derive the instantaneouspower from it. This predictor is inherently biased: E (cid:104) | ˆ h (cid:96) +1 | (cid:105) = E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α ∞ (cid:88) i =0 (1 − α ) i h (cid:96) − i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = α − α E (cid:2) | h | (cid:3) + 2 α ∞ (cid:88) j =0 j (cid:88) k =0 (1 − α ) j + k E (cid:2) h (cid:96) − j h ∗ (cid:96) − k (cid:3) (cid:54) = E (cid:2) | h | (cid:3) (13)nless α = 1 , in which case we would be simply using theprevious sample as a predictor.On the other hand, prediction applied directly over powerestimates is naturally unbiased. For the rest of this work, wewill assume that we operate directly over power samples. Forsimplicity, we will assume perfect knowledge of the previouspower sample γ (cid:96) = snr | h (cid:96) | , and focus on the uncertaintygiven by the time dynamics of h (cid:96) . Further discussions on thebias can be found in [8]. B. Special cases of α
1) Small α : When α << we can approximate the pre-dicted sample as ˆ γ (cid:96) +1 ≈ ˆ γ (cid:96) ≈ ˆ γ (cid:96) − · · · ≈ ˆ γ −∞ , where ˆ γ −∞ is the first prediction guess used to initialize the algorithm.Then, the MSE mse ≈ E (cid:2) | ˆ γ (cid:96) +1 − γ −∞ | (cid:3) (14)is minimized when ˆ γ −∞ = E [ γ (cid:96) ] , and in consequence mse = Var [ γ (cid:96) ] (15)by the definition of variance.In summary, setting a very low α ends up giving a constantvalue as a prediction, and its lowest MSE is given by thevariance of the process.
2) Almost-1 α : When α is almost equal to one, then ˆ γ (cid:96) +1 ≈ γ (cid:96) , and the MSE is given by mse ≈ E (cid:2) | γ (cid:96) − γ (cid:96) +1 | (cid:3) = 2 (cid:0) E (cid:2) | γ (cid:96) | (cid:3) − E (cid:2) γ (cid:96) γ ∗ (cid:96) +1 (cid:3)(cid:1) . (16)The predictor is the previous sample. The error decreases ascorrelation increases; in the limit where γ (cid:96) +1 ≈ γ (cid:96) (constantchannel), then mse = 0 . C. Fixed inter-block period
Let us start by assuming that data is transmitted at a constantrate (or that there exists a periodic probing of the channel,as discussed in Section I), so that the time distance betweenconsecutive blocks is always the same. In this case, τ (cid:96) = T s ∀ (cid:96) ,and in consequence ρ (cid:96) = ρ = J (2 πf d T s ) . The MSE is givenin closed form by the following lemma. Lemma 1.
With fixed inter-block period, the MSE of thesingle-pole predictor is given by mse = 2snr − α (cid:18) − αα − ρ − (cid:19) , < α < . (17) Proof:
See Appendix A.Note that, if α approaches zero without reaching it, thenthe mse approaches snr , which is the variance of snr | h (cid:96) | asanticipated in Section III-B1. Corollary.
The optimum value of α is given by α opt = 12 (cid:0) − ρ − (cid:1) = 12 (cid:16) − J (2 πf d T s ) − (cid:17) . (18) Proof:
Differentiating (17) with respect to α , we obtain ∂ mse ∂α = 2snr (2 − α ) ( ρ − (cid:0) ρ (2 α −
3) + 1 (cid:1) (1 + ( α − ρ ) . (19)Equating to zero and solving for α gives the result.When the argument of the Bessel function becomes small,then ρ tends to and the optimum α tends to as well,which is consistent with the analysis in Section III-B2: whencorrelation is high, all the importance is given to the previoussample. On the other hand, when the argument grows largeand ρ tends to zero, the optimum value of α would tend to alarge negative quantity. This, however, cannot be allowed, as α has to be between and for convergence; we shall clip thenegative values to zero when needed. This means that, whencorrelation is very low, the channel will be predicted alwaysby the same sample, the one used to initialize the algorithm.We shall remark that α opt will be between and , butbeware that this conclusion relies on our assumptions: othercorrelation models could potentially yield optimum valuesabove . D. Random inter-block period
For the random inter-block period, we need to average overboth the realizations of h (cid:96) and τ (cid:96) . Doing so, the MSE readsas mse = E τ (cid:20) E h (cid:2) | ˆ γ (cid:96) +1 − γ (cid:96) +1 | (cid:3)(cid:21) , (20)and its expression is given in the following lemma. Lemma 2.
The MSE of the single-pole predictor with expo-nential inter-block delay is given by mse = 2snr − α (cid:32) − αα − π K ( − π f T ) − (cid:33) (21) where K ( x ) denotes the complete elliptic integral of thefirst kind [9, Sec. 17.3] and T is the average time betweenconsecutive blocks.Proof: See Appendix B.
Corollary.
The value of α that minimizes the MSE is givenby α opt = 12 (cid:16) − π K ( − π f T ) − (cid:17) . (22)We will compare these expressions numerically in thefollowing section.IV. S IMULATION RESULTS
A. Numerical evaluation of the MSE
We first evaluate the MSE given by (17) and (21), as afunction of α , for different correlation levels.Figure 1 shows the results with fixed block separation T s ∈{ , , , } ms. The speed v is set to km/h, the carrierfrequency f c to GHz, and the SNR to dB; the dashed lineis the variance, which would be the MSE obtained by alwaysusing the mean as a prediction. We can see that, for delays α M S E Figure 1. MSE as a function of α for different values of T s ; the dashed lineshows the variance, and squares mark the minimum MSE. α M S E Figure 2. MSE as a function of α for different values of T ; the dashed lineshows the variance, and circles mark the minimum MSE. above m/s, there is no gain in terms of MSE with respectto just using the mean; furthermore, the gains with a delay of m/s would be reduced to at most %.Similarly, Figure 2 evaluates the expressions for the variableseparation case with T ∈ { , , , } ms. Here, for a meanblock separation of m/s, there is still almost % advantagein terms of MSE when doing prediction. This is because asufficient number of samples of the process τ (cid:96) will take valueswell below the mean, making prediction useful.Finally, we evaluate the optimum α for both cases inFigure 3 for different speeds (represented on the plot bydifferent values of f d ). It is worth noticing the sensitivity ofthe fixed interval case: a slight deviation in T s causes an abruptchange in α opt , whereas significant deviations are needed in T to see the same effect.
10 20 30 40 50 6000.20.40.60.8 T s (or ¯ T ) α o p t f d = 5.55 Hzf d = 11.11 Hzf d = 2.7 Hz Figure 3. α as a function of T s for the case of fixed intervals (solid lines),and as a function of T for variable intervals (dashed lines). B. Throughput simulation
We next report average throughput (average over time)results through Monte Carlo simulations. Unless otherwisestated, the simulation parameters are the same as in theprevious section. We use adaptive coding and modulation, andselect the MCS based on the predicted value. For comparison,we also consider the following techniques: • Perfect prediction: as an upper bound, we show theaverage throughput that would be obtained with perfectknowledge of the future channel state. • Previous sample: we test the case of simply using theprevious channel sample as a predictor (equivalent tofixing α = 1 ). • Fixed rate:
The same MCS C j ∗ is used in every trans-mission. It is selected as j ∗ = arg max j (cid:0) − P (cid:2) snr | h (cid:96) | ≤ S j (cid:3)(cid:1) R j = arg max j exp (cid:18) − S j (cid:19) R j . (23)Results are shown in Figure 4 for snr = 5 dB; the optimumvalue of α is used in every case. We can see that, for both fixedand variable delays, using the previous sample as a predictorperforms close to using the optimum α for low delays, andcomparably worse for higher delays; the difference seems tobe smaller in the random case. In either case, this kind ofprediction offers little or no advantage for (average) delaysabove m/s, even with a speed as low as km/h.V. C ONCLUSIONS
We have analyzed the performance of a channel power pre-dictor with fixed and random delays. Closed-form expressionsfor the MSE and the predictor parameters have been obtainedfor both cases. Results show that, for low speeds, predictionprovides an advantage for average delays up to about ms. T s (or T ) (ms) T h r o u g hpu t( bp s / H z ) Perfect predictionPrevious sampleIIR predictionFixed rateVariable intervalsFixed intervals
Figure 4. Throughput as a function of T s (fixed case) or T (variable case). snr = 5 dB. Extensions to more general predictors and channel models willbe the subject of future work.A
PPENDIX AA NALYTICAL
MSE
WITH FIXED INTERVALS
Let us start from the fact that E h (cid:2) γ (cid:96) γ ∗ (cid:96) + i (cid:3) = snr · (cid:18) E h (cid:2) h (cid:96) h ∗ (cid:96) + i (cid:3) (cid:19) , (24)as shown among others in [10, Eq. 14]. Substituting (24) in(11) we obtain mse = 22 − α (cid:32) E (cid:2) γ (cid:96) (cid:3) − α ∞ (cid:88) i =1 (1 − α ) i − × snr · (cid:16) E (cid:2) h (cid:96) h ∗ (cid:96) + i (cid:3) (cid:17)(cid:33) = 2snr − α (cid:32) − α ∞ (cid:88) i =1 (1 − α ) i − (cid:0) ρ i (cid:1)(cid:33) = 2snr − α (cid:32) − α ∞ (cid:88) i =1 (1 − α ) i − − α ∞ (cid:88) i =1 (1 − α ) i − ρ i (cid:33) = 2snr − α (cid:32) − α ∞ (cid:88) i =1 (1 − α ) i − − αα − ρ − (cid:33) . (25)Finally, noting that α ∞ (cid:88) i =1 (1 − α ) i − = (cid:40) α = 00 if 0 < α < (26)we finish the proof. A PPENDIX BA NALYTICAL
MSE
WITH RANDOM INTERVALS
If we average over both the channel realizations and arrivalprocess, we obtain: mse = E τ (cid:20) E h (cid:2) | ˆ γ (cid:96) +1 − γ (cid:96) +1 | (cid:3)(cid:21) = E τ (cid:34) − α (cid:32) E (cid:2) γ (cid:96) (cid:3) − α ∞ (cid:88) i =1 (1 − α ) i − E h (cid:2) γ (cid:96) γ ∗ (cid:96) − i (cid:3)(cid:33)(cid:35) = 2snr − α (cid:32) − α ∞ (cid:88) i =1 (1 − α ) i − E τ (cid:20) E h (cid:2) h (cid:96) h ∗ (cid:96) − i (cid:3) (cid:21)(cid:33) (27)where we have used (11), together with (24). Now, plugging(8), we obtain mse = 2snr − α − α ∞ (cid:88) i =1 (1 − α ) i − E τ i (cid:89) j =1 ρ (cid:96) − j = 2snr − α − α ∞ (cid:88) i =1 (1 − α ) i − i (cid:89) j =1 E τ (cid:2) ρ (cid:96) − j (cid:3) = 2snr − α (cid:32) − α ∞ (cid:88) i =1 (1 − α ) i − (cid:16) E τ (cid:2) ρ (cid:3) i (cid:17)(cid:33) (28)where we have used the fact that the ρ (cid:96) are i.i.d randomvariables. Defining ρ · = E τ (cid:2) ρ (cid:3) and using (17), we arriveat mse = 2snr − α (cid:32) − αα − ρ − (cid:33) (29)whenever α (cid:54) = 0 .It only remains to compute the expectation E τ (cid:2) ρ (cid:3) . Since τ is exponentially distributed: ρ = E τ (cid:2) ρ (cid:3) = 1 T (cid:90) ∞ J (2 πf d τ ) e − τ/T d τ = 2 π K (cid:16) − π f T (cid:17) (30)which follows from [11, Eq. 6.6112.4], and where K ( x ) denotes the complete elliptic integral of the first kind [9,Sec. 17.3]. The MSE finally reads as mse = 2snr − α (cid:32) − αα − π K ( − π f T ) − (cid:33) (31)which finishes the proof.R EFERENCES[1] R. Akl, S. Valentin, G. Wunder, and S. Stanczak, “Compensating forCQI aging by channel prediction: The LTE downlink,” in
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