## 1 Introduction  
Fixed-lag smoothers have been widely investigated because they provide optimal estimates, unlike filters and predictors [1; 2; 3; 4]. In practice, smoothers are often used to improve the performance of filters, provided delays can be tolerated. In smoothing problems, the objective is to estimate past or delayed states using past measurements up to the current time. Three types of smoothers exist: fixed-point, fixed-lag, and fixed-interval smoothers. The fixed-lag smoother is the most useful type because it offers more accurate and general solutions than filters. Thus, smoothers have been considered as a suitable choice in many applications where a short time delay is acceptable, such as denoising ECG signals [5], image processing [6], target tracking [7; 8; 9], system-on-chip testing [10], structural dynamics estimation [11; 12], and vibration analysis [13]. The structures of fixed-lag smoothers can be categorized as infinite impulse response (IIR) and finite impulse response (FIR) structures, based on the duration of the impulse responses. Owing to the IIR structure, the fixed-lag Kalman (FK) smoother has been widely used considering its optimality [1; 4; 14; 15]. However, certain potential problems may arise due to this IIR structure. Because the FK smoother is designed under the assumption of an accurate system model, optimality cannot be guaranteed when there exists a model mismatch. Moreover, because it uses all the information from initiation to the current time, undesirable errors can accumulate in the state due to modeling and numerical errors,such that the estimates might oscillate or diverge. To overcome the shortcomings of the IIR structure, receding-horizon fixed-lag (RHF) smoothers have been proposed for linear systems [5; 16; 17; 18; 19; 20; 21] and non-linear systems [22] as alternatives to FK smoothers. Because RHF smoothers estimate the state by using the most recent finite measurements, they exhibit good properties such as bounded-input bounded-output (BIBO) stability, fast tracking ability, and robustness against modeling uncertainties and numerical errors. However, previous RHF smoothers require vast amounts of computational power and memory, thereby preventing efficient implementations in real-time applications. Moreover, their complicated batch-form derivation is difficult to understand, and thus prevents further developments. Because these disadvantages are mainly caused by the high-dimensional matrix multiplications involved in batch calculations, one possible approach to solving this problem is to consider the RHF smoother. In [17], a guideline for choosing the appropriate horizon length of an RHF smoother is introduced for linear time-invariant systems; however, the structure of the proposed smoothing algorithm has a complicated batch form, making it difficult for application to large dimensional systems. In fact, the adjustment of horizon length is not a fundamental solution to the problem of reducing the computational cost because the horizon length is an important design parameter of RHF smoothers that can significantly affect the estimation performance. An alternative approach to reducing computational complexity and memory size involves separating the estimation horizon by changing the high-dimensional matrix multiplication into the sum of reduced-dimensional matrix multiplications. If the estimates are obtained from two separated and independent estimators, the computational cost can be reduced by parallel computing. Because RHF smoothers estimate the state at a fixed lag time, finite measurements on the overall horizon can be partitioned into two independent subsets of measurements based on the estimation time. For the separated two subsets of measurements, different receding horizon estimators can be implemented independently, and their estimates can be combined in an optimal manner using [PERSON]'s theorem [1]. A recursive RHF smoothing algorithm can provide a method for overcoming the disadvantages of computational complexity and memory size.