# Estimation of Compressible Channel Impulse Response for OFDM Modulated Transmissions
## 1 Introduction  
OFDM (Orthogonal Frequency Division Multiplexing) has become the primary modulation for physical layer design in many modern digital transmission systems since the beginning of the century. The main OFDM feature behind its popularity is the division of the entire frequency transmission spectrum into narrowband flat subchannels using DFT (Discrete Fourier Transform). When the modulation parameters such as the subchannel spacing, guard interval, and symbol lengths are well matched to the occupied frequency band, channel characteristics, and system mobility, the subchannels are orthogonal, and the transmission within them is mutually independent in consecutive OFDM symbols [1]. The subchannel orthogonality as a result of FFT makes data decoding in the individual subchannels straightforward with ZF (Zero Forcing) approach, provided that the channel frequency characteristic is known at the time when the OFDM symbols are received. In practice, the CSI (Channel State Information) is acquired prior to data reception by means of pilots, i.e., training complex symbols (e.g., belonging to QPSK constellation) known to the receiver and defined in the frequency domain. If all active subchannels of a single OFDM symbol convey pilots, they form a block pilots. On the other hand, the pilots transmitted only in some dedicated subchannels are named comb or/and scattered ones [2, 3]. It is worth mentioning that estimation precision of CSI is crucial not only for high data reception quality but also for effective users management in multiuser MIMO-OFDM systems [4]. The transmitted data are usually arranged in frames. Each frame starts with a synchronization preamble followed by one or more OFDM training symbols with block pilotsfor channel estimation. Comb pilots transmitted in the remaining OFDM symbols are necessary to keep synchronization and track channel variations using interpolation methods. An implementation example of the abovementioned frame structure can be found in 802.11 standards' family [5; 6]. A similar frame configuration seems to be appropriate in any simple OFDM transmission scheme including various proprietary solutions designed for IoT, industrial, or military-oriented applications. The good accuracy of the channel reconstruction with block pilots is one of the key factors that determines the overall transmission quality. Moreover, it can be even the sole channel reconstruction stage if the coherence time is larger than the frame duration (i.e., the channel is considered fixed during frame transmission). Considering the above system configuration, the block-pilot oriented methods implemented in the channel estimation stage are the main concern of this study. Because this initial processing is executed in receiver regardless of channel mobility, all tests were performed for fixed or slow-fading channel environments to simplify the simulations. This provides an in-depth properties understanding of the considered methods without excluding their application in mobile systems. The simplest estimation procedure of the frequency channel characteristic is based on the LS (Least Squares) method [7]. Each subchannel transfer function is estimated independently using the pilots transmitted and received only in that subchannel, but its variance is high due to the presence of noise during signal transmission. To reduce the noise impact, the LMMSE (Linear Minimum Mean Square Error) approach can be applied, but it requires statistical knowledge about the reconstructed channel and higher computational resources [8]. Both methods originally reconstruct the channel in the frequency domain; therefore, their estimates can be directly applied to restore the transmitted symbols using the ZF approach. In [9], a simple extension of the LS method is proposed to smooth its channel characteristic estimate in the frequency domain. It uses the fact that in a properly designed OFDM transmission, the maximum delay spread of a CIR (Channel Impulse Response) should not be longer then the guard period between symbols.