# Estimation of Compressible Channel Impulse Response for OFDM Modulated Transmissions
## 2 System and Channel Models  
Consider an OFDM symbol in the baseband. This consists of \(N\) frequency samples of complex values usually taken from a QPSK or QAM constellation sets, which describe binary data encoding rules. The symbol is transformed by \(N\)-point IFFT into the time domain. Before it is transmitted through channel with the sampling period of \(T_{S}\), the last \(M\) samples of the IFFT result are copied and appended to the beginning of the symbol to form a cyclic prefix. This facilitates symbol reception in the presence of linear channel distortion and time synchronization offset by preserving the mutual independence of the adjacent OFDM symbols in the time.  
Let \(F\) consecutive OFDM symbols form a single information frame, starting with a preamble \(s\) of \(P\)-element pseudorandom synchronization sequence followed by a training OFDM symbol with block pilots in all subchannels (see the frame structure in Figure 2). The training symbol is used for the reconstruction of a linear multipath channel described by the impulse response \(h\) of length \(L<M\) samples. Because, in these studies, the channel characteristic is considered constant during the single frame transmission, the obtained channel estimate is applied to decode the information in all OFDM data symbols of the given frame.  
Figure 2: Frame structure.  
The transmission environment is characterized by an impulse response of the wireless channel within \(B=10\) MHz channel band conforming with the paths models for indoor and outdoor-to-indoor transmissions presented in [20]. It is worth mentioning that the time separation between the subsequent paths is sometimes under the resolution of the sampling period \(T_{S}=1/B\). This enhances the compressibility feature of the channel and makes it impossible to resolve the individual paths in the estimated CIR. Other causes of compressibility are as follows: the delays of the individual paths differ slightly from multiples of the sampling period \(T_{S}\), or they are all shifted in time to simulate a time synchronization offset.  
The system model assumes operation in the baseband with perfect frequency synchronization (no interchannel interferences; therefore, subchannel orthogonality is preserved), whereas the frame timing \(t_{s}\) is acquired using correlation between preamble in received signal \(y\) and the original synchronization sequence \(s\):  
\[t_{s}=\operatorname*{arg\,max}_{n}\left(\left|\sum_{i=0}^{p-1}s^{*}(i)y(n+i) \right|\right)-\Delta. \tag{1}\]  
The parameter \(\Delta\) slightly advances the moment when the frame starts to be analyzed. It reduces a risk of ISI (intersymbol interferences) because individual OFDM symbols extracted from the frame structure have their origin in the region of cyclic prefix that is free from preceding symbol interference.  
Due to the subchannel orthogonality, the information transmitted in the respective subchannels can be processed independently and simultaneously. The signal received in \(k\)th subchannel is described as follows:  
\[Y(k)=X(k)H(k)+W(k), \tag{2}\]  
where \(X\) is the original information symbol, \(H\) is the frequency characteristic of a real channel, and \(W\) is a complex, zero-mean, white Gaussian noise signal. To restore the original information, ZF (Zero Forcing) or MMSE (Minimum Mean Square Error) equalization techniques can be applied. For the former, widely used due to its simplicity, the equalized data \(\hat{X}\) can be calculated in the receiver using the following formula:  
\[\hat{X}(k)=\frac{Y(k)}{\hat{H}(k)}. \tag{3}\]  
where \(\hat{H}\) represents a channel estimate determined with the block pilots. The above system description is outlined in Figure 3. In this paper, DFT-based channel estimation methods are under investigation. The next section presents their overview and the proposition of a new one.