# Estimation of Compressible Channel Impulse Response for OFDM Modulated Transmissions
## 4 Numerical Experiments and Performance Analysis  
### Simulation Parameters  
The proposed modification and the representative DFT-based reference methods were tested in a purpose-made OFDM modulated transmission system. The analysis was performed in baseband assuming perfect frequency synchronization. The frame synchronization was recovered using the standard preamble-based crosscorrelation technique (see Equation (1)), and the LS CIR estimate was the base for further processing both in the reference methods and in the proposed one. The frame synchronization fluctuations resulting in a cyclic shift of the CIR estimate were compensated by the detection of its the most significant part according to Formula (8) using a moving average filter of length \(M\). The block-pilot symbols were randomly generated from the QPSK unit-energy symbol constellation and transmitted in all \(N\) subchannels. The synchronization preamble was a double repetition of the sequence obtained as the result of IFFT transformation of an \(N\)-element random QPSK vector. The system noise was modeled as the AWGN. The major assumptions of the system model are summarized in Table 2.  
According to the assumed mobility parameter (maximum speed is \(v=10\) km/h), the channel characteristic can be considered fixed during frame transmission because the \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Carrier frequency & \(f_{C}=2\) GHz \\ Sampling frequency & \(f_{S}=10\) MHz \\ Active subchannels & \(N=128\) \\ FFT size & \(N=128\) \\ Cyclic prefix length & \(M=32\) (samples) \\ Synchronization sequence & \(P=256\) (samples) \\ Synchronization factor & \(\Delta=M/2\) \\ Block-pilot OFDM symbols per frame & 1 \\ Pilot constellation & QPSK \\ Data OFDM symbols per frame & \(F=50\) \\ Frame duration & \(T_{F}=P+(M+N)FT_{S}=825.6\)\(\upmu\)s \\ Data constellation & 16-QAM, 64-QAM \\ Data frames per transmission & 50 \\ Maximum speed & \(v=10\) km/h \\ Doppler spread & \(f_{d}=\frac{v}{2}f_{C}\approx 18.5\) Hz \\ Coherence time [23] & \(T_{C}\approx\frac{8.423}{f_{d}}=22.9\) ms \\ SNR range & \(\langle 0,25\rangle\) dB with 5 dB step \\ \hline \hline \end{tabular}
\end{table} 
Table 2: System model parameters. coherence time \(T_{C}\) is significantly larger then the frame duration \(T_{F}\). This communication scenario can be typical, for example, for sensor networks and related systems designed to handle at most pedestrian traffic (including recreational running too). The benchmark channel models used in the simulation analysis are enumerated in Table 3. They are a part of the recommendation for the evaluation of radio transmission systems operating in 2 GHz band [20]. The selected channels represent propagation environments typical for indoor transmission (Indoor A and B) and outdoor-to-indoor transmission when the base station is located outdoors. They cover cases of the most frequently occurring propagation conditions including also indoor-to-outdoor transmission scenarios on the basis of channel reciprocity property [24]. Every channel model is characterized by a tapped-delay line model in the time domain. In the mentioned recommendation, the particular taps are the coefficients of CIR directly. In the simulations presented here, a more realistic scenario was assumed. The taps were considered propagation paths, and the target CIR was a result of low pass filtering in baseband. The paths delay varied randomly about 3% around average value according to the model definitions. Additionally the shifted version of Indoor B model was proposed to enhance channel compressibility--the sampling moments (i.e., integer multiples od \(T_{S}\)) were not consistent with the paths delay. The quality metrics, MSE and BER, were evaluated in Monte Carlo experiments using 300 independent uncoded baseband transmissions over CIRs randomly generated by MATLAB built-in function. There was assumed a single-channel realization per transmission.