In this section, we review different algorithms used for the estimation of the carrier frequency, power, and oversampling factor of an observed signal component. Then the features based on autocorrelation and cyclic autocorrelation are presented for CP-OFDM signals to estimate the useful time interval, cyclic prefix duration, and the number of subcarriers in frequency selective channels.

### 2.1. Estimation of the Carrier Frequency

To estimate the carrier frequencies , with the number of signals components present in an observed spectrum, one can use a nonparametric approach based on a Fast Fourier Transform (FFT) [12]. This method is appropriate for narrowband signals above the noise floor provided that the frequency resolution is high enough. One can assume that the observed compound signal has been sampled at least twice the maximum bandwidth of interest :

with the received sampled data stream and its Power Spectral Density (PSD) ( is the number of samples). Note that the larger the number of samples, the higher the frequency resolution provided by the FFT grid. This is a crucial parameter to detect signals having a small bandwidth. However, if the FFT size is too large, the sequence of length can be divided into blocks of size , then one can perform FFTs of length and add the contribution of each block given by (1). The carrier frequencies , are then estimated by detecting the band-edges and , for all with a threshold between the noise level and the signal level, which is driven by the sensitivity of the receiver. The carrier frequencies are then estimated as

### 2.2. Estimation of the Oversampling Factor and Power Spectral Density

Assuming that the carrier frequencies have been estimated, then each individual signal component of interest can be downconverted to baseband and lowpass filtered (with decimation). The resulting digital baseband signal can then be modeled as a received sequence of length such that

where is the vector of transmitted symbols, which have been oversampled by a factor , the 's are the multipath channel coefficients with the number of channel taps, is the vector of Additive White Gaussian Noise (AWGN), is the receiver phase offset, and is the receiver frequency offset.

In order to calculate the oversampling factor , which corresponds to the ratio between the bandwidth of the lowpass filter and the bandwidth of the signal of interest, one can use again a nonparametric approach based on a Fast Fourier Transform (FFT) on the received sequence [11]:

with the PSD of the received signal. Then, the idea is to design a target filter which has the smallest Euclidian distance to . Knowing the total energy in the frequency domain, one can design the following target filter:

with the all-one vector of length and the all-zero vector of length . The optimization problem is to minimize over the following expression:

An exhaustive search on the index is performed to solve this optimization problem. Then, one can calculate the ratio between the bandwidth of the lowpass filter and the bandwidth of the signal of interest. The number of FFT points spanning the bandwidth of the lowpass filter corresponds to the number of samples , while the number of FFT points spanning the transmitter bandwidth corresponds to twice the cutoff frequency of the optimal target filter. Once the optimal is found, one can calculate the oversampling factor:

The estimated signal power (average of PSD over all frequency bins) is then defined as

Figure 1 shows an example of a received sequence of 4096 samples from a real CP-OFDM data measurement (with 26 tones going through an unknown channel and sampled at the receiver with a known sampling rate) after carrier estimation, downconversion, and lowpass filtering. The estimation of the oversampling factor and signal power is performed using the algorithm presented in this section, leading to and for this particular data measurement.

### 2.3. Estimation of the Useful Time Interval and the CP-Length for CP-OFDM Signals

The estimation of the useful time interval and the CP-length exploits autocorrelation or cyclic autocorrelation properties of the received sequence [9–11]. The autocorrelation of the received sequence which corresponds to the second-order/one-conjugate cyclic cumulant at zero cyclic frequency in the work of [6, 7] can be written as

with the shift index. One can assume that the autocorrelation function is cyclically shifted; that is, two vectors are concatenated in order to cope with the data outside the interval . For CP-OFDM, the last part of the OFDM symbol is copied at the beginning to prevent Inter Symbol Interference (ISI) after multipath propagation. Therefore, a peak in the autocorrelation function can be observed at delay . Figure 2 shows the ideal autocorrelation function for a transmitted CP-OFDM signal .

The autocorrelation function can be derived by replacing the received sequence model (3) into (9), leading to

with the variance of the transmitted signal and the variance of the AWGN. As stated by these equations, there are peaks due to the multipath coefficients when the channel is stationary over the observation window. Therefore, one can use a peak detection algorithm similar to [7] based on positive and negative slopes ("+−" corresponding to the event of a positive slope followed by a negative slope on Figure 3). One can assume that the maximum channel delay spread is smaller than the useful time interval , and therefore the peaks corresponding to the cyclic prefix insertion will appear as a second cluster of peaks at higher values of . Hence one can discard the peaks with the lowest shifts and keep the peaks with the highest shifts for the estimation of the useful time interval.

Figure 3 shows the technique used for estimating the largest peak. The useful time interval is estimated by an exhaustive search according to the following optimization problem:

corresponding to the search for the optimal index for which the difference between a peak (when a +− occurs) and its lowest previous point (its previous −+) is maximized. The choice of the modulus leads to an insensitivity of the autocorrelation feature to phase and frequency offsets (as the exponentials factors in (10) disappear).

The number of subcarriers can be determined as the ratio between the useful time interval and the transmitter sampling period [11]. Moreover, the ratio between the transmitter sampling period and the sampling period of the lowpass filter leads to the oversampling factor . Without loss of generality, the sampling period of the lowpass filter is normalized to unity. Therefore, once the useful time interval has been estimated, one can determine the number of subcarriers by

From [8], it is known that a CP-OFDM signal is cyclostationary with period . The cyclic autocorrelation is given by

From the model in (3), one can determine the cyclic period using the following optimization problem with an exhaustive search on :

The cyclic period can then be found by the following equation:

Finally, time and frequency offsets can be determined using cyclostationarity properties of CP-OFDM signals with prior information on the pulse shaping filter [8] or conventional autocorrelation methods without prior knowledge on the pulse shaping filter [15].