**Primary Synchronization Signal**

The PSS sequence in LTE downlink is one of three Zadoff-Chu sequences. Each ZC corresponds to one of three physical layer cell IDs in a group. The PSS is sent over the central 6 RBs. Out of the 72 subcarriers only the central 62 subcarriers are used. The remaining 10 subcarriers are kept reserved. The PSS is sent every 5 milliseconds, twice in a radio frame. In the case of FDD, the PSS is sent in the last OFDM symbol of every 1st and 11th slot of a frame.

**Secondary Synchronization Signal**

The SSS is also sent over the central 6 RBs. Like the PSS, only 62 subcarriers are used, the remaining 10 subcarriers are kept reserved. In the case of FDD, the SSS is sent in the symbol immediately preceding the symbol carrying PSS. In the case of TDD, the SSS is sent 3 symbols earlier than the PSS symbol. Unlike the PSS, the SSS is a binary sequence which is a interleaving of two length-31 binary sequences. The sequence is scrambled with another sequence which is generated.

**Detection of reference sequences in time-domain**

In the LTE, the PSS is a ZC sequence in the frequency domain, distributed over the central 62 subcarriers. The corresponding OFDM signal in the time domain is the inverse FFT of not only the ZC sequence coefficients, but also resource element values in other subcarriers outside the central 6 RBs. The time domain OFDM signal is therefore not a ZC sequence. Furthermore, the ZC sequence used as PSS has length 62. The IFFT size used in OFDMA is 2048 so as to accommodate the maximum bandwidth of 20 MHz. Even if we reserve all other subcarriers except the central 62 subcarriers, the time domain OFDM symbol is not a ZC sequence. Therefore, to detect a ZC sequence in the time domain, we need to use the equivalent time-domain representations.

Consider a vector of samples in the time domain. Let us denote it by . In order to check if contains the ZC sequence , we would first apply the FFT on , extract the coefficients corresponding to central 62 subcarriers, and then correlate them with . The absolute value of the correlation can be written as , where denotes the Discrete Fourier Transform (DFT) matrix of order , and is a rectangular matrix of size 62x2048 for extracting only the coefficients corresponding to central 62 subcarriers. The above term can be rewritten as , where denotes the IFFT of after padding zeros. Thus the ZC sequence can be equivalently represented by the length- reference sequence .

The ZC sequences have constant amplitude value of 1. Thus when a ZC sequence is multiplied by its complex conjugate element-wise, we get 1 at all 62 subcarriers