•This chapter points out embedded pilot-assisted delayed Doppler channel estimation
•Embedded pilot assisted delay time channel estimation
• Real-time OTFS software-defined radio implementation
1. Introduction
In this chapter, we will focus on the input-output relationship of OTFS pilot symbols for channel estimation. Assuming integer delays and Doppler taps, the delay-Doppler input-output relationship is:
where 
gi is the complex channel gain of the i-th path with integer delay.
Recall from Chapter 2 that in a broadband system, the actual channel delay displacement τi can be approximated as an integer multiple of the sampling period 1/Mdeltaf, that is, τi = li/Mdeltaf, where li∈Z. For large OTFS frames (large N values), the actual Doppler shift νi can also be approximated as an integer multiple of the Doppler resolution 1/NT, that is, νi = ki/NT, where ki∈Z. A large N results in an OTFS frame of long duration NT, which may increase the likelihood of intra-frame channel parameter changes, leading to channel estimation degradation. Therefore, in general, we consider N<M. However, for small values of N, the fractional Doppler effect is more prominent because it causes data symbols to leak into more than P delay-Doppler grid positions. Therefore, in this case, it is necessary to consider the influence of fractional Doppler in the input-output relationship.
Equation (7.1) describes the input-output relationship of an OTFS with only integer Doppler taps. For fractional-order Doppler shifts, we follow the notation in Chapters 2 and 4, where κi∈R represents the normalized Doppler shift. Then, the input-output relationship of fractional Doppler frequency shift is,
According to the approximation, each received Y[m,n] is the aggregation of information symbols along all paths. As can be seen from (7.3), the channel representation accuracy decreases. From the received symbol Y[m,n], if the channel parameters gi, τi and νi (and the corresponding taps li and κi) are known, we can use the algorithm in Chapter 6 to detect the data symbol X[m,n]. Therefore, the following channel estimation method must be used to obtain channel state information.
2. Embedded pilot delay-Doppler channel estimation
Let us consider the following system setup. At the transmitter, the OTFS frame consists of a pilot symbol, Ng protection symbol and MN−Ng−1 data symbol, as shown in the figure below
We arrange pilot, guard and data symbols in a delayed Doppler grid for transmission in OTFS frame transmission
With the help of zero guard symbols (Ng) , we arrange all symbols in such a way to ensure that there is no interference caused by channel delay and Doppler dispersion between pilot symbols and data symbols. Therefore, we have Ng = (2lmax + 1)*(4kmax + 1)−1 guard symbol with overhead as follows
Typically, in LTE channels, the overhead of pilot symbols and guard symbols is less than 1% of the data frame.
At the receiving end, we use the received symbols Y[m,n], mp≤mp≤+lmax, np−kmax≤nnp≤+kmax for channel estimation, while the remaining received symbols Y[m,n] are used for data detection, such as As shown in Figure b. By replacing m and n with mp + li and np + ki in (7.1), the pilot symbol of the i-th path received by the receiving end, (i = 1,...,P,) can be expressed as:
Our goal is to estimate the channel parameters (gi, li, ki) for i = 1,...,P, where the number of propagation paths P is unknown. We start from the received samples Y[mp + l, np + k], 0≤l≤lmax, −kmax≤k≤kmax, for channel estimation, and the remaining samples are used for data detection, as shown in Figures a and b. Show. The estimated delay-Doppler channel gain is:
However, in the presence of noise in (7.6), 
it may be mistaken for a channel path. Therefore, we propose path detection below a threshold-based channel estimation scheme.
Let b[l,k] represent whether there is a path with delay l and Doppler frequency shift k according to the threshold criterion, that is
The threshold T can be dynamically adjusted to obtain the best probability of false detection and/or path detection. The number of paths can be estimated as 
Then, the input-output relationship in (7.1) can be rewritten with the estimated channel parameters as
The estimated Doppler response corresponding to each delay tap defined in Chapter 4 can be written as
Then, the time-varying Doppler diffusion vector of each extension tap ˆνm, l∈CN×1 of all OTFS grid points can be calculated as
By phase rotation zk(m−l), the MN × MN delay-Doppler channel matrix H can be completely reconstructed using the information of ˆνl (see Chapter 4)
Chapter 4 Relevant Information Review: