MMSE channel estimation

In linear minimum mean square error estimation, the criterion function is statistical mean square error. First, the channel estimation value to be determined is expressed as:

Where w is the matrix of weight coefficients to be determined, so these weight coefficients are estimated. The criterion function of LMMSE estimation is as follows:

Find the partial derivative of w to make it equal to 0. Using the principle of orthogonality, we can get:

that is

Therefore, the minimum mean square error of g is

Finally, the estimated value of H is

The following figure shows the performance comparison between the channel estimation technology and the equalization technology under the EVA channel condition, in which the modulation mode is 16-QAM and the coding rate is 1/2 Turbo code.

It can be seen from the figure that the performance of LMMSE channel estimation is obviously better than that of LS channel estimation under multipath channel conditions. It can also be seen that in the case of low SNR, the performance has not been greatly improved, and the influence of noise is the main factor. Under the condition of AWGN channel, due to the existence of Turbo code, the performance of LS channel estimation has no bit error rate when the signal-to-noise ratio is 12dB. At this time, there is little difference between the curves of the two channel estimates. Therefore, under the channel condition of this system (the channel condition is indoor channel, and the signal-to-noise ratio is above 14dB), ls channel estimation is enough. In addition, because there is no interference between antennas, the performance curves of MMSE equalization and ZF equalization almost coincide.

Considering the possible timing error, it will lead to phase offset, which is a linear function of k. When using tracking pilot for channel estimation, the influence of phase noise is ignored, and when using linear interpolation, all phase offsets can be accurately estimated. At this time, linear interpolation is the most suitable in frequency domain. Under indoor channel conditions, the signal-to-noise ratio can be guaranteed to be greater than 14dB, and the channel coding can be completely decoded, so the channel estimation method in frequency domain adopts LS algorithm.

Under the condition of white Gaussian channel, if there is no timing error, the estimated channel response should be approximately flat in real part and zero in imaginary part. If there is a timing error, the estimated channel will have a certain phase deviation. The following figure shows the channel estimation results in two cases.

The above diagram is a constellation diagram of channel estimation values, and the following diagram is a time domain diagram. In the time domain diagram, the upper diagram is the real part of the channel estimation value after frequency domain interpolation, and the lower diagram is the imaginary part. The first 600 subcarriers have positive frequencies and the last 600 subcarriers have negative frequencies. The first picture is the case of accurate timing, which shows that there is still phase noise in the channel estimation value; In the second picture, there is an error in the time point. At this time, the introduced phase deviation will be superimposed on the phase noise. If the timing deviation is less than 3 samples, the phase deviation will be less than 2pi, and the timing deviation of this picture is exactly 3 samples.

The actual channel is an indoor channel, and the channel changes slowly in time domain, so the nearest neighbor interpolation method can be used to estimate other symbols without pilot insertion. At this time, it is considered that the channel response in one time slot is approximately unchanged.

Therefore, the channel estimation algorithm adopts linear interpolation in frequency domain and adjacent interpolation in time domain. No pilot information is inserted in the pilot located on the antenna port 2, so this characteristic can be used to estimate the signal-to-noise ratio, that is, the pilot on the sixth symbol of each time slot is used to calculate the signal power, and the pilot on the seventh symbol is used to calculate the noise power.

Considering the existence of certain phase noise, the constellation of each symbol has certain phase deviation, and one of the main phase noise is the influence of residual frequency offset. Assuming that the residual frequency offset after frequency offset compensation is f, these frequency offsets are accumulated into one time slot by using the above-mentioned time domain channel interpolation method. Assuming that the last residual frequency offset is f_delta, the residual phase offset of each symbol in a time slot is

Because the channel estimation in time domain is obtained by direct spread, the residual phase offset will increase due to the accumulation of time. The accumulation interval is one OFDM symbol length (plus CP). The phase offset of all OFDM symbols in the whole downlink subframe is shown in the following figure:

Each OFDM symbol undergoes the same phase deflection. When the residual frequency offset with ideal frequency offset compensation effect is 10Hz, the maximum phase offset is about 0.0 1pi. The constellation phase offset of each symbol in the next time slot of the actual channel is as follows.