Effect of DNN Approximation for Channel Estimation and Signal Detection on OFDM Applications

Deep learning and OFDM structure are determined in this section.

2.1 Deep learning structure

DNNs need a huge count of data and need to train before they are implemented offline. DNNs utilize a huge count of data and need to train before they are implemented offline. Therefore, there are two phases of distributing the DNN; the first phase supposes training as offline and the other phase supposes online distribution of the training net [11]. As we need a big number of data on the training, a channel that known model is anticipated to be utilized. Models of developed channels can define the realistic channels efficiently when full statistics of channels are given. To simulate and generate a big number of data with changing statistics of the channel is made easy by using the developed channel models.

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Figure 1. Deep learning structure

The pilot in OFDM symbols and the data in OFDM symbol came to DNN as inputs. The pilot and data symbols together are discretized into imaginary parts and real parts and applied network as input and so the count of input neurons can reproduce to be 256 neurons when we use 64 subcarriers. The output layer of DNN estimates and recovers the received data as bits and the Sigmoid activation function is used in this layer. As given Table 1 DNN parameters are batch size, number of class, max epochs, number of hidden units and input size.

Table 1. Training parameters of DNN

Deep learning applications have been successful in a large variety of fields via an important success developed, including opinion on computers, the processing of inherent language, speech identification and so on. Generally, DNNs that present deepness sample of ANNs raise the hidden layers' number to develop the ability to identify. In the network each layer is formed of multiple neurons, which is a nonlinear function of a summation of the neurons on the previous phase and illustrated in Figure 1.

Sigmoid and Relu functions can specify as the nonlinear function, determined as f_k(x)=1/(1+e^−x), and f_l(x)=max(0,x). So, through a series of non-linear transforms of input data (I) is formed the network output w is stated as Eq. (1) mathematically [3].

$z=f(D, Q)=f^{(D-1)} f^{(D-2)}\left(\ldots f^{(1)}(D)\right)$                        (1)

In Eq. (1), layers the number is D, and the neural networks’ weights are stood by Q. The weights on the neurons are defined as the parameters of the model, and on a training set, the optimal weights are usually obtained using acknowledged wanted outputs [15].

2.2 OFDM structure

In the transmitter block, the bit flow is corresponded to a matrix of dimension Nt × Nf, if the count of antennas on the transmitter side is assumed Nt and the count of subcarriers is assumed Nf. Figure 2 shows a block of an OFDM system under nonlinear distortion. In the process of data transmission first, any modulation is performed, then Inverse Fast Fourier Transform (IFFT) is implemented, and then a CP is added, and finally, the RF is amplified and sent the data. The transmitted signal has non-linearity distorted owing to the amplifier. The matrix includes the QAM (Quadrature Amplitude Modulation) or PSK (Phase-Shift Keying) symbols, all row indicates an antenna mark, and all column shows a subcarrier. Then each row modulates with OFDM utilizing traditional techniques [16].

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Figure 2. Block diagram of OFDM

In Figure 2, the OFDM structure, the transmission channel is regarded to keep non-changed for two successive OFDM symbols, one of these includes generally pilots, while the others contain the data, and these two symbols are jointly described as a frame. The pilot location should maintain stability along all steps of the training and propagation to realize estimation of channel effectively. When the signal has been carried throughly the channel and accepted by the receiver antenna, the transmitted symbol is transformed into a parallel data flow thus the CP is extracted and FFT operation is applied [10].

The receiver side makes the inverse processes of the transmit side. First, receiver removes the cyclic prefix and then transformation of FFT and demodulation processes are implemented, and finally, an original bit stream is obtained.

Considering multi path channel defined with complicated random factors {h(m)}(m=0….,M−1),  the received signal y(m) clarifies in Eq. (2).

$y(m)=x(m) \otimes h(m)+w(m)$                        (2)

Eq. (2) shows the convolution as circular between x(m) is described transmitted signal and h(m) is defined channel response and w(m) is introduced the AWGN (Additive White Gaussian Noise) is added to this.

$Y(k)=X(k) H(k)+W(k)$                        (3)

The receiving signal on the frequency area is given with Eq. (3) following extracting the CP and enforcing the DFT process and the DFT of y(m), x(m), h(m) and w(m) are Y, X, H, and W. The receiving data symbols formed a pilot build, a data build is assumed as input in our work, and the transmitted data is recovered by the purposed DNN method.

Equation 3 can write as vector form in Eq. (4).

$\bar{Y}=X \bar{H}+\bar{W}$                        (4)

When X is assumed a matrix of input, $\bar{Y}$  is assumed output matrix, cost function in LS Channel Estimation can be written as Eq. (5).

$C=\|\bar{Y}-X \bar{H}\|^2$                        (5)

$\hat{H}_{L S}=\left(X^H X\right)^{-1} X^H Y$                        (6)

The solution of LS estimators is given in Equation 6 and because of invertible X, Equation 6 can rewrite as Eq. (7) as a conclusion Eq. (8) is obtained.

$\hat{H}_{L S}=\left(X^H\right)^{-1} X^{-1} X^H Y$                       (7)

$\hat{H}_{L S}=X^{-1} Y$                      (8)

LS estimators have the advantage of their simplicity and make calculations with very low complexity with no information about statistics of the channel.

The mean value of coefficients of channel can define as E{H(k)}=0 and variance can determine as $E\left\{|H(k)|^2\right\}=L \sigma_h^2$ . Also, the mean and variance of AWGN channel can write as E{W(k)}=0 and $E\left\{|W(k)|^2\right\}=N \sigma_n^2$  respectively.

L assumes number of multi-paths and N assumes number of subcarriers. Channel and noise variances are given by $\sigma_h$  and $\sigma_n$  respectively. MMSE estimators are defined in Eq. (9).

$\hat{H}_{M M S E}$ $(k)=R_{H(k) Y(k)} $ $R_{Y(k) Y(k)} $ $Y(k)$                       (9)

Cross covariance of H(k) and Y(k) are illustrated with $R_{H(k) Y(k)}$  and given in Eqns. (10) and (11).

$R_{H(k) Y(k)}=E\left\{H(k) Y^*(k)\right\}=L \sigma_h^2 X^*(k)$                      (10)

$R_{Y(k) Y(k)}=E\left\{Y(k) Y^*(k)\right\}$$=L \sigma_h^2\left|X^*(k)\right|^2 N \sigma_n^2$                     (11)

$\hat{H}_{M M S E}$ $(k)$$=L \sigma_h^2 X^*(k) \frac{1}{L \sigma_h^2|X(k)|^2+N \sigma_n^2} Y(k)$                      (12)

MMSE channel estimation equation is given in Eq. (12) by using Eqns. (10) and (11).

There are two stages as shown in Figure 3 to do an efficient DNN application on both symbol detection and channel estimation. For the offline training phase, we train the model with the received OFDM symbols, which are created with diverse data sequences, and for different channel cases. For the online spreading phase, the DNN approximation creates the output, which rescues the transmitting data with no estimate of the wireless channel clearly.

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Figure 3. Online deployment and Offline training of OFDM block diagram

Many channel models have improved that define the real channels greatly considering statistics of channel. The training data is acquired using these channel models by simulation. In all steps of the simulation, first, we generate a random data sequence for the transmitted signals and compose the matching OFDM skeleton with a pilot symbol series. Then we simulate the available random channel related to the models of the channel. OFDM signal that is received includes the valid distortion of the channel and the noise of channel based on the OFDM skeleton. The training data is collected from the receiver symbol and the actually transmitter symbol. The receiver symbol from the pilot build and one data build is admitted as the input of the pattern of deep learning. To make smaller the gap among the neural network output and the transmitting symbol, the model of deep learning is trained [17-21].