Learn Modulation and Demodulation in the AWGN Channel with Deep Neural Networks by Autoencoders and End-to-end Training

This code is provided as supplementary material of the lecture Machine Learning and Optimization in Communications (MLOC).

This code illustrates

End-to-end-learning of modulation scheme and demodulator in an AWGN channel with time-varying batch size



In [1]:

    
import tensorflow as tf
import numpy as np
import matplotlib
import matplotlib.pyplot as plt
from ipywidgets import interactive
import ipywidgets as widgets

Here, we consider the simple AWGN channel.



In [2]:

    
M = 16

EbN0 = 11

# noise standard deviation
sigma_n = np.sqrt((1/2/np.log2(M)) * 10**(-EbN0/10))

Helper function to compute Bit Error Rate (BER)



In [3]:

    
# helper function to compute the symbol error rate
def SER(predictions, labels):
    return (np.sum(np.argmax(predictions, 1) != labels) / predictions.shape[0])

Here, we define the parameters of the neural network and training, generate the validation set and a helping set to show the decision regions



In [4]:

    
# validation set. Training examples are generated on the fly
N_valid = 100000

# number of neurons in hidden layers at transmitter
hidden_neurons_TX_1 = 50
hidden_neurons_TX_2 = 50
hidden_neurons_TX_3 = 50
hidden_neurons_TX_4 = 50

# number of neurons in hidden layers at receiver
hidden_neurons_RX_1 = 50
hidden_neurons_RX_2 = 50
hidden_neurons_RX_3 = 50
hidden_neurons_RX_4 = 50



y_valid = np.random.randint(M,size=N_valid)
y_valid_onehot = np.eye(M)[y_valid]

# meshgrid for plotting
ext_max = 1.8  # assume we normalize the constellation to unit energy than 1.5 should be sufficient in most cases (hopefully)
mgx,mgy = np.meshgrid(np.linspace(-ext_max,ext_max,400), np.linspace(-ext_max,ext_max,400))
meshgrid = np.column_stack((np.reshape(mgx,(-1,1)),np.reshape(mgy,(-1,1))))

This is the main function of TensorFlow that generates the computation graph. We have a single interface to the outside (a tf.placeholder which is the batch size. Here the idea is to vary the batch size during training. In the first iterations, we start with a small batch size to rapidly get to a working solution. The closer we come towards the end of the training we increase the batch size. If keeping the abtch size small, it may happen that there are no misclassifications in a small batch and there is no incentive of the training to improve. A larger batch size will most likely contain errors in the batch and hence there will be incentive to keep on training and improving.

Here, the data is generated on the fly inside the graph, by using TensorFlows random number generation. As TensorFlow does not natively support complex numbers (at least in early versions), we decided to replace the complex number operations in the channel by a simple rotation matrix and treating real and imaginary parts separately.

We use the ELU activation function inside the neural network and employ the Adam optimization algorithm.



In [5]:

    
# generate graph
graph = tf.Graph()

with graph.as_default():    
    # placeholder for training data (passed from external)
    tf_batch_size = tf.placeholder(tf.int32, shape=())

    # the validation dataset, we only have labels
    tf_valid_labels = tf.constant(y_valid_onehot, dtype=tf.float32)

    # temporary identity matrix for one-hot vector conversion
    tf_onehot_conversion = tf.constant(np.eye(M), dtype=tf.float32)
    
    # the mesgrid for plotting the decision region
    tf_meshgrid = tf.constant(meshgrid, dtype=tf.float32)
    
    # define weights
    Ws = {
        'T1' : tf.Variable(tf.truncated_normal([M,hidden_neurons_TX_1], stddev=0.8)),
        'T2' : tf.Variable(tf.truncated_normal([hidden_neurons_TX_1, hidden_neurons_TX_2], stddev=0.8)),
        'T3' : tf.Variable(tf.truncated_normal([hidden_neurons_TX_2, hidden_neurons_TX_3], stddev=0.8)),
        'T4' : tf.Variable(tf.truncated_normal([hidden_neurons_TX_3, hidden_neurons_TX_4], stddev=0.8)),
        'T5' : tf.Variable(tf.truncated_normal([hidden_neurons_TX_4, 2], stddev=0.8)),
        'R1' : tf.Variable(tf.truncated_normal([2,hidden_neurons_RX_1], stddev=0.8)),
        'R2' : tf.Variable(tf.truncated_normal([hidden_neurons_RX_1, hidden_neurons_RX_2], stddev=0.8)),
        'R3' : tf.Variable(tf.truncated_normal([hidden_neurons_RX_2, hidden_neurons_RX_3], stddev=0.8)),
        'R4' : tf.Variable(tf.truncated_normal([hidden_neurons_RX_3, hidden_neurons_RX_4], stddev=0.8)),
        'R5' : tf.Variable(tf.truncated_normal([hidden_neurons_RX_4, M], stddev=0.8))
        }

    bs = {
        'T1' : tf.Variable(tf.truncated_normal([hidden_neurons_TX_1], stddev=0.8)),
        'T2' : tf.Variable(tf.truncated_normal([hidden_neurons_TX_2], stddev=0.8)),
        'T3' : tf.Variable(tf.truncated_normal([hidden_neurons_TX_3], stddev=0.8)),
        'T4' : tf.Variable(tf.truncated_normal([hidden_neurons_TX_4], stddev=0.8)),
        'T5' : tf.Variable(tf.truncated_normal([2], stddev=0.8)),
        'R1' : tf.Variable(tf.truncated_normal([hidden_neurons_RX_1], stddev=0.8)),
        'R2' : tf.Variable(tf.truncated_normal([hidden_neurons_RX_2], stddev=0.8)),
        'R3' : tf.Variable(tf.truncated_normal([hidden_neurons_RX_3], stddev=0.8)),
        'R4' : tf.Variable(tf.truncated_normal([hidden_neurons_RX_4], stddev=0.8)),
        'R5' : tf.Variable(tf.truncated_normal([M], stddev=0.8))
    }
        
    

    def network_transmitter(batch_labels):
        nn = tf.nn.elu(tf.matmul(batch_labels, Ws['T1'])+bs['T1'])
        nn = tf.nn.elu(tf.matmul(nn, Ws['T2'])+bs['T2'])
        nn = tf.nn.elu(tf.matmul(nn, Ws['T3'])+bs['T3'])
        nn = tf.nn.elu(tf.matmul(nn, Ws['T4'])+bs['T4'])
        nn = tf.matmul(nn, Ws['T5'])+bs['T5']
        return nn

    def network_receiver(inp):
        nn = tf.nn.elu(tf.matmul(inp, Ws['R1'])+bs['R1'])
        nn = tf.nn.elu(tf.matmul(nn, Ws['R2'])+bs['R2'])
        nn = tf.nn.elu(tf.matmul(nn, Ws['R3'])+bs['R3'])
        nn = tf.nn.elu(tf.matmul(nn, Ws['R4'])+bs['R4'])
        logits = tf.matmul(nn, Ws['R5'])+bs['R5']
        return logits
    
    def channel_model(modulated):
        # just add noise, nothing else
        received = modulated + tf.random_normal(shape=(tf_batch_size,2), stddev=sigma_n)
        return received
      
    def autoencoder(batch_labels):        
        # compute output
        encoded = network_transmitter(batch_labels)
        
        # compute normalization factor and normalize channel output
        norm_factor = tf.sqrt(tf.reduce_mean(tf.square(encoded)) * 2 )                            
        modulated = encoded / norm_factor    
                
        received = channel_model(modulated)
        
        logits = network_receiver(received)
        return logits
    
    

    # generate random data
    batch_temp = tf.random_uniform(shape=[tf_batch_size], dtype=tf.int32, minval=0, maxval=M)
    # convert to one-hot representation
    batch_labels = tf.nn.embedding_lookup(tf_onehot_conversion, batch_temp)

    logits = autoencoder(batch_labels)

    loss = tf.reduce_mean(tf.nn.softmax_cross_entropy_with_logits_v2(labels=batch_labels, logits=logits))    

    # use Adam optimizer
    optimizer = tf.train.AdamOptimizer().minimize(loss)
 
    # get constellation     
    constellation_unnormalized = network_transmitter(tf_onehot_conversion)
    norm_factor = tf.sqrt(tf.reduce_mean(tf.square(constellation_unnormalized)) * 2 )
    constellation = constellation_unnormalized / norm_factor

    # compute channel output of validation and decision of validation
    valid_modulated = network_transmitter( tf_valid_labels) / norm_factor
    valid_received = channel_model(valid_modulated)        
    valid_prediction = tf.nn.softmax(network_receiver(valid_received))
    
        
    # mesh prediction for plotting     
    mesh_prediction = tf.nn.softmax(network_receiver(tf_meshgrid))









    



WARNING:tensorflow:From /home/schmalen/anaconda3/envs/lecture_MLOC/lib/python3.6/site-packages/tensorflow/python/framework/op_def_library.py:263: colocate_with (from tensorflow.python.framework.ops) is deprecated and will be removed in a future version.
Instructions for updating:
Colocations handled automatically by placer.

Now, carry out the training as such. First initialize the variables and then loop through the training. Here, the epochs are not defined in the classical way, as we do not have a training set per se. We generate new data on the fly and never reuse data.



In [6]:

    
num_epochs = 30
batches_per_epoch = 100

# increase batch size while learning from 100 up to 10000
batch_size_per_epoch = np.linspace(100,10000,num=num_epochs)

validation_SERs = np.zeros(num_epochs)
validation_received = []
decision_region_evolution = []
constellations = []

with tf.Session(graph=graph) as session:
    # initialize variables
    tf.global_variables_initializer().run()

    print('Initialized')
    for epoch in range(num_epochs):
        for step in range(batches_per_epoch):
            feed_dict = {tf_batch_size : batch_size_per_epoch[epoch] }
            
            # run an optimization step
            _,l = session.run([optimizer, loss], feed_dict=feed_dict)
        
        # compute validation BER        
        valid_out = valid_prediction.eval(feed_dict={ tf_batch_size : N_valid })
        validation_SERs[epoch] = SER(valid_out, y_valid)
        print('Validation SER after epoch %d: %f (loss %f)' % (epoch, validation_SERs[epoch], l))                
        
        # store received validation data
        validation_received.append( valid_received.eval(feed_dict={ tf_batch_size : N_valid }) )
        
        # store constellation
        constellations.append(constellation.eval())
        # store decision region for generating the animation
        decision_region_evolution.append(mesh_prediction.eval())









    



Initialized
Validation SER after epoch 0: 0.392480 (loss 5.799330)
Validation SER after epoch 1: 0.157030 (loss 0.768917)
Validation SER after epoch 2: 0.081710 (loss 0.329736)
Validation SER after epoch 3: 0.050150 (loss 0.167261)
Validation SER after epoch 4: 0.029940 (loss 0.113798)
Validation SER after epoch 5: 0.022930 (loss 0.120639)
Validation SER after epoch 6: 0.017980 (loss 0.070293)
Validation SER after epoch 7: 0.014310 (loss 0.045015)
Validation SER after epoch 8: 0.011520 (loss 0.061462)
Validation SER after epoch 9: 0.010080 (loss 0.040354)
Validation SER after epoch 10: 0.009580 (loss 0.040808)
Validation SER after epoch 11: 0.008240 (loss 0.056751)
Validation SER after epoch 12: 0.007670 (loss 0.028068)
Validation SER after epoch 13: 0.007130 (loss 0.031828)
Validation SER after epoch 14: 0.006220 (loss 0.024985)
Validation SER after epoch 15: 0.006060 (loss 0.022244)
Validation SER after epoch 16: 0.005490 (loss 0.023023)
Validation SER after epoch 17: 0.004750 (loss 0.021163)
Validation SER after epoch 18: 0.004660 (loss 0.017400)
Validation SER after epoch 19: 0.003970 (loss 0.017365)
Validation SER after epoch 20: 0.004010 (loss 0.016985)
Validation SER after epoch 21: 0.003760 (loss 0.013356)
Validation SER after epoch 22: 0.003560 (loss 0.010697)
Validation SER after epoch 23: 0.003340 (loss 0.013571)
Validation SER after epoch 24: 0.003320 (loss 0.014782)
Validation SER after epoch 25: 0.002860 (loss 0.006900)
Validation SER after epoch 26: 0.002940 (loss 0.008635)
Validation SER after epoch 27: 0.002950 (loss 0.009830)
Validation SER after epoch 28: 0.002560 (loss 0.013795)
Validation SER after epoch 29: 0.002800 (loss 0.007740)

Plt decision region and scatter plot of the validation set. Note that the validation set is only used for computing BERs and plotting, there is no feedback towards the training!



In [7]:

    
cmap = matplotlib.cm.tab20
base = plt.cm.get_cmap(cmap)
color_list = base.colors
new_color_list = [[t/2 + 0.5 for t in color_list[k]] for k in range(len(color_list))]

# find minimum SER from validation set
min_SER_iter = np.argmin(validation_SERs)
ext_max_plot = 1.05*max(max(abs(validation_received[min_SER_iter][:,0])), max(abs(validation_received[min_SER_iter][:,1])))
print('Minimum SER obtained: %1.5f' % validation_SERs[min_SER_iter])
print(constellations[min_SER_iter])









    



Minimum SER obtained: 0.00256
[[ 0.13155846  0.43206117]
 [ 0.3316632  -0.9176315 ]
 [-1.4141717  -0.29890186]
 [-0.86112845  1.0356466 ]
 [-1.0116309  -0.88188   ]
 [ 0.54143023  0.9130588 ]
 [-0.1566072   0.979529  ]
 [-0.61200047  0.4758839 ]
 [-0.8135951  -0.27923715]
 [-0.24470621 -0.04198585]
 [ 0.88629234 -0.55038446]
 [ 0.8334197   0.2929301 ]
 [-0.25486514 -0.61509377]
 [-0.3918109  -1.2078713 ]
 [-1.1936462   0.3103261 ]
 [ 0.38037696 -0.18909855]]



In [8]:

    
%matplotlib inline
plt.figure(figsize=(19,6))
font = {'size'   : 14}
plt.rc('font', **font)
plt.rc('text', usetex=True)
    
plt.subplot(131)
plt.scatter(constellations[min_SER_iter][:,0], constellations[min_SER_iter][:,1], c=range(M), cmap='tab20',s=50)
plt.axis('scaled')
plt.xlabel(r'$\Re\{r\}$',fontsize=14)
plt.ylabel(r'$\Im\{r\}$',fontsize=14)
plt.xlim((-ext_max_plot,ext_max_plot))
plt.ylim((-ext_max_plot,ext_max_plot))
plt.grid(which='both')
plt.title('Constellation',fontsize=16)

plt.subplot(132)
#plt.contourf(mgx,mgy,decision_region_evolution[-1].reshape(mgy.shape).T,cmap='coolwarm',vmin=0.3,vmax=0.7)
plt.scatter(validation_received[min_SER_iter][:,0], validation_received[min_SER_iter][:,1], c=y_valid, cmap='tab20',s=4)
plt.axis('scaled')
plt.xlabel(r'$\Re\{r\}$',fontsize=14)
plt.ylabel(r'$\Im\{r\}$',fontsize=14)
plt.xlim((-ext_max_plot,ext_max_plot))
plt.ylim((-ext_max_plot,ext_max_plot))
plt.title('Received',fontsize=16)

plt.subplot(133)
decision_scatter = np.argmax(decision_region_evolution[min_SER_iter], 1)
plt.scatter(meshgrid[:,0], meshgrid[:,1], c=decision_scatter, cmap=matplotlib.colors.ListedColormap(colors=new_color_list),s=4)
plt.scatter(validation_received[min_SER_iter][0:4000,0], validation_received[min_SER_iter][0:4000,1], c=y_valid[0:4000], cmap='tab20',s=4)
plt.axis('scaled')
plt.xlim((-ext_max_plot,ext_max_plot))
plt.ylim((-ext_max_plot,ext_max_plot))
plt.xlabel(r'$\Re\{r\}$',fontsize=14)
plt.ylabel(r'$\Im\{r\}$',fontsize=14)
plt.title('Decision regions',fontsize=16)

#plt.savefig('decision_region_AWGN_AE_EbN0%1.1f_M%d.pdf' % (EbN0,M), bbox_inches='tight')

Generate animation and save as a gif.



In [33]:

    
%matplotlib notebook
%matplotlib notebook
# Generate animation
from matplotlib import animation, rc
from matplotlib.animation import PillowWriter # Disable if you don't want to save any GIFs.

font = {'size'   : 18}
plt.rc('font', **font)

fig = plt.figure(figsize=(14,6))
ax1 = fig.add_subplot(1,2,1)
ax2 = fig.add_subplot(1,2,2)

ax1.axis('scaled')
ax2.axis('scaled')

written = False
def animate(i):
    ax1.clear()
    ax1.scatter(constellations[i][:,0], constellations[i][:,1], c=range(M), cmap='tab20',s=50)

    ax2.clear()
    #ax2.scatter([0,0.02],[0.02,0], c=[1,2], cmap='tab20',s=100)
    #decision_scatter = np.argmax(decision_region_evolution[i], 1)
    decision_scatter = np.argmax(decision_region_evolution[i], 1)
    ax2.scatter(meshgrid[:,0], meshgrid[:,1], c=decision_scatter, cmap=matplotlib.colors.ListedColormap(colors=new_color_list),s=4)
    ax2.scatter(validation_received[i][0:4000,0], validation_received[i][0:4000,1], c=y_valid[0:4000], cmap='tab20',s=4)
    
    #plt.scatter(meshgrid[:,0] * ext_max,meshgrid[:,1] * ext_max, c=decision_scatter, cmap=matplotlib.colors.ListedColormap(colors=new_color_list),s=4, marker='s')
    #plt.scatter(X_valid[0:4000,0]*ext_max, X_valid[0:4000,1]*ext_max, c=y_valid[0:4000], cmap='tab20',s=4)
    ax1.set_xlim(( -ext_max_plot, ext_max_plot))
    ax1.set_ylim(( -ext_max_plot, ext_max_plot))
    ax2.set_xlim(( -ext_max_plot, ext_max_plot))
    ax2.set_ylim(( -ext_max_plot, ext_max_plot))
    ax1.set_title('Constellation', fontsize=14)
    ax2.set_title('Decision regions', fontsize=14)
    
    ax1.set_xlabel(r'$\Re\{r\}$',fontsize=14)
    ax1.set_ylabel(r'$\Im\{r\}$',fontsize=14)
    ax2.set_xlabel(r'$\Re\{r\}$',fontsize=14)
    ax2.set_ylabel(r'$\Im\{r\}$',fontsize=14)

    
anim = animation.FuncAnimation(fig, animate, frames=min_SER_iter+1, interval=200, blit=False)
fig.show()
#anim.save('learning_AWGN_AE_EbN0%1.1f_M%d.gif' % (EbN0,M), writer=PillowWriter(fps=5))



In [ ]: