This code is provided as supplementary material of the lecture Machine Learning and Optimization in Communications (MLOC).
This code illustrates
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
Specify the parameters of the transmission as the fiber length $L$ (in km), the fiber nonlinearity coefficienty $\gamma$ (given in 1/W/km) and the total noise power $P_n$ (given in dBM. The noise is due to amplified spontaneous emission in amplifiers along the link). We assume a model of a dispersion-less fiber affected by nonlinearity. The model, which is described for instance in [1] is given by an iterative application of the equation $$ x_{k+1} = x_k\exp\left(\jmath\frac{L}{K}\gamma|x_k|^2\right) + n_{k+1},\qquad 0 \leq k < K $$ where $x_0$ is the channel input (the modulated, complex symbols) and $x_K$ is the channel output. $K$ denotes the number of steps taken to simulate the channel Usually $K=50$ gives a good approximation.
[1] S. Li, C. Häger, N. Garcia, and H. Wymeersch, "Achievable Information Rates for Nonlinear Fiber Communication via End-to-end Autoencoder Learning," Proc. ECOC, Rome, Sep. 2018
In [2]:
# Length of transmission (in km)
L = 5000
# fiber nonlinearity coefficient
gamma = 1.27
Pn = -21.3 # noise power (in dBm)
Kstep = 50 # number of steps used in the channel model
# noise variance per step
sigma_n = np.sqrt((10**((Pn-30)/10)) / Kstep / 2)
def simulate_channel(x, Pin, constellation):
# modulate bpsk
input_power_linear = 10**((Pin-30)/10)
norm_factor = 1 / np.sqrt(np.mean(np.abs(constellation)**2)/input_power_linear)
modulated = constellation[x] * norm_factor
temp = np.array(modulated, copy=True)
for i in range(Kstep):
power = np.absolute(temp)**2
rotcoff = (L / Kstep) * gamma * power
temp = temp * np.exp(1j*rotcoff) + sigma_n*(np.random.randn(len(x)) + 1j*np.random.randn(len(x)))
return temp
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
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# set input power
Pin = 4
input_power_linear = 10**((Pin-30)/10)
# number of points in constellation
M = 16
# 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
# assume that the worst case constellation is the one where all points lie on a straight line starting at the center and then are spreaded equidistantly. In this case, this is the scaling factor of the constellation points and we assume that there is an (M+1)th point which defines ext_max
ext_max = np.sqrt(M*input_power_linear)
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.
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# 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):
# simulate the channel
for i in range(Kstep):
power = tf.norm(modulated, axis=1) ** 2
rotcoff = (L / Kstep) * gamma * power
# rotation matrix corresponding to exp(1j*rotcoff)
temp = tf.stack([modulated[:,0] * tf.cos(rotcoff) - modulated[:,1]*tf.sin(rotcoff), modulated[:,0]*tf.sin(rotcoff)+modulated[:,1]*tf.cos(rotcoff)], axis=1)
modulated = temp + tf.random_normal(shape=(tf_batch_size,2), stddev=sigma_n)
return modulated
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)) / input_power_linear * 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)) / input_power_linear * 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))
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.
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num_epochs = 150
batches_per_epoch = 350
# 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())
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)
print('Minimum SER obtained: %1.5f' % validation_SERs[min_SER_iter])
ext_max_plot = 1.05*max(max(abs(validation_received[min_SER_iter][:,0])), max(abs(validation_received[min_SER_iter][:,1])))
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_AE_Pin%d.pdf' % Pin,bbox_inches='tight')
Generate animation and save as a gif.
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%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_decision_AE_Pin%d_varbatch.gif' % Pin, writer=PillowWriter(fps=5))
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