This example illustrates how a backpropagation neural network can be used in digital signal processing problems. A simple backpropagation filter will be created to remove undesired noise from an input signal. For this example we’ll contaminate a signal with high frequency noise. To create the model, we will need both a contaminated (for input) and uncontaminated (for our target) signal.

Traditional linear signal processing would employ a low-pass filter of the following form:

S(k+n/2) = F[ x(k), x(k+1), ...., x(k+n) ]

where S is the resulting filtered signal, k is the kth point in the time series and n is the number of samples being processing to compute the filtered point. F is the filter. We’ll model the backpropagation low-pass filter with the same inputs that are used in linear signal processing theory. We will use 9 input nodes (i.e., n = 9) to filter the contaminated signal (i.e. t-4, t-3, t-2, t-1, t, t+1, t+2, t+3, t+4) and one output node to represent the filtered of clean signal (at t). The data has been created in the file DigitalFilter.dat. Figure 1 shows the contaminated signal below.

The network files DigitalFilter.net (untrained) and DigitalFilterTrained.net are available for examination. Running the converged network displays the excellent job the model does in filtering the high frequency noise from the desired lower frequency signal (see bottom figure). The training was concluded prior to finding a global minimum in the test set error and it is likely that this model will not overtrain. One minor problem seen with this model is that the test set exposed the model to signal values outside the range of the training data. For this reason, the model's output signal is slightly clipped at the higher values in the test set. This simply points out that input ranges should be adequately covered in the training data. Backpropagation/Neural Network filters have proven to be an excellent alternative to traditional digital filters in the fast growing field of digital signal processing..

Figure 1. Contaminated Signal (training set in green, test set in red)

Figure 2. Target (red) and Trained Signals (green/blue)