4: A lowpass filter can be constructed with an Adaline having two weights Consider a simple case of the removal of a random noise from a

constant signal. The constant signal level is C = 3, and the random noise signal has a constant power, — — 0.025. Assume that the random noise is completely with the constant input signal. Calculate the optimum weight vector and the mean squared error in the output after the optimum weight vector has been found. By finding the eigenvalues of the matrix, R, determine the maximum value of the constant for use in the LMS algorithm.

2.3 APPLICATIONS OF ADAPTIVE

SIGNAL PROCESSING

Up to now, we have been concerned with the Adaline minus the threshold condition on the output. In Section 2.4, on the Madaline, we will replace the threshold condition and examine networks of Adalines. In this section, we will

L

2.3 Applications of Adaptive Signal Processing

look at a few examples of adaptive signal processing using only the ALC portion of the Adaline.

2.3.1 Echo Cancellation in Telephone Circuits

You may have experienced the phenomenon of echo in telephone conversations: you hear the words you speak into the mouthpiece a fraction of a second later in the earphone of the telephone. The echo tends to be most noticeable on long- distance calls, especially those over satellite links where transmission delays can be a significant fraction of a second.

Telephone circuits contain devices called hybrids that are intended to isolate incoming signals from outgoing signals, thus avoiding the echo effect. Unfortunately, these circuits do not always perform perfectly, due to causes such as impedance mismatches, resulting in some echo back to the speaker. Even when the echo signal has been attenuated by a substantial amount, it still may be audible, and hence an annoyance to the speaker.

Certain echo-suppression devices rely on relays that open and close circuits in the outgoing lines so that incoming voice signals are not sent back to the speaker. When transmission delays are long, as with satellite communications, these echo suppressors can result in a loss of parts of words. This choppy- speech effect is perhaps more familiar than the echo effect. An adaptive filter can be used to remove the echo effect without the choppiness of the relays used in other echo suppression circuits [9, 7J.

Figure is a block diagram of a telephone circuit with an adaptive filter used as an echo-suppression device. The echo is caused by a leakage of the incoming voice signal to the output line through the hybrid circuit. This leakage adds to the output signal coming from the microphone. The output of the adaptive filter, is subtracted from the outgoing signal, s + where s is the outgoing pure voice signal and is the noise, or echo caused by leakage of the incoming voice signal through the hybrid circuit. The success of the echo cancellation depends on how well the adaptive filter can mimic the leakage through the hybrid circuit.

Notice that the input to the filter is a copy of the incoming signal, n, and that the error is a copy of the outgoing signal,

E s + - y (2.16)

We assume that y is correlated with the noise, but not with the pure voice signal, s. If the quantity, y, is nonzero, some echo still remains in the out- going signal. Squaring and taking expectation values of both sides of Eq. gives

= + + - (2.17)

= + - (2.18)

Equation (2.18) follows, since s is not correlated with either y or resulting in the last term in Eq. being equal to zero.

Voice s noise, Hybrid circuit Adaptive filter To earphone Figure 2.15 1 Adaptive filter r Hybrid circuit e To earphone Voice signal

This figure is a schematic of a telephone circuit using an

adaptive filter to cancel echo. The adaptive filter is depicted as a box; the slanted arrow represents the adjustable weights.

The signal power, {.s2}, is determined by the source of the voice

say, some amplifier at the telephone switching station local to the sender. Thus, is not directly affected by changes in The adaptive filter attempts to minimize and, in doing so, minimizes ((n' - the power of the uncanceled noise on the outgoing line.

Since there is only one input to the adaptive filter, the device would be configured as a transverse filter. Widrow and Stearns [9] suggest sampling the incoming signal at a rate of 8 KHz and using 128 weight values.

2.3.2 Other Applications

Rather than go into the details of the many applications that can be addressed by these adaptive filters, we refer you once again to the excellent text by Widrow and Stearns. In this section, we shall simply suggest a few broad areas where adaptive filters can be used in addition to the echo-cancellation application we have discussed.

Figure shows an adaptive filter that is used predict the future value of a signal based on its present value. A second example is shown in Figure In this example, the adaptive filter learns to reproduce the output from some plant based on inputs to the system. This configuration has many uses as an adaptive control system. The plant could represent many things, including a human operator. In that case, the adaptive filter could learn how to respond to changing conditions by watching the human operator. Eventually, such a device might result in an automated control system, leaving the human free for more important

Another useful application of these devices is in adaptive beam-forming antenna arrays. Although the term antenna is usually associated with electro-

2.3 Applications of Adaptive Signal Processing 71

Current signal

Prediction of

current signal

Past signal

Figure 2.16 This schematic shows an adaptive filter used to predict signal

values. The input signal used to train the network is a delayed

value of the actual signal; that is, it is the signal at some past time. The expected output is the current value of the signal. The adaptive filter attempts to minimize the error between its output and the current signal, based on an input of the signal value from some time in the past. Once the filter is correctly predicting the current signal based on the past signal, the current signal can be used directly as an input without the delay. The filter will then make a prediction of the future signal value.

Input signals

Prediction of plant output

Figure This example shows an adaptive filter used to model the output from a system, called the plant. Inputs to the filter are the same as those to the plant. The filter adjusts its weights based on the difference between its output and the output of the plant.

magnetic radiation, we broaden the definition here to include any spatial array of sensors. The basic task here is to learn to steer the array. At any given time, a signal may be arriving from any given direction, but antennae usually are directional in their reception characteristics: They respond to signals in some directions, but not in others. The antenna array with adaptive filters learns to adjust its directional characteristics in order to respond to the incoming signal no matter what the direction is, while reducing its response to unwanted noise signals coming in from other directions.

Of course, we have only touched on the number of applications for these devices. Unlike many other neural-network architectures, this is a relatively mature device with a long history of success. In the next section, we replace the binary output condition on the ALC circuit so that the latter becomes, once again, the complete Adaline.

2.4 THE MADALINE

As you can see from the discussion in Chapter 1, the Adaline resembles the perceptron closely; it also has some of the same limitations as the perceptron. For example, a two-input Adaline cannot compute the XOR function. Com- bining Adalines in a layered structure can overcome this difficulty, as we did in Chapter 1 with the perceptron. Such a structure is illustrated in Figure 2.18.