How do adaptive filters work

Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more. What Does an Adaptive Filter Do? Ask Question. Asked 9 years, 8 months ago. Active 1 year, 8 months ago.

Viewed 8k times. Can anyone explain me how it is being used in real day applications. Improve this question. Royi Prashant Singh Prashant Singh 2 2 gold badges 5 5 silver badges 11 11 bronze badges.

Add a comment. Active Oldest Votes. Improve this answer. I have heard that adaptive filters are implemented through LMS algorithm. Just getting a pointer to what the LMS algorithm looks like won't tell you a whole lot. If you write some software to do it and it doesn't work, you'll be hard-pressed to figure out the problem. With that said, Wikipedia has a decent page on the LMS filter.

So does it track the frequencies and phases of the mains signal and subtract them, leaving desired signals at those frequencies? Or does it null out anything at those frequencies with notch filters? Can you think of a better example? Andrew Polar Andrew Polar 4 4 bronze badges. Sign up or log in Sign up using Google.

Where p l is the impulse respond of the unknown plant, By choosing each w 1 n close to each p l , the error will be minimized.

For using white noise as the excitation signal, minimizing e n will force the w 1 n to approach p l , that is,. When the plan is time varying, the adaptive algorithm has the task of keeping the modelling error small by continually tracking time variations of the plant dynamics. Usually, the input signal is awideband signal, in order to allow the adaptive filter to converge to a good model of the unknownsystem.

In thecases where the impulse response of the unknown system is of finite length and the adaptive filteris of sufficient order, the MSE becomes zero if thereis no measurement noise or channel noise. In practical applications the measurement noise isunavoidable, and if it is uncorrelated with the input signal, the expected value of the adaptive-filtercoefficients will coincide with the unknown-system impulse response samples.

The output errorwill of course be the measurement noise Diniz, Some real world applications of the system identification scheme include control systems and seismic exploration. The linear prediction estimates the values of a signal at a future time. This model is wide usually in speech processing applications such as speech coding in cellular telephony, speech enhancement, and speech recognition. In this configuration the desired signal is a forward version of the adaptive filter input signal.

When the adaptive algorithm convergences the filter represents a model for the input signal, this model can be used as a prediction model. The linear prediction system is shown in figure 2. Adaptive filter for linear prediction. These parameters are part of the coding informationthat is transmitted or stored along with other information inherent to the speech characteristics, suchas pitch period, among others.

The adaptive signal predictor is also used for adaptive line enhancement ALE , where the input signalis a narrowband signal predictable added to a wideband signal. After convergence, the predictoroutput will be an enhanced version of the narrowband signal. Yet another application of the signal predictor is the suppression of narrowband interference in awideband signal.

The input signal, in this case, has the same general characteristics of the ALE. The inverse modeling is an application that can be used in the area of channel equalization, for example it is applied in modems to reduce channel distortion resulting from the high speed of data transmission over telephone channels. High-speed data transmission through channels with severe distortion can be achieved in several ways, one way is to design the transmit and receive filters so that the combination of filters and channel results in an acceptable error from the combination of intersymbol interference and noise; and the other way is designing an equalizer in the receiver that counteracts the channel distortion.

The second method is the most commonly used technology for data transmission applications. Figure 3 shows an adaptive channel equalizer, the received signal y n is different from the original signal x n because it was distorted by the overall channel transfer function C z , which includes the transmit filter, the transmission medium, and the receive filter.

Adaptive Channel equalizer. Therefore the equalizer must be designed by. In practice, the telephone channel is time varying and is unknown in the design stage due to variations in the transmission medium. Thus it is needed an adaptive equalizer that provides precise compensation over the time-varying channel.

The adaptive filter requires the desired signal d n for computing the error signal e n for the LMS algorithm. An adaptive filter requires the desired signal d n for computing the error signal e n for the LMS algorithm. Since the adaptive filter is located in the receiver, the desired signal generated by the transmitter is not available at the receiver.

The desired signal may be generated locally in the receiver using two methods. During the training stage, the adaptive equalizer coefficients are adjusted by transmitting a short training sequence. This known transmitted sequence is also generated in the receiver and is used as the desired signal d n for the LMS algorithm.

After the short training period, the transmitter begins to transmit the data sequence. In the data mode, the output of the equalizer x n is used by a decision device to produce binary data. Assuming that the output of the decision device is correct, the binary sequence can be used as the desired signal d n to generate the error signal for the LMS algorithm.

Adaptive filtering can be a powerful tool for the rejection of narrowband interference in a direct sequence spread spectrum receiver. Figure 4 illustrates a jammer suppression system.

In this case the output of the filter y n , is an estimate of the jammer, this signal is subtracted from the received signal x n , to yield an estimate of the spread spectrum.

To enhance the performance of the system a two-stage jammer suppressor is used. The adaptive line enhancer, which is essentially another adaptive filter, counteracts the effects of finite correlation which leads to partial cancellation of the desired signal.

The number of coefficients required for either filter is moderate, but the sampling frequency may be well over KHz. Jammer suppression in direct sequence spread spectrum receiver. In certain situations, the primary input is a broadband signal corrupted by undesired narrowband sinusoidal interference. The conventional method of eliminating such sinusoidal interference is using anotch filter that is tuned to the frequency of the interference Kuo et al. To design the filter, we need the precisefrequency of the interference.

The adaptive notch filter has the capability to track the frequency of theinterference, and thus is especially useful when the interfering sinusoid drifts in frequency.

A single-frequency adaptive notch filter with two adaptive weights is illustrated in figure 5 ,where the input signal is a cosine signal as. For a sinusoidal signal, two filter coefficients are needed.

The reference input is used to estimate the composite sinusoidal interfering signal contained in theprimary input d n. The center frequency of the notch filter is equal to the frequency of the primary sinusoidal noise. Therefore, the noise at that frequency is attenuated.

This adaptive notch filter providesa simple method for eliminating sinusoidal interference. Adaptive Notch Filter.

The noise cancellers are used to eliminate intense background noise. This configuration is applied in mobile phones and radio communications, because in some situations these devices are used in high-noise environments. Figure 6 shows an adaptive noise cancellation system. Adaptive noise canceller system. The ambient noise is processed by the adaptive filter to make it equal to the noise contaminating the speech signal, and then is subtracted to cancel out the noise in the desired signal.

In order to be effectively the ambient noise must be highly correlated with the noise components in the speech signal, if there is no access to the instantaneous value of the contaminating signal, the noise cannot be cancelled out, but it can be reduced using the statistics of the signal and the noise process. Figure 7 shows a voice signal with noise; those signals were used in noise canceller system implemented on a digital signal processor.

The desired signal is a monaural audio signal with sampling frequency of 8 KHz. The noise signal is an undesired monaural musical piece with a sampling frequency of 11 KHz.

As it can be seen in the image the desired signal is highly contaminated, so in this structure it must be used a fast adaptation algorithm in order to reach the convergence and eliminate all the unwanted components from the desired signal. Signals used in the noise canceller system.

The frequency analysis of the signals used in the noise canceller system can be seen on the spectrograms of the figure 8. The figure shows that the output signal has some additionalfrequency components with respect to the input signal.

Spectrograms of the signals used in the noise canceller system. The output of the noise canceller is the error signal, the figure 9 shows the error signal obtained when it is used an LMS algorithm. With the spectrogram of the signal it is shown that all the undesired frequency components were eliminated.

The adaptive noise canceller system is used in many applications of active noise control ANC , in aircrafts is used to cancel low-frequency noise inside vehicle cabins for passenger comfort. Most major aircraft manufacturers are developing such systems, mainly for noisy propeller-driven airplanes.

Another application is active mufflers for engine exhaust pipes, which have been in use for a while on commercial compressors, generators, and such. With the price for ANC solutions dropping, even automotive manufacturers are now considering active mufflers as a replacement of the traditional baffled muffler for future production cars. The resultant reduction in engine back pressure is expected to result in a five to six percent decrease in fuel consumption for in-city driving.

Another application that has achieved widespread commercial success are active headphones to cancel low-frequency noise. The active headphones are equipped with microphones on outside of the ear cups that measure the noise arriving at the headphones.

For feedforward ANC, the unit also includes a microphone inside each ear cup to monitor the error - the part of the signal that has not been canceled by the speakers in order to optimize the ANC algorithm. Very popular with pilots, active headphones are considered essential in noisy helicopters and propeller-powered airplanes.

The filter coefficients adjustment with this algorithm is performed until the MSE is minimized. This adaptive algorithm is the most used due its simplicity in gradient vector calculation, which can suitably modify the cost function [ 11 ], [ 17 ]. The NLMS algorithm employs the method of maximum slope, where the convergence factor presents a compromise between convergence speed and accuracy, i.

The adaptive NLMS algorithm takes the following form:. This algorithm eliminates the strong dependence of data input, and the convergence algorithm depends directly of the input signal power to absorb large variations in the signal x[k].

This algorithm is used when the environment is very dynamic and requires speed response. RLS algorithm computes and update recursively coefficients when new samples of the input signal are received, and is intended to exploit the autocorrelation matrix data structure to reduce the number of operations to a computational complexity [ 21 ], [ 22 ].

A simple least square estimate of the weight filter vector w[k] is:. Where the vector of optimal coefficients w[k] is obtained from the autocorrelation matrix calculation R N [k] between the input signal x[k]. The infinite memory of RLS algorithm averages the value of each coefficient to ensure the best approximation of steady-state ratios and significantly improves the final performance of echo cancellation.

In practice this amount is necessary because the weight cannot be updated until the arrival of the next sample. The vector K N [k] is called Kalman gain and can be generated recursively without inverting the matrix R -1 N [k]. In this algorithm, the coefficients is updated for each sample at time k, this is done by taking into account the N previous entries [ 1 ], [ 21 ]. The adaptation process seeks to minimize the variance of that error signal.

It's important to use wideband noise as an input signal in order to identify the characteristics of the unknown system over the entire frequency range from zero to half the sampling frequency. C DSP has behavior specifications such as: floating point calculation, MHz clock frequency 4. DSK C has four audio stereo jacks for: microphone input, line input, speaker output and line output.

The sampling rate of AIC23 Codec can be configured for input and output independently and support a wide range of frequencies from 8 to 96 kSps. Simulink uses a block based approach to algorithm design and implementation. Once the desired functionality has been captured and simulated, can be generated code for the DSP. Here creates and edits the CCS project with the code. The link for CCS is used to invoke the code building process to build an executable.

This code can then be downloaded on the DSP target from where it runs. The codec setting is necessary for the signals acquisition in the DSK C, for this reason it was configured to work at 8 kHz sampling rate to guarantee the Nyquist theorem for cutoff frequency of input signals, both the Gaussian Noise Signal an the FIR filter response unknown system which were designed at sample frequency of 8 kHz [ 31 ] - [ 33 ].

Design specifications for the fixed filter were: order filter 50, windowing method used Kaiser, inferior cut off frequency 1. Once the digital filter coefficients were obtained, its mathematical model was calculated and exported to Simulink file.

The block estimates the weights or coefficients needed to minimize the error between the output signal y[k] and the desired output signal d[k] [ 34 ].

The signal to filter should be connected to the Input Terminal. This input can be a scalar random signal or a data channel. In this case the input sig nal is a White Gaussian Noise. The Desired Signal must have the same type and size of the input signal; the unknown system response Fixed FIR Filter corresponds to the desired signal.

The Output Terminal is where the filtered signal is removed. The Error Terminal provides the result of subtracting the output signal of the desired signal. The design parameters considered the commitment performance versus complexity. Figure 5. If is necessary to keep the power consumption in the smallest possible levels and the application does not requires real-time execution, the best option is to implement an adaptive LMS filter and Normalized LMS NLMS.

Moreover, a better choice for applications that require real-time execution and fast convergence falls on the RLS adaptive filter. The identification system architecture of Fig. In summary, the implementation method of adaptive algorithm in the DSK platform involves the following steps [ 30 ], [ 36 ]:. Figure 7.

The experimental results using the setup identification system given in Section 3 are illustrated by the graphs in Figs. The adaptive identification system implemented was validated by four performance criterions: The identification system implemented was validated by four performance criterions: Temporal Analysis using the learning curve calculation, Mean Square Error estimation and the algorithm errors computation; Frecuencial Analysis using the Fast Fourier Transform and its spectrogram analysis; Computational Complexity through measurement the clock cycles and time execution of the tested algorithms; and finally the precision of filter adaptive weights estimation [ 37 ]-[ 46 ].

A shorter filter length was required for obtaining the desired identification. Each of the five step-sizes was interesting: on one hand, the larger the step-size, the faster the convergence.

But on the other hand, the smaller the step-size, the better the steady state square error. Where y[k] is the predicted output for the adaptive filter and N is the number of samples used in the identification process. The MSE graph of the filtered output signal by the adaptive filter with respect to the filter input indicates how fast reaches the Least Square Error LSE , and therefore defines the filter convergence rate.

The MSE quantifies the difference between the estimated model identified and the real model. For obtaining MSE, both power error signal and power input signal in a number of samples is calculated. In order to get better insight, Fig. The convergence speed evaluation was done by defining the point at which the graph has not significant changes in the MSE along the samples. From Fig. Although RLS algorithm converges faster is important to note that its computational complexity was superior due that the correlation matrix inversion was involved.

In order to compare these algorithms easily, the best parameters in above implementation results are selected. In Fig. Under the same filter length for the adaptive algorithms, at first glance the results of Fig. In addition the least error value not was reached by the LMS algorithm. It's important to state that the minimum error is conditioned by the characteristics of the data transfer channel, in this experience was used a Jack Stereo 3.

According results in Fig. It was observed that with increase in number of training sessions, the MSE value steadily decreases. It means that the adaptive filters trained with the adaptive algorithms were tracking the system properties. The performance of the adaptive filters was appreciated by comparing the error signal, i.

The adaptive algorithm convergence is reached when there is no significant change in the Error along several samples. The best behavior is obtained for the adaptive algorithm who reaches before to this point. The algorithms errors results are indicated in Fig. The reason is that the LMS algorithm only uses the transient data to minimize the square error, while for RLS algorithm a group of data is used.

As RLS uses more available information under certain restraints, its convergence speed is much faster than LMS algorithm. The mean and standard deviation of the error signals were calculated too, in order to characterize the adaptive algorithms performances.

The corresponding values are indicated in Table III.