Lab 6. Advanced Filter Design in Matlab

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1 E E Lab June 30, 2006 Lab 6. Advanced Filter Design in Matlab Introduction This lab will briefly describe the following topics: Median Filtering Advanced IIR Filter Design Advanced FIR Filter Design Adaptive Filters Multirate Filters Median Filtering Filitering using LTI systems is limited to designs that compute outputs from linear combinations of current and previous values of the inputs and the outputs. Sometimes, however, linear techniques are insufficient to perform the required filtering. A simple example of a nonlinear filtering technique is median filtering. The Signal Processing Toolbox function medfilt1 implements one-dimensional median filtering by applying a sliding window to a sequence. The filter replaces the center value in the window with the median value of all the points within the window. y = medfilt1(x,n) applies an order n (window length n) one-dimensional median filter to vector x. The function considers the signal to be 0 beyond the endpoints. The output y has the same length as x. y = medfilt1(x,n,blksz) uses a for loop to compute blksz (block size) output samples at a time. Use blksz << length(x) if you are low on memory, since medfilt uses a working matrix of size n blksize. By default, blksz = length(x) for fastest execution with sufficient memory. Median filters are often used for noise reduction. The advantage of median filters over linear filters such as averaging filters comes from the robustness of the median with respect to outliers. Outlying data has much less effect on the center of a data set when the center is measured by the median rather than the mean. As a result, median filters are useful for removing salt and pepper noise involving extreme outliers while otherwise preserving the shape of the signal. c 2006GM

2 A two-dimensional median filtering function, medfilt2, is available in the Image Processing Toolbox. x = [ ]; y = medfilt1(x,1) y = medfilt1(x,2) y = medfilt1(x,3) edit saltysig saltysig(5) saltysig(10) saltysig(20) Advanced IIR Designs The Filter Design Toolbox includes three functions for the design of IIR filters that optimally meet filter requirements. iirlpnorm uses a least-p th norm unconstrained optimization algorithm for IIR filters that have arbitrary shape magnitude response curves. The algorithm differs from the traditional IIR design algorithms in several respects: The designs are done directly in the z-domain. There is no need for bilinear transformation. The numerator and the denominator order can be different. You can design IIR filters with arbitrary magnitude response in addition to the basic lowpass, highpass, bandpass, and bandstop. iirlpnormc uses a least-p th norm optimization algorithm as well, but this version is constrained to let you restrict the radius of the poles of the IIR filter. iirgrpdelay uses a least-p th constrained optimization algorithm to let you design allpass IIR filters that meet a prescribed group delay specification. The Demos section of the documentation for the Filter Design Toolbox includes examples for each of the functions: Least P-Norm Optimal IIR Filter Design Constrained Least P-Norm IIR Filter Design IIR Filter Design Given a Prescribed Group Delay

3 Try the demos on IIR filter design in the documentation for the Filter Design Toolbox. Advanced FIR Designs The Filter Design Toolbox also includes a function for the design of FIR filters that optimally meet filter requirements. firgr provides a minimax, or equiripple algorithm that allows you to design the following real FIR filters: Types 1 through 4 linear phase Minimum phase Maximum phase Minimum order, even or odd Extra-ripple Maximal-ripple Constrained-ripple Single-point band Forced gain Arbitrarily shaped frequency response The Demos section of the documentation for the Filter Design Toolbox includes the following examples: Minimax FIR Filter design Least Pth-Norm optimal FIR Filter Design FIR Nyquist (Lth Band) Filter Design Constrained Equiripple FIR Filter Design Constrained-Band Equiripple FIR Filter Design Efficient Narrow Transition Band FIR FIlter Design Halfband FIR Filter Design Try the demos on FIR filter design in the documentation for the Filter Design Toolbox. Adaptive Filters Adaptive filters are self-learning systems. As an input signal enters the filter, the coefficients adjust themselves to achieve a desired result. The result might be System identification Identify the response of an unknown system such as a communications channel or a telephone

4 line. The filter adapts once, then stays fixed. Inverse system identification Develop a filter with a responsse that is the inverse of an unknown system. You can overcome echoes in modem connections and local telephone lines by inserting an inverse adaptive filter and using it to compensate for the induced noise on the lines. Noise cancellation Perform active noise cancellation where the filter adapts in real time to keep the error small. Compare this to system identification. Prediction Predict a signal s future values. Adaptive Filters: Block Diagram The block diagram below represents a general adaptive filter. An adaptive FIR filter designs itself based on the characteristics of the input sinal and a signal that represents the desired behavior of the filter on the input. The filter does not require any other frequency response specifications. The filter uses an adaptive algorithm to reduce the error between the output signal y(n) and the desired signal d(n). When perormance criteria for e(n) have achieved their minimum values through the iterations of

5 the adapting algorithm, the adaptive filter is finished and its coefficients have converged to a solution. The output from the adaptive filter should approximate the desired signal d(n). If the characteristics of the input (the filter environment) are changed, the filter adapts by generating a new set of coefficients. Adaptive filter functions in the Filter Design Toolbox implement the largest block in the figure, using an appropriate technique for the adaptive algorithm. Inputs are x(n) and initial values for a and b. Choosing an Adaptive Filter Two main considerations are involved in choosing an adaptive filter: the job to do and the filter algorithm to use. The suitability of an adaptive filter design is determined by Filter Consistency Does filter performance degrade when the coefficients change slightly as a result of quantization or the use of fixed-point arithmetic? Will excessive noise in the signal hurt the performance of the filter? Filter Performance Does the filter provide sufficient identification accuracy or fidelity, or does it provide sufficient signal discrimination or noise cancellation? Tools Do tools exist to make the filter development process easier? Better tools can make it practical to use more complex adaptive algorithms. DSP Requirements Can the filter perform its job within the constraints of the application? Does the processor have sufficient memory, throughput, and time to use the proposed adaptive filtering approach? Can you use more memory to reduce the throughput, or use a faster signal processor? Simulating filters using the functions in the Filter Design Toolbox is a good way to investigate these issues. Beginning with a least mean squares (LMS) filter, which is relatively easy to implement, might provide sufficient information for your evaluation. This can also form a basis from which you can study and compare more complex adaptive filters. At some point, you must test your design with real data and evaluate its performance.

6 Adaptive Filters in the Filter Design Toolbox Adaptive filters are implemented in the Filter Design Toolbox as adaptive filter (adaptfilt) objects. This object-oriented approach to adaptive filtering involves two steps: 1. Use a filter constructor method to create the filter. 2. Use the filter method associated with adaptfilt objects to apply the filter to data. The filter method for adaptfilt objects takes precedence over the filter function in Matlab when input arguments involve adaptfilt objects. There are over two dozen adaptive filter constructor methods in the Filter Design Toolbox. They fall into the following five categories: Least mean squares (LMS) based FIR adaptive filters Recursive least squares (RLS) based FIR adaptive filters Affine projection (AP) FIR adaptive filters FIR adaptive filters in the frequency domain (FD) Lattice based (L) FIR adaptive filters For example, ha = adaptfilt.lms(length,step,leakage,coeffs,states); constructs an FIR LMS adaptive filter object ha. The input arguments for each constuctor method are described in the documentation. To apply the filter to a signal x, use y = filter(ha,x) Note that in the context of adaptive filtering, Kalman filters are equivalent to RLS filters. Other nonadaptive Kalman filters are available in the Control System Toolbox. edit afodemo afodemo System Identification One common application of adaptive filters is to identify an unknown black box system. Applications are as diverse as finding the response of an unknown communications channel and finding the frequency response of an auditorium to design for echo cancellation.

7 In a system identification application, the unknown system is placed in parallel with the adaptive filter. In this case the same input x(n) feeds both the adaptive filter and the unkown system. When e(n) is small, the adaptive filter response is close to the response of the unknown system. edit filterid filterid(1e2,0.5) filterid(5e2,0.5) filterid(1e3,0.5) filterid(5e2,0.1) filterid(5e2,0.5) filterid(5e2,1) Sketch the block diagram of an adaptive filter system to be used for system identification. Inverse System Identification By placing the unknown system in series with an adaptive filter, the filter becomes the inverse of the unknown system when e(n) becomes small. The process requires a delay inserted in the desired signal d(n) path to keep the data at the summation synchronized and the system causal. Without the delay element, the adapting algorithm tries to match the output y(n) from the adaptive filter to an input x(n) that has not yet reached the adaptive elements because it has to pass through the unknown system. Including a delay equal to the delay caused by the unknown system prevents this condition. Plain old telephone systems (POTS) commonly use inverse system identification to compensate for the copper transmission medium. When you send data or voice over telephone lines, the losses due to capacitances distributed along the wires behave like a filter that rolls off at higher frequencies (or data rates). Adding an adaptive filter with a response that is the inverse of the wire response, and adapting in real time, compensates for the rolloff and other anomalies, increasing the available frequency range and data rate for the telephone system. Sketch the block diagram of an adaptive filter system to be used for inverse system identification.

8 Noise Cancellation In noise cancellation, adaptive filters remove noise from a signal in real time. To do this, a signal Z(n) correlated to the noise is fed to the adaptive filter. As long as the input to the filter is correlated to the unwanted noise accompanying the desired signal, the adaptive filter adjusts its coefficients to reduce the value of the difference between y(n) and d(n), removing the noise and resulting in a clean signal in e(n). Notice that the error signal converges to the input data signal in this case, rather than converging to zero. edit babybeat babybeat Sketch the block diagram of an adaptive filter system to be used for system identification. Prediction Predicting signals might seem to be an impossible task. Some limiting assumptions must be imposed by assuming that the signal isperiodic and either steady or slowly varying over time. Accepting these assumptions, an adaptive filter attempts to predict future values of the desired signal based on past values. The signal s(n) is delayed to create x(n), the input to the adaptive filter. s(n) also drives the d(n) input. When s(n) is periodic and the filter is long enough to remember previous values, this structure, with the delay in the input signal, can perform the prediction. You might use this structure to remove a periodic component from stochastic noise signals. Sketch the block diagram of an adaptive filter system to be used for system identification. Multirate Filters Multirate filters alter the sample rate of the input signal during the filtering process. They are useful in both rate conversion and filter bank applications. Like adaptive filters, multirate filters are implemented in the Filter Design Toolbox as multirate filter (mfilt) objects. Again, this object-oriented approach to filtering involves two steps:

9 1. Use a filter constructor method to create the filter. 2. Use the filter method associated with mfilt objects to apply the filter to data. The filter method for mfilt objects (and all filter objects) takes precedence over the filter function in Matlab when input arguments involve mfilt objects. There are approximately a dozen multirate filter constructor methods in the Filter Design Toolbox. For example, hm = mfilt.firdecim(3) creates an FIR decimator with a decimation factor of 3. The only input argument needed is the decimation factor. Input arguments for each constructor method are described in the documentation. To apply the filter to a signal x, use y = filter(hm,x) Multirate filters can also be designed interactively in the FDATool. Click the sidebar button Create a Multirate Filter. edit multidemo multidemo Interactively design the filter in the demo in the FDATool. (The material in this lab handout was put together by Paul Beliveau and derives principally from the MathWorks training document MATLAB for Signal Processing, 2006.) c 2006GM

Lab 8. Signal Analysis Using Matlab Simulink

E E 2 7 5 Lab June 30, 2006 Lab 8. Signal Analysis Using Matlab Simulink Introduction The Matlab Simulink software allows you to model digital signals, examine power spectra of digital signals, represent