Figure 1 is a block diagram of system identification using adaptive filtering. The objective is to change (adapt) the coefficients of an FIR filter, W W, to match as closely as possible the response of an unknown system, H H. The unknown system and the adapting filter process the same input signal x⁢n x n and have outputs d⁢n d n (also referred to as the desired signal) and y⁢n y n .

Figure 1: System identification block diagram.

Simulate the system identification block diagram shown in Figure 1.

Previously in MATLAB, you used the filter command or the conv command to implement shift-invariant filters. Those commands will not work here because adaptive filters are shift-varying, since the coefficient update equation changes the filter's impulse response at every sample time. Therefore, implement the system identification block on a sample-by-sample basis with a do loop, similar to the way you might implement a time-domain FIR filter on a DSP. For the "unknown" system, use the fourth-order, low-pass, elliptical, IIR filter designed for the IIR Filtering: Filter-Design Exercise in MATLAB.

Use Gaussian random noise as your input, which can be generated in MATLAB using the command randn. Random white noise provides signal at all digital frequencies to train the adaptive filter. Simulate the system with an adaptive filter of length 32 and a step-size of 0.02 0.02. Initialize all of the adaptive filter coefficients to zero. From your simulation, plot the error (or squared-error) as it evolves over time and plot the frequency response of the adaptive filter coefficients at the end of the simulation. How well does your adaptive filter match the "unknown" filter? How long does it take to converge?

Once your simulation is working, experiment with different step-sizes and adaptive filter lengths.

Use the same "unknown" filter as you used in the MATLAB simulation.

Although the coefficient update equation is relatively straightforward, consider using the lms instruction available on the TI processor, which is designed for this application and yields a very efficient implementation of the coefficient update equation.

To generate noise on the DSP, you can use the PN generator from the Digital Transmitter: Introduction to Quadrature Phase-Shift Keying, but shift the PN register contents up to make the sign bit random. (If the sign bit is always zero, then the noise will not be zero-mean and this will affect convergence.) Send the desired signal, d⁢n d n , the output of the adaptive filter, y⁢n y n , and the error to the D/A for display on the oscilloscope.

When using the step-size suggested in the MATLAB simulation section, you should notice that the error converges very quickly. Try an extremely small μ μ so that you can actually watch the amplitude of the error signal decrease towards zero.

If your project requires some modifications to the implementation here, refer to Haykin and consider some of the following questions regarding such modifications: