6.2 Lab 5: prelab  (Page 3/4)

Many extremely efficient structures for IIR filter implementations exist. Two of special note are thefollowing:

• All-pass filter implementations with $N$ multiplies for an order $N$ filter (instead of $\mathrm{2N}$ multiplies for a cascade realization). These are structurally all-pass, meaningthat they remain all-pass even when their coefficients are quantized. (S. Mitra, Digital Signal Processing,a Computer Based Approach, 2nd Ed., pp. 378-382).
• Parallel all-pass Realization of IIR transfer functions with $N$ multiplies for an $N$ 'th order filter. (S. Mitra, Digital Signal Processing, a ComputerBased Approach, 2nd Ed., pp. 401-405, and Sec. 9.9, pp. 629-633).

Questions

• Generate an elliptic LPF using the command: [b,a]=ellip(4,.5,10,.1); Using MATLAB commands, generate a cascaded second-order system implementation and a lattice implementation (don'tworry about normalization if you don't want) of this system and compare their advantages and disadvantages -especially as they relate to implementation on the C5400.
• How close to the unit circle are the poles of the system from question 1? Does this concern you? Explore how muchthe poles moved in the 2 implementations of part 1.
• Use the grpdelay command to view the group delay for the filter in the passband. Is itapproximately linear?
• Why does minimizing the $L$-norm equate to minimizing the $L_{\mathrm{\infty }}$ -norm maximum error over a given frequency range?

Simulation

Simulate the IIR system in MATLAB. Compute the response of the system with appropriate test inputs. Make sure toinclude side-effects due to finite precision in your simulation.

Multi-rate/multi-stage

Reading exercise

Read through the following resources:

• "Optimum FIR Digital Filter Implementations for Decimation, Interpolation, and Narrow-Band Filtering," byCrochiere and Rabiner. This is colloquial paper on the topic of multi-stage filter implementation. The paper isavailable here .
• Course notes on multi-stage filter implementation by Prof. Mark Fowler from Binghamton University. The notes are available here .

Design exercises

Given the filter specification given in the filter specification , answer the following questions:

• What is the maximum decimation factor that can be used?
• What is the average number of MACs per input sample that are required for a single stageimplementation?
• What are the appropriate decimation and interpolation factors for a a two stageimplementation?
• What are the appropriate pass-band and stop-band frequencies and maximum ripple for the overall filter ateach stage? Your answer will demonstrate that the use of multiple filter stages along with multi-rate signalprocessing can achieve a overall filter of lower order than just a single stage filter.
• Estimate the filter order for each stage. We recommend using the MATLAB command remezord . This algorithm frequently underestimates the filter order needed, but gives you agood starting point. Verify that the filter specifications are met, i.e. pass-band and stop-bandripple and pass-band and stop-band band edge locations. Do this by passing the arguments returned by remezord to the MATLAB command remez . Observe the frequency response of the system described by the filtercoefficients returned by \texttt{remez} using the MATLAB command freqz . If the specifications are not met, increase the order of the filter until thespecifications are met.
• Determine the average number of MACs per sample for the two-stage implementation. Which is more efficient,the single stage approach or the multistage approach?