IIR Filters

digital signal processing

1. IIR FILTERS Digital Signal processing Marmik Kothari LY EC- 140410111027 SVIT VASAD

2. Basics of IIRs Two convolutionsare involved: one with the previous inputs, and one with the previous outputs. In each case the convolving function is called the filter coefficients. If such a filter is subjected to an impulse (a signal consisting of one value followed by zeroes) then its output need not necessarily become zero after the impulse has run through the summation. So the impulse response  of such a filter can be infinite in duration. Such a filter is called an Infinite Impulse Response filter or IIR filter

3.   The impulse response need not necessarily be infinite: if it were, the filter would be unstable. In fact for most practical filters, the impulse response will die away to a negligibly small level . One might argue that mathematically the response can go on for ever, getting smaller and smaller: but in a digital world once a level gets below one bit it might as well be zero. The  Infinite Impulse Response refers to the ability of the filter to have an infinite impulse response and does not imply that it necessarily will have one: it serves as a warning that this type of filter is prone to feedback and instability.

4. DIERCT FORM I The block diagram is in two halves : and since the results from each half are simply added together it does not matter in which order they are calculated . So the order of the halves can be swapped:

5. This is called direct form 2 . Its advantage is that it needs less  delay elements . And since delay elements require hardware (for example, processor registers) the direct form 2 requires less hardware and so is more efficient than direct form I. direct form 2 is also called  canonic , which simply means 'having the minimum number of delay elements '.

6. CASCADE AND PARALLEL The outputs from each second order section are simply added together. If  scaling  is required, this is done separately for each section. It is possible to scale each section appropriately, and by a different scale factor, to minimise  quantisation error . In this case another extra multiplier is required for each section, to scale the individual section outputs back to the same common scale factor before adding them. The order in which parallel sections are calculated does not matter, since the outputs are simply added together at the end. In the cascade form, the output of one section forms the input to the next:

7. In practice, the propagation of errors is crucial to the success of an IIR filter so the order of the sections in the cascade, and the selection of which filter coefficients  to group in each section, is vital: sections with high gain are undesirable because they increase the need for  scaling  and so increase quantisation errors it is desirable to arrange sections to avoid excessive scaling To reduce the gain of each section we note that: Poles  cause high gain (bumps in the frequency response ) Zeroes cause low gain (dips in the frequency response) the closer to the unit circle , the greater the effect This suggests a way to group poles and zeroes in each section to avoid high gain sections:

8. IIR filters are very sensitive to  quantisation  errors. The higher the order of the filter, the more it suffers from quantisation effects: because the filter is more complex, and so the errors accumulate more. In fact, IIR filters are so sensitive to quantisation errors that it is generally unrealistic to expect anything higher than a second order filter to work. This is why IIR filters are usually realised as second order sections. Most analogue filters (except for Bessel filters) are also usually realised as second order sections, which is a convenient excuse but not the real one.