Digital Signal Processing: IIR Filters Design PDF

Digital Signal Processing: IIR Filters Design and Implementation **IIR (Infinite Impulse Response) Filters: Overview** IIR filters are digital filters with an infinite impulse response, meaning their response to an impulse input theoretically continues indefinitely. **Advantages of IIR Filters:**...

Digital Signal Processing: IIR Filters Design and Implementation **IIR (Infinite Impulse Response) Filters: Overview** IIR filters are digital filters with an infinite impulse response, meaning their response to an impulse input theoretically continues indefinitely. **Advantages of IIR Filters:** - Lower order filters compared to FIR filters - Less computational requirements - Can directly mimic analog filter characteristics - Efficient implementation for low-frequency applications **Disadvantages of IIR Filters:** - Non-linear phase response - Potential stability issues - Round-off errors can accumulate - More complex design process **Design Methods from Analog Prototypes** 1. **Butterworth Filters** - Maximally flat magnitude response - Transfer function:\ [\$\\mid H(j\\omega) \\mid\^{2} = \\frac{1}{1 + (\\omega/\\omega\_{c})\^{2N}}\$]{.math.inline}\ where [*N*]{.math.inline} is the filter order and [*ω*~*c*~]{.math.inline} is the cutoff frequency 1. **Chebyshev Filters** - Ripples in passband (Type I) or stopband (Type II) - Better rolloff than Butterworth - Transfer function (Type I):\ [\$\\mid H(j\\omega) \\mid\^{2} = \\frac{1}{1 + \\epsilon\^{2}T\_{N}\^{2}(\\omega/\\omega\_{c})}\$]{.math.inline}\ where [*T*~*N*~]{.math.inline} is the Chebyshev polynomial **Digital IIR Filter Design Methods** 1. **Impulse Invariance Method** - Maps analog impulse response to digital domain - Relationship: [*h*\[*n*\] = *Th*~*a*~(*nT*)]{.math.inline} - Suffers from aliasing at high frequencies 1. **Bilinear Transformation** - Maps s-plane to z-plane using:\ [\$s = \\frac{2}{T}\\frac{1 - z\^{- 1}}{1 + z\^{- 1}}\$]{.math.inline} - Warping effect must be pre-compensated - No aliasing issues **IIR Filter Realization Structures** 1. **Direct Form I** - Direct implementation of difference equation - Structure:\ [\$y\\lbrack n\\rbrack = \\sum\_{k = 0}\^{M}{}b\_{k}x\\lbrack n - k\\rbrack - \\sum\_{k = 1}\^{N}{}a\_{k}y\\lbrack n - k\\rbrack\$]{.math.inline} 1. **Direct Form II** - More efficient than Direct Form I - Combined delay elements - Potential for overflow in intermediate calculations 1. **Cascade Form** - System function decomposed into second-order sections - Better numerical properties - [*H*(*z*) = *H*~*k*~(*z*)]{.math.inline} 1. **Parallel Form** - Parallel combination of first and second-order sections - Good for fixed-point implementation - [\$H(z) = \\sum\_{k = 1}\^{K}{}H\_{k}(z)\$]{.math.inline} 1. **Ladder Form** - Based on analog ladder filters - Excellent numerical properties - Low sensitivity to coefficient quantization **Implementation Considerations:** - Choose structure based on: - Computational efficiency - Numerical accuracy requirements - Hardware constraints - Stability considerations Note: When implementing these filters, careful attention must be paid to: - Coefficient quantization effects - Limit cycles - Overflow prevention - Scaling considerations