EEE4176 Application of Digital Signal Processing Lecture: FIR Filter Design Assistant Prof. Yangmo Yoo Dept. of Electronic Engineering SOGANG UNIVERSITY 2011.9 EEE4176 Application of Digital Signal Processing

EEE4176 Application of Digital Signal Processing FIR Filters Linear Phase FIR Filters Generally, both the constant phase and constant group delay filters are referred to as LINEAR PHASE FILTER. Impulse response of FIR filter Order of the filter = N Length of the filter = N+1 Cf) 2011.9 EEE4176 Application of Digital Signal Processing

Linear Phase FIR Filters Type-I , : even order / odd length; : Even Symmetric Frequency Response Amplitude function: : even symmetric, period 2011.9 EEE4176 Application of Digital Signal Processing

Linear Phase FIR Filters Type-II , : odd order / even length; : Even Symmetric Frequency Response Amplitude function: : even symmetric, period ※ : Not suitable for HPF and BRF 2011.9 EEE4176 Application of Digital Signal Processing

Linear Phase FIR Filters Type-III , : even order / odd length; : Odd Symmetric Frequency Response Amplitude function: : off symmetric, period ※ : Not suitable for LPF, HPF and BRF ※ Used for differentiators and Hilbert transformers. 2011.9 EEE4176 Application of Digital Signal Processing

Linear Phase FIR Filters Type-IV , : odd order / even length; : Odd Symmetric Frequency Response Amplitude function: : off symmetric, period ※ : Not suitable for LPF, and BRF ※ Used for differentiators and Hilbert transformers. 2011.9 EEE4176 Application of Digital Signal Processing

Linear Phase FIR Filters Summary of Linear-Phase filter types 2011.9 EEE4176 Application of Digital Signal Processing

FIR filter design: Impulse Response Truncation Method IRT filter design procedure Write the desired (or ideal) amplitude response Phase: either 0(symmetric) or (anti-symmetric) : Symmetric about n=0 Choose the filter's phase characteristics integer or fractional group delay initial phase 0 or Choose the filter order N, then the Ideal desired filter is described as\ Compute the impulse response of the ideal filter response using the IDFT: In most cases, can be expressed in a closed form. Truncate the impulse response to get 2011.9 EEE4176 Application of Digital Signal Processing

FIR filter design: Impulse Response Truncation Method Differentiators : Type III or IV ※ Odd N is preferred !! (See Fig. 9.9), Type IV ※ T = 1 => Frequency range limited due to the finite sampling rate 2011.9 EEE4176 Application of Digital Signal Processing

FIR filter design: Impulse Response Truncation Method Hilbert Transformers Note: 2011.9 EEE4176 Application of Digital Signal Processing

FIR filter design: Impulse Response Truncation Method Digital Hilbert transformer Ideal Hilbert transformer Causal filter design 2011.9 EEE4176 Application of Digital Signal Processing

FIR filter design: Impulse Response Truncation Method Digital Hilbert transformer N: Even vs. Odd ? ※ Type III is preferred. Ideal noncausal impulse response: h(n) = 0 for n = 0 for even n h(n) = -h(n) (h(n) = -h(N-n) 2011.9 EEE4176 Application of Digital Signal Processing

Hilbert Transformer Applications: PW Doppler 2011.9 EEE4176 Application of Digital Signal Processing

Hilbert Transformer Applications: PW Doppler 2011.9 EEE4176 Application of Digital Signal Processing

FIR filter design: Window Method Design procedure Determine the spec Design the ideal impulse response using IRT method. Band edge freq of ideal filter = N ? Windowing The length of is Window selection Select a Window with properties meeting your requirements. Depends also on Filter spec: Limitations in length, ripple, attenuation, etc. => How to determine N ? 2011.9 EEE4176 Application of Digital Signal Processing

FIR filter design: Window Method Filter Length For sharper transition and/or lower attenuation and ripple, use longer window Choose N -> design and check response -> change N and repeat the design 2011.9 EEE4176 Application of Digital Signal Processing

FIR filter design: Window Method Kaiser windows Empirical formulas for design parameters Matlab approach to FIR filter design using a Kaiser window: See pp. 298 FIR filter design examples  Read text (pp. 298 - 303) 2011.9 EEE4176 Application of Digital Signal Processing

FIR filter design: Window Method Advantages of window method Simple, robust, good performance Disadvantages of window method Not optimum in the sense that often over-spec ripples are not uniform • Filter length N is not optimum, i.e., minimum 2011.9 EEE4176 Application of Digital Signal Processing

FIR filter design: Least-Squares Design Formulation Define a weight function and the weighted freq-domain error as Define the integral of weighted square freq-domain error as Find 's for minimum (D) (F) (E) 2011.9 EEE4176 Application of Digital Signal Processing

FIR filter design: Least-Squares Design Formulation Substituting (D) and (F) in (E) gives Solve for 2011.9 EEE4176 Application of Digital Signal Processing

Least-Squares Linear-Phase FIR Filter Design 1/3 FIR Filter with length of L+1 Even symmetry in the impulse response  sum of cosines: Matrix form 2011.9 EEE4176 Application of Digital Signal Processing

Least-Squares Linear-Phase FIR Filter Design 2/3 Design problem can be written as Desired frequency response at the specific frequencies 2011.9 EEE4176 Application of Digital Signal Processing

Least-Squares Linear-Phase FIR Filter Design 3/3 Least Squares Optimization 2011.9 EEE4176 Application of Digital Signal Processing

EEE4176 Application of Digital Signal Processing OPTIMAL FIR FILTERS Chebyshev approximation Optimum design criterion in the sense that the weighted approximation error between the desired freq response and the actual freq response is spread evenly across the passband and evenly across the stopband of the filter minimizing the maximum error. Equi-ripple FIR filter Optimal filter in the sense stated above Minimize maximum error. Same error in pass and stop bands. Sharpest FIR filter or Smallest length filter for a given spec. Precise control of the critical filter frequencies Frequency sampling method You specify the frequency response for a set of frequencies Computer gives the impulse response for that freq. response 2011.9 EEE4176 Application of Digital Signal Processing

EEE4176 Application of Digital Signal Processing OPTIMAL FIR FILTERS Mathematical background 1. Define the desired amplitude response and the weighting function on a compact subset of , i.e., the set (union of passbands and stopbands). 2. Define the weighted error function as 3. Choose that minimizes 2011.9 EEE4176 Application of Digital Signal Processing

EEE4176 Application of Digital Signal Processing OPTIMAL FIR FILTERS Parks-McClellan algorithm Minmax solution : are fixed. are variable. Use Remez exchange algorithm as a computational procedure based on the alteration theorem Used to design Multi-band filters (LPF, HPF, BPF, BRF), differentiator, Hilbert Inputs : Filter length , filter type, # of bands, edge frequencies, Desired filter response, weighting factors Alteration theorem The function is optimal in the minimax sense if and only if there exist at least K+2 frequencies in such that At least K + 2 extrema in sign. 2011.9 EEE4176 Application of Digital Signal Processing

EEE4176 Application of Digital Signal Processing OPTIMAL FIR FILTERS Parks-McClellan algorithm Example) K = 7 case 2011.9 EEE4176 Application of Digital Signal Processing

Practical FIR Filter Design with MATLAB From Practical FIR Filter Design in MATLAB by Ricardo A. Losada (Mathworks)

Practical FIR Filter Design with MATLAB 2011.9 EEE4176 Application of Digital Signal Processing

Practical FIR Filter Design with MATLAB 2011.9 EEE4176 Application of Digital Signal Processing

Practical FIR Filter Design with MATLAB 2011.9 EEE4176 Application of Digital Signal Processing

Practical FIR Filter Design with MATLAB 2011.9 EEE4176 Application of Digital Signal Processing

Practical FIR Filter Design with MATLAB 2011.9 EEE4176 Application of Digital Signal Processing

Practical FIR Filter Design with MATLAB 2011.9 EEE4176 Application of Digital Signal Processing