Introduction Ideal Filtering Numerical filtering : Z transform Finite Impulse Response Filters (FIR) Infinite Impulse Response Filter (IIR) Choice of the numerical filter Signal processing: numerical filtering Matthieu Kowalski Univ Paris-Sud L2S (GPI) Matthieu Kowalski Signal processing: numerical filtering 1 / 50 Introduction Ideal Filtering Numerical filtering : Z transform Finite Impulse Response Filters (FIR) Infinite Impulse Response Filter (IIR) Choice of the numerical filter 1 Introduction 2 Ideal Filtering Definitions Ideal filters 3 Numerical filtering : Z transform 4 Finite Impulse Response Filters (FIR) Definition FIR filter synthesis 5 Infinite Impulse Response Filter (IIR) General method 6 Choice of the numerical filter Specifications Classical filters Matthieu Kowalski Signal processing: numerical filtering 2 / 50 Introduction Ideal Filtering Numerical filtering : Z transform Finite Impulse Response Filters (FIR) Infinite Impulse Response Filter (IIR) Choice of the numerical filter Definitions Ideal filters Plan 2 Ideal Filtering Definitions Ideal filters Matthieu Kowalski Signal processing: numerical filtering 3 / 50 Introduction Ideal Filtering Numerical filtering : Z transform Finite Impulse Response Filters (FIR) Infinite Impulse Response Filter (IIR) Choice of the numerical filter Definitions Ideal filters Filtering Definition A filter is a linear time invariant system. It can be written as a convolution. Impulse response Let S a filter. The impulse response h of S is the system output to the unit impulse (Dirac). Hence h = S(δ) Analogical signals. For all signal x Z +1 y = S(x) = h ? x = x ? h y(t) = h(u)x(t − u) du −∞ Numerical signals. For all signal x +1 X y = S(x) = h ? x = x ? h yn = hk xn−k k=−∞ For finite signals, the convolution is circular : the signals are periodic with the same period. Matthieu Kowalski Signal processing: numerical filtering 4 / 50 Introduction Ideal Filtering Numerical filtering : Z transform Finite Impulse Response Filters (FIR) Infinite Impulse Response Filter (IIR) Choice of the numerical filter Definitions Ideal filters Realizable filters { 1 Realizable filter A filter with impulse respons h is realizable iff it is stable and causal. Remark 1 If a filter is stable, then it admits a Fourier transform. Reciprocally, if a filter admit an invertible Fourier transform, then it is stable. If a filter admits an invertible Fourier transform, it can be non causal (and then non realizable). Matthieu Kowalski Signal processing: numerical filtering 5 / 50 Introduction Ideal Filtering Numerical filtering : Z transform Finite Impulse Response Filters (FIR) Infinite Impulse Response Filter (IIR) Choice of the numerical filter Definitions Ideal filters Realizable filters { 2 Stable filters A filter with an impulse response h is stable iff +1 X = jhk j < +1 k=−∞ Causal filter A filter with an impulse response h is causal iff h is causal, ie hk = 0 8k < 0 Realizable filter A filter with an impulse response h is causal iff h is realizable iff it is stable and caucal. Matthieu Kowalski Signal processing: numerical filtering 6 / 50 Introduction Ideal Filtering Numerical filtering : Z transform Finite Impulse Response Filters (FIR) Infinite Impulse Response Filter (IIR) Choice of the numerical filter Definitions Ideal filters Filtering and Fourier transform Frequency response or complex Gain The frequency response, or complex gain, of a filter is its Fourier transform (when it exists). Filtering in the frequency domain Let S a filter with an impulse response h and x a signal. One has y = S(x) = h ? x if h and x admit a Fourier transform, we have in the frequency domain y^ = h^ · x^ Filtering a signal acts directly on its spectrum Matthieu Kowalski Signal processing: numerical filtering 7 / 50 Introduction Ideal Filtering Numerical filtering : Z transform Finite Impulse Response Filters (FIR) Infinite Impulse Response Filter (IIR) Choice of the numerical filter Definitions Ideal filters Plan 2 Ideal Filtering Definitions Ideal filters Matthieu Kowalski Signal processing: numerical filtering 8 / 50 Introduction Ideal Filtering Numerical filtering : Z transform Finite Impulse Response Filters (FIR) Infinite Impulse Response Filter (IIR) Choice of the numerical filter Definitions Ideal filters Ideal low pass filter { 1 Definition The frequency response of an ideal low pass filter with cutting frequency ν0 is given by : ( pb 1 si jνj < ν0 h^(ν) ν0 = 0 sinon Frequency response 1 0.8 0.6 0.4 Amplitude 0.2 0 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Frequence (Hz) Matthieu Kowalski Signal processing: numerical filtering 9 / 50 Introduction Ideal Filtering Numerical filtering : Z transform Finite Impulse Response Filters (FIR) Infinite Impulse Response Filter (IIR) Choice of the numerical filter Definitions Ideal filters Ideal low pass filter { 2 Impulse response pb sin(2πν n) h ν0 = 0 n πn Impulse response 0.2 0.15 0.1 0.05 Amplitude 0 -0.05 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Temps (s) Matthieu Kowalski Signal processing: numerical filtering 10 / 50 Introduction Ideal Filtering Numerical filtering : Z transform Finite Impulse Response Filters (FIR) Infinite Impulse Response Filter (IIR) Choice of the numerical filter Definitions Ideal filters Ideal low pass filter { 2 1 ν pb Z 2 Z 0 ν0 i2πnν i2πnν hn = h^(ν)e dν = e dν 1 − 2 −ν0 1 ν0 = ei2πnν i2πn −ν0 ei2πnν0 − e−i2πnν0 = i2πn sin(2πν0n) =0.2 πn 0.15 0.1 0.05 Impulse response Amplitude 0 -0.05 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Temps (s) Matthieu Kowalski Signal processing: numerical filtering 11 / 50 Introduction Ideal Filtering Numerical filtering : Z transform Finite Impulse Response Filters (FIR) Infinite Impulse Response Filter (IIR) Choice of the numerical filter Definitions Ideal filters Ideal high pass filter { 1 Definition The frequency response of an ideal high pass filter with cutting frequency ν0 is given by : ( ph 0 si jνj < ν0 h^(ν) ν0 = 1 sinon pb = 1 − h^(ν) ν0 Frequency response 1 0.8 0.6 0.4 Amplitude 0.2 0 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Frequence (Hz) Matthieu Kowalski Signal processing: numerical filtering 12 / 50 Introduction Ideal Filtering Numerical filtering : Z transform Finite Impulse Response Filters (FIR) Infinite Impulse Response Filter (IIR) Choice of the numerical filter Definitions Ideal filters Ideal high pass filter { 2 Impulse response ph ν0 h0 = 0 ph pb sin(2πν n) h ν0 = −h ν0 = − 0 n 6= 0 n k πn Impulse response 0.3 0.25 0.2 0.15 0.1 Amplitude 0.05 0 -0.05 -0.1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Temps Matthieu Kowalski Signal processing: numerical filtering 13 / 50 Introduction Ideal Filtering Numerical filtering : Z transform Finite Impulse Response Filters (FIR) Infinite Impulse Response Filter (IIR) Choice of the numerical filter Definitions Ideal filters Ideal band pass filter { 1 Definition The frequency response of an ideal high pass filter with cutting frequencies ν0 and ν1 is given by : 8 1 si ν < ν < ν <> 0 1 pbandeν ;ν h^(ν) 0 1 = 1 si − ν0 < −ν < −ν1 :>0 sinon pb pb = h^(ν) ν1 − h^(ν) ν0 Frequency response 1 0.8 0.6 0.4 Amplitude 0.2 0 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Frequence (Hz) Matthieu Kowalski Signal processing: numerical filtering 14 / 50 Introduction Ideal Filtering Numerical filtering : Z transform Finite Impulse Response Filters (FIR) Infinite Impulse Response Filter (IIR) Choice of the numerical filter Definitions Ideal filters Ideal band pass filter { 2 Impulse response pbande pb pb sin(2πν n) sin(2πν n) h ν0;ν1 = h ν1 − h ν0 = 1 − 0 k k k πn πn Impulse response 0.3 0.2 0.1 0 Amplitude -0.1 -0.2 -0.3 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Temps (s) Matthieu Kowalski Signal processing: numerical filtering 15 / 50 Introduction Ideal Filtering Numerical filtering : Z transform Finite Impulse Response Filters (FIR) Infinite Impulse Response Filter (IIR) Choice of the numerical filter Definitions Ideal filters Ideal band cut pass filter { 1 Definition The frequency response of an ideal high pass filter with cutting frequencies ν0 and ν1 is given by : 8 0 si ν < ν < ν <> 0 1 cbandeν ;ν h^(ν) 0 1 = 0 si − ν0 < −ν < −ν1 :>1 sinon pbande = 1 − h^(ν) ν0;ν1 pb pb = 1 − h^(ν) ν1 + h^(ν) ν0 Frequency response 1 0.8 0.6 0.4 Amplitude 0.2 0 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Frequence (Hz) Matthieu Kowalski Signal processing: numerical filtering 16 / 50 Introduction Ideal Filtering Numerical filtering : Z transform Finite Impulse Response Filters (FIR) Infinite Impulse Response Filter (IIR) Choice of the numerical filter Definitions Ideal filters Ideal band cut pass filter { 2 Impulse response pbande cbandeν0;ν1 ν0;ν1 hk = 1 − hk pb pb ν0 ν1 = 1 − hk + hk sin(2πν n) sin(2πν n) = 1 − 1 + 0 πn πn Impulse response 1.3 1.2 1.1 1 0.9 Amplitude 0.8 0.7 0.6 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Temps (s) Matthieu Kowalski Signal processing: numerical filtering 17 / 50 Introduction Ideal Filtering Numerical filtering : Z transform Finite Impulse Response Filters (FIR) Infinite Impulse Response Filter (IIR) Choice of the numerical filter Z-transform : definition z transform Let s = fsngn2Z be a numerical signal (i.e.