SIGNALS AND SIGNAL PROCESSING

Transcription

1 SIGNALS AND SIGNAL PROCESSING Leture 2 (IR/IIR) Toomas Ruuben Contat data Toomas Ruuben truuben@lr.ttu.ee Home page of the ourse: Toomas Ruuben (TTÜ RSTI) ilters with inite Impulse Response (IR) ilters with inite Impulse Response (IR) IR filter have a sampled impulse response h i with finite duration Output of the filter ( n) is expressed as a onvolution from the impulse response and the sampled input signal x( n). Output of the filter is expressed as: N ( n) h( i) x( n i) i Inverted struture ma be used in ase of the following formula n ( n) x( i) h( n i) i n ( N ) Corresponding topolog is the following 3 4

2 requen response of the IR filters requen response of the IR filters Consider the first formula of ( n) Spetrum of the output signal of the filter is expressed as: N ( n) h( i) x( n i) i Y T X Convolution ma be alulated b using ourier s transform if impulse response ja input signal are sampled with the same frequen. t / Taking ourier transform from the impulse response h(n), we get frequen response of the sstem T N h( n) exp( j2π fn / ) n Output signal of the filter from the spetrum: ( n) Y exp( j2π fn / ) df requen response of the filter ma be also alulated via z- transform. z exp j2π f / Spetrum of the input signal is expressed as: X x( n) exp( j2π fn / ) n To explain this, let analse the filter formula more deepl 5 6 requen response of the IR filters requen response of the IR filters Strutual formula of the IR filter: N ( n) h( i) x( n i) ( n) h( ) x( n) + h( ) x( n ) + h( 2) x( n 2) + Taking ourier transform from the both sides of the strutual formula, and taking into aount that signal dela in frequen domain is defined b z transform, we get the spetum of the filter output signal i ( N ) x( n ( N ) ) + h h( ) X + h( ) X exp( j2π f )+ h( 2 ) X exp( j2π f 2/ ) +... Y /... + h( N ) X exp( j2π f( N ) ) / ( 2 N ) 2 X f h + h z + h z + + h N z rom the previous formula we an derivate the formula of the frequen response via z transform: N n T f Y f / X f h( n) z h( n) exp( j2π fn / ) n N Next let s look at the frequen and phase response of the filter if all values of the impulse response are equal with one () ( i), i n Taking into aount the formula of series sum we have T 2 ( N ) + z + z + + z h z z N exp exp ( j2π fn / ) ( j2π f / ) 7 8

3 requen response of the IR filters B modifing the previous formula we get ( ( sinπ fn / T ( f ) N exp jπ f N ) / ) N sinπ f / ( jφ ) T, f / 2, N exp < A rom here we an extrat the amplitude f and phase response of the filter : sin ( π fn / ) T f A N sinπ f / φ f π( N ) t The frequen response of the IR filter T A f is a periodial funtion with period. The phase response of the IR filter is linear whih means the dela of the singal in disretes. T A φ 9 requen response of the IR filters rom the formula: T A sin N ( π fn / ) sin( π f / ) we an find the width of the frequen response on the zero level from the equation: sin π f N / f N / π, f / N., π The frequen response of the IR filter in ase of h ( i), i IR filter with the smmetrial stuture The idea is to eliminate the dela between input and output signals. Let s shift the time referene to the entral point of the impulse response. or that we make summing in the formula of frequen response as: ( )/2 N T f h n exp j 2π fn /. n ( N ) or larit reasons, let s look at the situation where N is an even number. The amount of alulation an be redued when the impulse response of the IR filter is an even or an odd funtion n. h ( n ) h s IR filter with the smmetrial stuture In ase of even funtion h n h n h n h n and filter output signal is expressed as ( )/ 2 N ( ( + ) + ( )) n h i x n i x n i i Similar formula ma be obtained in ase of odd impulse responses h ( n) h( n) h( n) h( n) s In suh ases the input signals shifted towards eah other need to be substrated. The strutural sheme is on the following slide. s s 2

4 IR filter with the smmetrial stuture The struture of the IR filter if N is an odd. Here τ ( N ) / 2 requen response of the filter with the smmetrial struture Let s use the formula: ( N )/ 2 n ( N )/ 2 T f h n exp j2π fn /. and assume that the impulse response is an even funtion h n h n In this ase the frequen response will reveal as following: beause ( N f )/ 2 T T h( n) os 2π fn / n ( N )/ 2 ( N f ) / 2 n ( N )/ 2 h n sin 2π fn / 3 4 requen response of the filter with the smmetrial struture requen response of the filter with the smmetrial struture The impulse and frequen responses of the filter are assoiated with eah other b ourier transform. The impulse response an be derivated from the frequen response as following: / 2 h ( n) T os( 2π fn / ) d f. / 2 If h n h n, then also T and IR filter an be T( f) snthesized as following: ( )/ 2 N 2 os( 2π / ), T f h n fn n / 2 2 h n T f os 2π fn / d f ; Linear phase response and lak of distortions are most important properties of the IR filters. or that, the transmission funtion should be real. That means that the oefiient of the impulse response h( n) should be real numbers. The frequen response of the linear phase filter will reveal 2π τ / T f T f e j f That impulse response is obtained b taking the ourier transform from the frequen response funtion: / 2 j 2π f( n )/ h( n) τ T e df / 2 T 5 6

5 requen response of the filter with the smmetrial struture IIR (Infinite Impulse Response) filters Supposing that the dela τ is disrete with the step /. B replaing n τ m nm+ τ we get / 2 h m T f e f / 2 j 2π fm ( + τ) d / 2 2 T f os 2π fm / d f. IIR filter is based on adder and two IR filters. The output signal of IIR filter an be alulated with the following formula: N N H B ( ) + ( ) n h i x n i b e n e I e The strutural sheme of the IIR filter in a general wa: B shifting the time referene to the entral point of the impulse response, we an eliminate the linear phase hanges. 7 8 IIR filters IIR filters A more detailed sheme of IIR filter in the input of a IR filter: The modified struture of IIR filter with one dela register. 9 2

6 IIR filters The omparison of strutures: IIR filters The main risk of IIR filter is the possibilit to lose its stabilit. To prevent that from happening, the first ondition is: b Taking into aount the struture of the IIR filter and swithing into frequen domain, the spetrum of the signal in the output of the IIR filter and the frequen response funtion reveals as following: H X B Y, Y + ( B ) H X, Y T Y X H B 2 22 IIR filter Based on the previous, the frequen response funtion of the IIR filter an be seen prinipall as the multipliation of the frequen response of prefiltering part and frequen response of the feedbak sstem. T f T f T f T f T f Here And H H B B H, / ( ) T f H f T f B f IIR filter loses its stabilit if the following ondition appears: B B f / TB f N B ( ) n b e n e e IIR filter or retaining the stabilit of IIR filter the following ondition must be fulfilled: N B b e n e < e Assuming that the input signal is unipolar n, stabilit an be assured also b fulfilling a more simple but rigid ondition: N B e In ase of a bipolar output signal n the stabilit of the IIR filter an be assured b the following equation: N B e Let s look at some pratial assignments in Matlab environment. b e b e < < 23 24

7 MATLAB examples (IIR) MATLAB examples (IIR) [M,Wn]buttord(Wp,Ws,Rp,Rs) [M,Wn]hebord(Wp,Ws,Rp,Rs) [M,Wn]heb2ord(Wp,Ws,Rp,Rs) [M,Wn]ellipord(Wp,Ws,Rp,Rs) Here:..Wp - pass band (normalized b f s / 2) Ws -stop band (normalized b f s / 2) Rp -biggest waviness in pass band (db) Rs -minimal attenuation in stop band (db) M -ilter order Wn -Predistorted frequen Calulating the filter oeffiients A and B [B,A]butter(M,Wn) [B,A]heb(M,Rp,Wn) [B,A]heb2(M,Rs,Wn) [B,A]ellip(M,Rp,Rs,Wn) MATLAB examples (IIR) MATLAB examples (IIR) How to define filter tpes Example: LP: Wp., Ws.2 HP: Wp.2, Ws. B: Wp [.2.7], Ws [..8] S: Wp [..8], Ws [.2.7] To filter she signal x, following ommand ma be used Wp [6 2]/5; Ws [5 25]/5; Rp 3; Rs 4; [n,wn] buttord(wp,ws,rp,rs); %n6, Wn [b,a] butter(n,wn); freqz(b,a,28,) title('n6 Butterworth Bandpass ilter') filter(b,a,x) 27 28

8 MATLAB examples (IIR) 29