4, Superposition. 4. (Discrete System) 4-. x(n) y(n) (mapping) (Transformation). y(n)=t[x(n)]. 4-2..
) (Linearity) y(n) = T[x(n)] y2(n) = T[x2(n)] y(n) = T[ax(n)+bx2(n)] = T[ax(n)]+T[bx2(n)] = ay(n)+by2(n),., superposition. 4-3. superposition
superposion Superposition.. Superposition. superposition. Superposition. x(n) x[n], x[n], x2[n], y(n), y2(n), y2(n). (synthesis) (addition) y(n). x(n). 4-4. superposition
2) (Time Invariant) y(n)=t[x(n)] (time invariant shift invariant). y(n-k)=t[x(n-k)] k k. 4-5. 3) (Causality) causal noncausal. y(n)=t[x(n-k)], if k causal if k < noncausal
4.2 (Convolution) (Correlation) 4.2. ). h(n), x(n), y(n) y(n) x(n) h(n). 4-6. y n x nh n N y n k h k x n k *. 2). x(n) h(n) y(n) x(n) h(n).. a. [x(n),h(n)] folding b. (shift) c. x(n) h(n), d. 2, 3
4-7.
3) rinsing ramp sine. 4-8..
4-9. 4) End M M-.. 4-. End
5) a) a[n]*b[n] = b[n]*a[n] b) (a[n]*b[n])*c[n] = a[n]*(b[n]*c[n]) c) a[n]*b[n]+a[n]*c[n]=a[n]*(b[n]*c[n])
6) MATALB MATLAB conv.m. y=conv(x,h). [ ] h=[-.4 -.4] x=[ ]. "stem". [ ] h=[-.4 -.4]; subplot(3,,),stem(h,'filled') axis([-2 5 -.5.2]) x=[ ]; subplot(3,,2),stem(x,'filled') axis([-2 5 -.5.2]) y=conv(h,x); subplot(3,,3),stem(y,'filled') axis([-2 5 -.5 ])
.5 -.5-2 - 2 3 4 5 6 7 8 9 2 3 4 5 6.5 -.5-2 - 2 3 4 5 6 7 8 9 2 3 4 5.5 -.5-2 - 2 3 4 5 6 7 8 9 2 3 4 5 conv,. conv_m n. [ 2] h=[-.4 -.4], n 2, x=[ ], n2 8. "stem". [ ] function [y, ny] = conv_m(x,nx,h,nh) % modified convolution routine for signal processing % ------------------------------------------ % [y,ny] = conv_m(x,nx,h,nh) % [y,ny] = convolution result % [x,nx] = first signal % [h,nh] = second signal % nyb = nx()+nh(); nye = nx(length(x)) + nh(length(h)); ny = [nyb:nye]; y = conv(x,h); n=[:2]; h=[-.4 -.4]; subplot(3,,),stem(n,h,'filled') axis([-5 5 -.5.2]);
n2=[-3:5]; x=[ ]; subplot(3,,2),stem(n2,x,'filled') axis([-5 5 -.5.2]) [y,n]=conv_m(h,n,x,n2); subplot(3,,3),stem(n,y,'filled') axis([-5 5 -.5 ]).5 -.5-5 -4-3 -2-2 3 4 5 6 7 8 9 5.5 -.5-5 -4-3 -2-2 3 4 5 6 7 8 9 5.5 -.5-5 -4-3 -2-2 3 4 5 6 7 8 9 5 4.2.2 (Correlation) folding. (cross correlation) (auto correlation). (cross correlation), (auto correlation).. )
.. (echo), (correlation).. matched filtering. 4-. x(n),, t(n), y(n),. (threshold value).. 4-2.
2) (delay) 54.. 54,. 3) (power spectral density : p.s.d). x(n) (Fourier Tranform) x(n) p.s.d. p.s.d. 4) MATALB MATLAB "xcorr.m". Rxy=xcorr(x,y) Rxx=xcorr(x). [ 3] x=[3,, 7,, -, 4, 2], -3 nx 3 xcorr" Rxx [ ], "stem". x = [3, 6, -7,, -, 4, 8 2-6]; nx=[-4:4]; % given signal x(n) x=x; Rxx = xcorr(x,x); % crosscorrelation % Rxx = xcorr(x); subplot(2,,),stem(nx,x); subplot(2,,2),stem(rxx); xlabel('n') ylabel('rxx');title('crosscorrelation')
5-5 - -4-3 -2-2 3 4 3 Crosscorrelation 2 Rxx - 2 4 6 8 2 4 6 8 n
4 MATLAB,.., 2. 3. () : [y,ny] = conv_m(x,nx,h,nh) : Rxx=xcorr(x), Rxy=xcorr(x,y) [x,nx] = sigfold(x,nx); % obtain x(-n) [rxy,nrxy] = conv_m(y,ny,x,nx); % crosscorrelation MATLAB. (2) stem plot.
[ ] x=u(n), (-5 n 2), h=[ -.7.5 -.3.2 -.], ( n2 5) y=x*h. "stem". [ 2] x=δ(n), (-5 n 2), h=[ -.7.5 -.3.2 -.], ( n2 5) z=x*h. "stem".
[ 3] x(n)=[:-:], y(n)=x(n-3)+w(n) ( w(n).), ) Rxx y(n) Rxy.( conv_m ) 2) w(n)=.2w(n), w(n)=.4w(n), w(n)=.6w(n), Rxy. 3) y(n)=w(n) Rxy. 4). "stem".
[ 4]. x(n) (echo) e(n), y(n) x(n). y(n). y(n)=x(n)+e(n) e(n)= x(n-k) k (delay) x(n). x(n)=cos(.3πn)+.6cos(.5πn), =., k=5, 2 Rxx Ryy, Rxy.. "stem".
5 5. (Impulse Response) x(n)=δ(n), y(n) h(n). y n x nh n nh n h n h(n) (Impulse Response). 5-. 5.2 a) (Identity).5-2 - 2 3 4 5 6 7.5 2 3 4 5 6 7 8.5-2 2 4 6 8 2 4 6
b) (Amplification & Attenuation).5.5-2 - 2 3 4 5 6 7.5 2 3 4 5 6 7 8.5.5-2 2 4 6 8 2 4 6 amplification.2. -2-2 3 4 5 6 7.5 2 3 4 5 6 7 8.2. -2 2 4 6 8 2 4 6 Attenuation c) (Shift).5-2 - 2 3 4 5 6 7.5 2 3 4 5 6 7 8.5-2 2 4 6 8 2 4 6
d) (Echo).5-2 - 2 3 4 5 6 7.5 2 3 4 5 6 7 8.5.5-2 2 4 6 8 2 4 6 5.3 LTI(Linear Time Invariant),,. IIR(Infinite Impuse Reponse). (Linear Difference Equation) IIR,..,.
M N. IIR(Infinite Impuse Reponse),. FIR(Finite Impuse Reponse). zero-state zero-input. zero-state. zero-input.. zero-state.,. MATLAB filter. y=filter(b,a,x) b=[b, b,..., bm]; a=[a, a,..., an]; b a b, a. 5.4. LTI
.,. LTI.
5 MATLAB,.., 2. 3. () : delta = impseq(n,n,n2) y = filter(b,a,delta) : x = y = filter(b,a,x) MATLAB. (2) stem plot.
[ ], y(n) - y(n-) +.9y(n-2) = x(n) [ ] ) n 2 h(n),. 2) n 2 s(n),. "stem". %y(n) - y(n-) +.9y(n-2) = x(n) a=[,-,.9];b=; % Part a) x=impseq(,,2);n=[:2]; h=filter(b,a,x); subplot(2,,);stem(n,h) axis([,2,-.2,.2]) title('impulse Response');xlabel('n');ylabel('h(n)') % Part b) x=stepseq(,,2); s=filter(b,a,x); subplot(2,,2);stem(n,s) axis([,2,,2.5]) title('step Response');xlabel('n');ylabel('s(n)').5 Impulse Response h(n) s(n) -.5-2 4 6 8 2 n Step Response 2.5 2.5.5 2 4 6 8 2 n
[ ], ) y(n)=x(n)+.5y(n-)-.75y(n-2) 2) y(n)=.2x(n)-.4x(n-)+.5x(n-2)-.2x(n-3)+.4x(n-4) a) n 2 h(n), "stem" b) x=sin(2pi2t)+sin(2pit)+sin(2pi5t) y(n), "subplot" "plot".