) (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 superposition

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1 4, Superposition. 4. (Discrete System) 4-. x(n) y(n) (mapping) (Transformation). y(n)=t[x(n)]

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 superposition

3 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) superposition

4 2) (Time Invariant) y(n)=t[x(n)] (time invariant shift invariant). y(n-k)=t[x(n-k)] k k ) (Causality) causal noncausal. y(n)=t[x(n-k)], if k causal if k < noncausal

5 4.2 (Convolution) (Correlation) 4.2. ). h(n), x(n), y(n) y(n) x(n) h(n) 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

6 4-7.

7 3) rinsing ramp sine

8 4-9. 4) End M M End

9 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])

10 6) MATALB MATLAB conv.m. y=conv(x,h). [ ] h=[ ] x=[ ]. "stem". [ ] h=[ ]; subplot(3,,),stem(h,'filled') axis([ ]) x=[ ]; subplot(3,,2),stem(x,'filled') axis([ ]) y=conv(h,x); subplot(3,,3),stem(y,'filled') axis([ ])

11 conv,. conv_m n. [ 2] h=[ ], 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=[ ]; subplot(3,,),stem(n,h,'filled') axis([ ]);

12 n2=[-3:5]; x=[ ]; subplot(3,,2),stem(n2,x,'filled') axis([ ]) [y,n]=conv_m(h,n,x,n2); subplot(3,,3),stem(n,y,'filled') axis([ ]) (Correlation) folding. (cross correlation) (auto correlation). (cross correlation), (auto correlation).. )

13 .. (echo), (correlation).. matched filtering. 4-. x(n),, t(n), y(n),. (threshold value)

14 2) (delay) ,. 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')

15 Crosscorrelation 2 Rxx n

16 4 MATLAB,.., () : [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.

17 [ ] x=u(n), (-5 n 2), h=[ ], ( n2 5) y=x*h. "stem". [ 2] x=δ(n), (-5 n 2), h=[ ], ( n2 5) z=x*h. "stem".

18 [ 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".

19 [ 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".

20 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) a) (Identity)

21 b) (Amplification & Attenuation) amplification Attenuation c) (Shift)

22 d) (Echo) LTI(Linear Time Invariant),,. IIR(Infinite Impuse Reponse). (Linear Difference Equation) IIR,..,.

23 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 LTI

24 .,. LTI.

25 5 MATLAB,.., () : delta = impseq(n,n,n2) y = filter(b,a,delta) : x = y = filter(b,a,x) MATLAB. (2) stem plot.

26 [ ], 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) n Step Response n

27 [ ], ) 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".

Chapter4.hwp

Chapter4.hwp Ch. 4. Spectral Density & Correlation 4.1 Energy Spectral Density 4.2 Power Spectral Density 4.3 Time-Averaged Noise Representation 4.4 Correlation Functions 4.5 Properties of Correlation Functions 4.6