Digital signal processing a practical approach

digital signal processing a computer-based approach and digital signal processing course description
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Dr.JakeFinlay,Germany,Teacher
Published Date:22-07-2017
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EDISP (LTIlect) (English) Digital Signal Processing DT systems, LTI November 26, 2015DT systems A DT system: an operator mapping an input sequence xn into an output sequence yn. yn= Tfxng A rule (formula) for computing y(n) Examples: from x(n) y(n)= 3 x(n) xn yn Tfg x(n)+ x(n 1) y(n)= 2 . M1 Implementations: 1 y(n)= x(n k) å M PC program k=0 matlab m-file ¥ y(n)= h(k) x(n k) å custom VLSI or FPGA k=¥ programmable digital signal 2 y(n)= x(n) processor EDISP (LTIlect) (English) Digital Signal ProcessingDT systems, LTI November 26, 2015 2 / 29Linear & time-invariant DT systems Linearityproperty Tfa x n+a x ng=a Tfx ng+a Tfx ng 1 1 2 2 1 1 2 2 in other words: if x n y n 1 1 x n y n 2 2 then ax n ay n (scaling, homogeneity) 1 1 x n+ x n y n+ y n (additivity) 1 2 1 2 Timeinvariance(shiftinvariance) If Tfxng= yn then 8n ; Tfxn n g= yn n 0 0 0 Shift does not modify result System properties do not change EDISP (LTIlect) (English) Digital Signal ProcessingDT systems, LTI November 26, 2015 3 / 29Linear systems - examples y(n)= 3 x(n) – is linear; it is also memoryless x(n)+x(n1) y(n)= (not memoryless): 2 a x (n)+a x (n)+a x (n1)+a x (n1) 1 1 2 2 1 1 2 2 Tfa x (n)+a x (n)g = = 1 1 2 2 2 a x (n)+a x (n1) a x (n)+a x (n1) 1 1 1 1 2 2 2 2 = + = 2 2 x (n)+x (n1) x (n)+x (n1) 1 1 2 2 =a +a = 1 2 2 2 =a y (n)+a y (n) cnd 1 1 2 2 2 (notL) y(n)=(x(n)) because 2 2 2 Tfx (n)+ x (n)g=(x (n)+ x (n)) =(x (n)) +(x (n)) +2 x (n)x (n) 1 2 1 2 1 2 1 2 EDISP (LTIlect) (English) Digital Signal ProcessingDT systems, LTI November 26, 2015 4 / 29shift example Input signals xn k. Responses Tfxn kg of TI system Tf:g 1 1 y(n) x(n) 0 -2 -1 0 1 2 3 4 5 n 0 -2 -1 0 1 2 3 4 5 n 1 1 y(n k), k =+1 x(n k), k =+1 0 -2 -1 0 1 2 3 4 5 n 0 -2 -1 0 1 2 3 4 5 n 1 1 y(n k), k =2 x(n k), k =2 0 -2 -1 0 1 2 3 4 5 n 0 -2 -1 0 1 2 3 4 5 n EDISP (LTIlect) (English) Digital Signal ProcessingDT systems, LTI November 26, 2015 5 / 29Other properties: causality, stability causality y(n ) depends only on x(n); n n ( important in real-time implementations, 0 0 unimportant for off-line processing) EDISP (LTIlect) (English) Digital Signal ProcessingDT systems, LTI November 26, 2015 6 / 29Other properties: causality,stability stability bounded input causes bounded output BIBO bounded9B : 8n jx(n)j B ¥ x x EDISP (LTIlect) (English) Digital Signal ProcessingDT systems, LTI November 26, 2015 7 / 29Examples Decimator(compressor) y(n)= x(Mn) L, but not TI (prove it) 1-storderdifference forward: y(n)= x(n+ 1) x(n) noncausal backward: y(n)= x(n) x(n 1) causal Accumulator n y(n)= x(k) å k=¥ unstable; (hint: feed it with un) EDISP (LTIlect) (English) Digital Signal ProcessingDT systems, LTI November 26, 2015 8 / 29LTI systems: impulse response hn= Tfdng impulse response of Tf:g hn characterizes completely system Tf:g – we may compute its response for any input xn. x(1)d(n+ 1) Decompose xn into weighted sum x(n) x(0)d(n) of impulsesdn k x(4)d(n 4) ¥ 0 xn= xkdn k å -2 -1 0 1 2 3 4 5 n k=¥ Superpose responses (use LTI properties) ¥ yn= xkhn k å k=¥ this is aconvolutionsum EDISP (LTIlect) (English) Digital Signal ProcessingDT systems, LTI November 26, 2015 9 / 29Convolution example EDISP (LTIlect) (English) Digital Signal ProcessingDT systems, LTI November 26, 2015 10 / 29Convolution example EDISP (LTIlect) (English) Digital Signal ProcessingDT systems, LTI November 26, 2015 10 / 29Convolution example EDISP (LTIlect) (English) Digital Signal ProcessingDT systems, LTI November 26, 2015 10 / 29Convolution example EDISP (LTIlect) (English) Digital Signal ProcessingDT systems, LTI November 26, 2015 10 / 29Convolution example EDISP (LTIlect) (English) Digital Signal ProcessingDT systems, LTI November 26, 2015 10 / 29Convolution properties ¥ yn= xkhn k å k=¥ we denote as yn= xn hn Propertiesof“” “” is commutative: xn hn= hn xn “” distributes over addition xn(h n+ h n)= xn h n+ xn(h n) 1 2 1 2 Convolution of a signal xn with given fixed hn is linear Systemand hn ¥ causality, hn= 0; n 0. A hint: y(n)= h(k)x(n k) å k=¥ ¥ stability, S= jh(k)j¥ å k=¥ EDISP (LTIlect) (English) Digital Signal ProcessingDT systems, LTI November 26, 2015 11 / 29Linear difference equations . . . describe an important class of LTI systems. N M a y(n k)= b x(n k); a = 1 (traditionally) å k å k 0 k=0 k=0 or y(n) = a  y(n 1) a  y(n 2)::: a  y(n N)+ 1 2 n +b  x(n) +b  x(n 1)+ b  x(n 2)+:::+ b  x(n M) 0 1 2 n Note: if, for a given input x n, an output sequence y n satisfies given difference p p equation, yn= y n+ y n p h N will also satisfy the equation, if y n is a solution to a y(n k)= 0 (homogenous h å k k=0 equation). EDISP (LTIlect) (English) Digital Signal ProcessingDT systems, LTI November 26, 2015 12 / 29Difference equation – example An equation: y(n)= a y(n 1)+ x(n) with input  x(n) y(n) x(n)= 0; n 0 +  x(n)=6 0; n 0.    H a delay H H y(0) = ay(1)+ x(0) Initial condition: y(1)=a y(1) = a y(0)+ x(1) Let xn=dn y(2) = a y(1)+ x(2) y(0) = aa+ 1 ::: 2 y(1) = a(aa+ 1)= a a+ a 3 2 y(2) = a a+ a ::: n+1 n y(n) = a a+ a EDISP (LTIlect) (English) Digital Signal ProcessingDT systems, LTI November 26, 2015 13 / 29Difference equation – impulse response (example continued y(n)= a y(n 1)+ x(n)  x(n) y(n) Initial condition: y(1)=a +  xn=dn  n+1 n  Solution: y(n)= a a+ a  a delay H H H Find a homogenous part Stability: a = 0:7 n 1 a: a ¥ n 0 a 1: a 0 0 1 2 3 4 5 6 7 8 n 1 a 0: a 0 n a1: a ??? a =1:1 0 1 2 3 4 5 6 7 8 EDISP (LTIlect) (English) Digital Signal ProcessingDT systems, LTI November 26, 2015 14 / 29Z-transform – what and why jnq DTFT – a transform based on periodic decomposition (basis: e sequences) responses of most LTI systems – made of short sequences (FIR) and decaying n jnq complex exponentials r e (IIR) n jnq jq n n r e =(re ) = z can be a good basis Z-transform is a tool for analyzing transient signals, such as an impulse response of a system. EDISP (LTIlect) (English) Digital Signal ProcessingDT systems, LTI November 26, 2015 15 / 29Z-transform definition Z – a generalization of DTFT, similar toL as a generalization of CTFT ¥ n X(z)= x(n)z å n=¥ jq DTFT is equal to X(z) at unit circle z = e n jq Convergence: same as for DTFT of xn r (substitute z = re ) ¥ n x(n)r ¥ å n=¥ n 1 example: un is not absolutely summable; u(n) r can be, ifjr j 1 Z(un) is convergent for r 1. EDISP (LTIlect) (English) Digital Signal ProcessingDT systems, LTI November 26, 2015 16 / 29