ADVANCED DIGITAL SIGNAL PROCESSING

Transcription

1 ADVANCED DIGITAL SIGNAL PROCESSING PROF. S. C. CHAN ( sccha@eee.hku.hk, Rm. CYC-702) DISCRETE-TIME SIGNALS AND SYSTEMS MULTI-DIMENSIONAL SIGNALS AND SYSTEMS RANDOM PROCESSES AND APPLICATIONS ADAPTIVE SIGNAL PROCESSING 1

2 DISCRETE-TIME SIGNALS AND SYSTEMS DISCRETE-TIME SIGNALS AND SYSTEMS Z-TRANSFORM DISCRETE-TIME AND DISCRETE FOURIER TRANSFORMS FILTER DESIGN REFERENCES: A.V. OPPENHEIM AND R.W. SCHAFER, DISCRETE-TIME SIGNAL PROCESSING. ENGLEWOOD CLIFFS, NJ: PRENTICE-HALL, INC.,

3 DISCRETE-TIME SIGNALS AND SYSTEMS DISCRETE-TIME SIGNALS DISCRETE-TIME SYSTEMS LINEAR AND TIME-INVARIANT SYSTEMS IMPULSE RESPONSE 3

4 DEFINITION Cotiuous-time sigals: sigals defied alog a cotiuum of times x (t). Discrete-time sigals: sigals defied at discrete times x [] Digital sigals: both time ad amplitude are discrete ( x [] amplitude quatized to say 16-bits) Advatages of digital/discrete-time systems: 1) Sigals are represeted as strigs of 0 ad 1. High oise immuity. Trasmissio ad storage (copy!!) are almost free from errors, if it is doe properly. 2) Sophisticated, flexible, ad accurate processig (imagie computig with resistors, capacitors, ad operatioal amplifiers!). 3) Efficiet realizatio usig digital sigal processors (microprocessors doig real-time processig), very large scale itegratio (VLSI) circuits, etc... 4

5 1. DISCRETE-TIME SIGNALS Discrete-time sigals are represeted mathematically as sequeces of umbers: { x [ ]}, < < (1.1) where is a iteger. Such sequeces may arise from periodic samplig of a aalog sigal x a (t). x[ ] = x ( T ), < < (1.2) a T is the period i secods. 5

6 1.1. BASIC SEQUENCES AND OPERATORS 1) y [] is a delayed sequece of x [] if y [ 0 where 0 is a iteger. ] = x[ ] (1.3) 2) Uit sample sequece (or impulse) 0, 0 δ [ ] = (1.4) 1, = 0 3) Uit Step sequece 1, 0 u [ ] = (1.5) 0, < 0 4) Expoetial ad siusoidal sequeces x [ ] = Aα (1.6) 6

7 Ay sequece, x [], ca be expressed as a sum of scaled ad delayed impulses k = x [ ] = x[ k] δ [ k] (1.7) Example: u[ ] = δ [ k] = δ[ k]. k = 0 k = 7

8 For complex α ad A, α jω = α e 0 ad A jφ = A e (1.8) x [ ] = Aα = A α cos( ω0 + φ) + j A α si( ω0 + φ) (1.9) The sequece oscillates with a expoetially growig evelop if α > 1 or with a expoetially decayig evelop if α < 1. Whe α = 1, we obtai the complex expoetial sequece x ] = A cos( ω + φ) + j A si( ω + ), (1.10) [ 0 0 φ where ω 0 ad φ are the frequecy ad phase of the complex expoetial or siusoidal, respectively. 8

9 2. DISCRETE-TIME SYSTEMS A discrete-time system is defied mathematically as a trasformatio or operator that maps a iput sequece with values x [] ito a output sequece with values y []. y [ ] = T{ x[ ]} (2.1) x[] T [ ] y[] Example 2.1 The Ideal Delay System y ] = x[ ], < < (2.2) [ d d is a fixed iteger called the delay of the system. The iput is shifted right by d samples. 9

10 Example 2.2 Movig Average M 2 1 y [ ] = x[ k] (2.3) M + M k = M 1 The system averages the ( M + M 1) samples from + M 1 to M MEMORYLESS SYSTEMS A system is referred to as memoryless if the output y [] at every value of depeds oly o the iput x [] at the same value of. Example y [ ] = ( x[ ]) (2.4) 10

11 2.2 LINEAR SYSTEMS Liear systems satisfy the priciple of superpositio: Let y [ ] ad y [ ] be the resposes of a system whe the iputs are 1 2 respectively x [ ] ad x [ ]. The system is liear if ad oly if 1 2 T x [ ] + x [ ]} = T{ x [ ]} + T{ x [ ]} (2.5) (additive property) { ad T { ax[ ]} = at{ x[ ]} = ay[ ] (scalig property) where a is a arbitrary costat. x 0[ ] y ] = T{ x [ ]} 0[ y 1[ ] = T{ x1[ ]} x + [ ]

12 or equivaletly T ax [ ] + bx [ ]} = at{ x [ ]} + bt{ x [ ]} (2.6) { Priciple of superpositio allows us to add the cotributios from idividual iputs together, ad there are o couplig effects betwee them. EXERCISE: Show that systems of Examples 2.1 ad 2.2 are liear systems while that i Example 2.3 is oliear. (Show that they either satisfy or do ot satisfy (2.6)). Example 2.4 Accumulator y [ ] = x[ k]. k = The output y [] is the sum of all previous iputs up to the curret time istat. 12

13 2.3 TIME-INVARIANT SYSTEMS A time-ivariat system is oe for which a time shift of the iput sequece causes a correspodig shift i the output sequece, i.e. If y [ ] = T{ x[ ]}, the y[ 0] = T{ x[ 0 ]} for all iteger 0. (2.7) x [] y [ ] = T{ x[ ]} [ 0 0 x[ 0 ] y ] = T{ x[ ]} 0 0 Example 2.5 Dowsampler (Left as a exercise) y [ ] = x[ M], < < (2.8) M is a positive iteger. It discards ( M 1) samples every M samples. It is ot time-ivariat uless M = 1. 13

14 2.2.4 CAUSALITY A system is causal if for every choice of 0, the output sequece value at idex = 0 depeds oly o the iput sequece values for 0. This implies that: If x 1[ ] = x 2[ ] for 0, the y 1[ ] = y2[ ] for 0. Example 2.6 Forward differece system y[ ] = x[ + 1] x[ ] (2.9) The system is o-causal (fiitely o-causal) because it ivolves a future iput sample x [ +1]. No-causal systems are more difficult to implemet (e.g. y [ ] = 0.9 y[ + 1] + x[ ] - solvig a differece equatio with a proper iitial coditio!). 14

15 2.5 STABILITY A system is stable i the bouded-iput bouded-output (BIBO) sese if ad oly if every bouded iput sequece produces a bouded output sequece. The iput x [] is bouded if there exists a fixed positive fiite value B x such that x [ ] B < for all. (2.10) x Stability requires that for every bouded iput there exists a fixed positive fiite value B y such that : y [ ] B < for all. (2.11) y 15

16 Examples 2.1, 2.2, 2.3, ad 2.5 are stable systems. The accumulator of Example 2.4 is ustable because for x [ ] = u[ ] (the uit step iput), y[ ] 0, < 0 = u[ k] = k = ( + 1), 0 (2.12) There is o fixed fiite value B y such that for all. ( + 1) By < 16

17 3. LINEAR TIME-INVARIANT SYSTEMS A liear system is completely characterized by its impulse respose. Let [ ] = T{ δ [ k]} be the respose of the system to ( k), a h k impulse occurrig at = k. I geeral, h k () depeds o both ad k. Cosider the output of a system T{.} to x[]: y [ ] = T x[ k] δ [ k] (3.1) Use (1.7) ad (2.1) k = From the priciple of superpositio, we have { [ k] } = y [ ] = x[ k] T δ x[ k] h [ ] (3.2) k = k = With the additioal costrait of time-ivariat ( y[ 0] = T{ x[ 0 ]}), we have [ ] = h( k) ad h k k δ 17

18 y[ ] = x[ k] h[ k] = x[ ] h[ ] = k (3.3) * deotes discrete-time covolutio. Direct Computatio of discrete-time covolutio It ca also be computed efficietly usig discrete Fourier trasform (DFT). How about if the impulse respose is of ifiite legth? 18

19 FINITE-DURATION IMPULSE RESPONSE (FIR) SYSTEMS If the impulse respose h[] of a LTI system is of fiite duratio, i.e. h [ ] 0 - < N N <, 1 2 it is called a fiite-duratio impulse respose (FIR) filter or system. M y[ ] = h[ k] x[ k] (3.4) k = 0 The impulse respose is h[]=0, <0 (causal), h[0]=3, h[1]=5, h[2]=7,h[3]=9, h[4]=2, h[5]=4. h[]=0, >5. M is called the order/degree of the system. M+1 is the filter legth. For causal system, h [ ] = 0 for < 0 19 Structure (sigal flow graph) of a orecursive filter.. Other commoly used ames of FIR filters are o-recursive filters ad movig average (MA) filters.

20 INFINITE-DURATION IMPULSE RESPONSE (IIR) SYSTEMS If the impulse respose h[] of a LTI system is of ifiite duratio, it is called ifiite-duratio impulse respose (IIR) systems. Example 1: The impulse respose of the accumulator is ifiite i duratio, which belogs to the class of IIR systems. Example 2: y [ ] = a y[ 1] + x[ ]. The filter output is obtaied through a recurrece relatio, istead of from the discrete-time covolutio. The impulse respose is h[]=0, <0 (causal), h[0]=1, h[1]=a, h[2]= 2 a,. The impulse respose is equal to h[ ] a u[ ] what coditio is it stable? Structure (sigal flow graph) of a simple recursive filter. =. Is it stable? Uder 20

21 4. LINEAR CONSTANT COEFFICIENT DIFFERENCE EQUATIONS A importat subclass of LTI systems are those with iput x [] ad output y [] satisfy a N th -order liear costat-coefficiet differece equatio of form: N k = 0 a k M y[ k] = b x[ k], 1 N k = 1 k = 0 k k = 0 a (4.1) 1 = y[ ] = a y[ k] + b x[ k] (4.2) k Feedback Feedfoward M k Although the impulse respose is of ifiite duratio ad its output is still give by the discrete-time covolutio of x[] ad h[], it is ot employed to compute the system output. Feedforward (orecursive) part Feedback (recursive) part Structure of a recursive (IIR) digital etwork 21

22 MATLAB simulatio The MATLAB file cotais data for =1:200 x()=si(/12)+.6*rad; % this geerates the artificial data set of a radom % Gaussia distributed radom variable added to a siusoid ed save C14data.mat x load C14data.mat % this calls up the data file of this ame ad % hece places the variable x i the MATLAB workplace y=zeros(size(x)); % this iitializes the output values to be zero for =2:200 % we start at =2 so that the idex of the first % elemet of the output array to be addressed is oe y()=x()+.9*y(-1); ed subplot(2,1,1); plot(x); ylabel( x[] ) subplot(2,1,2); plot(y); ylabel( y[] ),xlabel( ) 22

23 MATLAB simulatio.. Effect of simple recursive filterig (a) iput sigal (b) output sigal. The oise is reduced ad the siusoid is ehaced. A filter ca do a lot more 23

24 5. PROPERTIES OF LTI SYSTEMS 1) Commutative: x[ ] h[ ] = h[ ] x[ ] (5.1) Lettig m = k i (3.3) leads to the desire results. 2) Distributive: x [ ] ( h [ ] + h [ ]) = x[ ] h [ ] + x[ ] h [ ] (5.2) 3) Systems i Cascade: h [ 1 2 ] = h [ ] h [ ] (5.3) Cosider the respose of the system to a impulse System i cascade. 24

25 4) Systems i Parallel h ] = h [ ] + h [ ] (5.4) [ 1 2 Cosider the respose of the system to a sigle impulse. 25

26 6. STABILITY OF LTI SYSTEMS (the proof ca be omitted for 1 st readig) Liear time-ivariat systems are stable if ad oly if the impulse respose is absolutely summable, i.e. if Proof: = S h[ k] < (6.1) k = Sufficiet: From (3.3), k = y [ ] = h[ k] x[ k] < h[ k] x[ k] (6.2) k = If x [] is bouded so that x[ ] Bx, the x k = y [ ] B h[ k] (6.3) Therefore, = S h[ k] < implies the system is stable. k = 26

27 Necessary: Sice a ustable system does ot ecessary give a ubouded output for every iput. We must show that if S =, the a bouded iput ca be foud that will cause a uboud output. The sequece is * h [ ], h[ ] 0 x [ ] = h[ ], * complex cojugate. (6.4) 0, h[ ] = 0 is bouded by uity. However, y[0] = x[ k] h[ k] = h[ k] h[ k k = k = ] 2 = S Thus, if S =, the system is ustable. For the system to be stable S <. For the ideal delay, movig average, forward differece, ad backward differece examples, it is clear that S < sice their impulse resposes are oly of fiite duratio. 27

28 FIR systems are always be stable as log as each of the impulse respose is fiite i magitude. For IIR systems, it is easier to ifer the stability from the poles of their z- trasform. 28