離散時(shí)間信號(hào)處理DSPIntroduction

1、1usstdigital signal processing毛 倩辦公室：儀表一館113email: tel:m) 55271601(o)introduction3 digital signal processing (dsp) is used in a wide variety of applications.telephone & telegramradaraudio signal processingmultimediasystemimage processingmobile telephonecommunication systemdigitaltvintro

2、duction4introduction signals a signal can be defined as a function that conveys information. signals are presented mathematically as functions of one or more independent variables.for example:a speech signal would be represented mathematically as a function of one time variable-f(t); - one-dimension

3、al (1-d) signal 一維信號(hào)a picture would be represented mathematically as a brightness function of two spatial variables-f(x,y). - two-dimensional (2-d) signal 二維信號(hào)a color video signal (a rgb television signal) is a 3-d signal. -multidimensional (m-d) signal 多維信號(hào)5introduction what is digital signal proce

4、ssing? digital signal processing is the science to process signals by digital means. this includes a wide variety of goals: filtering, transformation, recognition, enhancement, compression, and much more. dsp is one of the most powerful technologies that will shape science and engineering in the twe

5、nty-first century. suppose we attach an analog-to-digital converter to a computer, and then use it to acquire a chunk of real world data.6introduction 簡(jiǎn)單的說，數(shù)字信號(hào)處理是利用計(jì)算機(jī)或?qū)Ｓ锰幚碓O(shè)備，以數(shù)值計(jì)算的方法對(duì)信號(hào)進(jìn)行采集 、變換、綜合、估值與識(shí)別等加工處理，借以達(dá)到提取信息和便于應(yīng)用的目的。7introduction digital signal processing includes two meanings: processing

6、 analog signals in a digital way. processing digital signals. advantages: high reliability（可靠性） high agility (靈活性好，易于實(shí)現(xiàn)系統(tǒng)性能） high precision（高精度） low cost （成本低） 8introductionmain tools: discrete-time signal representations. discrete transforms and their fast algorithms (z transform, dft&fft). design

7、and implementation of digital filter (iir（無限長(zhǎng)單位沖激響應(yīng)）&fir （有限長(zhǎng)單位沖激響應(yīng)）filter). multirate systems（多率系統(tǒng)）, filter banks（濾波器組）, and wavelets（小波）. implementation of digital signal processing systems.9references 程佩青，數(shù)字信號(hào)處理教程，清華大學(xué)出版社 胡廣書，數(shù)字信號(hào)處理理論、算法與實(shí)現(xiàn)，清華大學(xué)出版社 高西全，丁玉美，闊永紅，數(shù)字信號(hào)處理原理、實(shí)現(xiàn)及應(yīng)用，電子工業(yè)出版社 chapter 2 dis

8、crete-time signals and system112.1 discrete-time signals: sequences the independent variable of a signal may be either continuous or discrete. continuous-time signals are those that are defined at continuous times. discrete-time signals are those that are defined at discrete times.tamplitudetamplitu

9、decontinue-time signaldiscrete-time signal122.1 discrete-time signals notations a discrete-time signal can be represented as where t is time interval between samples. each sample of sequence x(nt) is determined by the amplitude of signal at instant nt. for example where t is 0.03. another notation i

10、s a sequence of numbers. for example, the sequence x can be represented aswhere z is the set of integer numbers（整數(shù)集）, and x(n) is referred to as the “nth sample” of the sequence. for example a convenient notation for the sequence x just is x(n). ( ),x nnz),(znntx),.3(),2(),1 (),0(.,xxxx),.09. 0(),06

11、. 0(),03. 0(),0(.,xxxx132.1 discrete-time signals graph discrete-time signals are often depicted graphically.x(n) or x(nt) n or nt14unit sample sequence（單位抽樣序列） the definition of the unit impulse(n) n0115delayed unit sample sequence （延時(shí)單位抽樣序列） the definition of the delayed unit sample sequence(n m)

12、n01m16unit step sequence （單位階躍序列） the definition of the unit stepu(n) n010)2() 1()()()() 1()()()(mnnnmnnununununu(n)的后向差分17cosine function the definition of the cosine function is , whose angular frequency（角頻率） is rad/sample.n0)162cos(n)(nx18exponential sequence （實(shí)指數(shù)序列） the definition of the real ex

13、ponential function is n0n19unit ramp（單位斜坡序列） the definition of the unit rampr(n) n0202.1 discrete-time signals an arbitrary sequence can be expressed as a sum of scaled, delayed unit impulses. the unit step u(n) can be expressed as and the unit ramp r(n) can be expressed as kknkxnx)()()(0)()(kknnu0)

14、()(kknnnr21example: generate the signal with impulse sequence-3 -2 -10 1 2 3 4 53a2a6ax(n)n)6()2()3()(623nanananx3an0)3(3na0n2a)2(2na06an)6(6na22periodic sequence a sequence x(n) is defined to be periodic if and only if there is an integer n0 such that x(n) = x(n + n) for all n. in such a case, n is

15、 called the period of the sequence. note, not all discrete cosine functions are periodic. if 2/ is an integer（整數(shù)） or a rational number（有理數(shù)), this sequence will be periodic; if 2/ is an irrational number（無理數(shù)）, this cosine function will not be periodic at all. knknnnnn22),(cos()cos(z23example determin

16、e whether following discrete signal is periodic or not. if it is, calculate the period of the signal. ( )cos(0.8510)x nn solution: if the signal is periodic, then we have then so system is periodic, its period is 40 (when k=17). ()cos0.85 ()10cos(0.85100.85)( )cos(0.8510)x nnnnnnx nnknkn1740285. 024

17、2.2 discrete-time systems definition: a system is defined mathematically as a unique transformation or operator that maps an input sequence x(n) into an output sequence y(n). this can be denoted as y(n) = tx(n) where t expresses a discrete-time system.discrete-time systemy(n)x(n)excitationresponset2

18、52.2.1 memoryless systems a system is referred to as memoryless if the output y(n) at every value of n depends only on the input x(n) at the same value of n.262.2.2 linearity systems linearity（線性）if y1(n) and y2(n) are the responses when x1(n) and x2(n) are the inputs respectively, then a system is

19、linear if and only ifta x(n) = a t x(n) and t x1(n)+ x2(n) = t x1(n)+ t x2(n)for any constants a and b. example: y(n)=tx(n)=3x(n)+4 ta x(n) =3a x(n)+4 atx(n) =3a x(n)+4a ta x(n) at x(n) so it is not a linearity system.272.2.3 time-invariant systems time invariance（時(shí)不變性）a discrete-time system is time

20、 invariant if and only if, for any input sequence x (n) and integer n0, thent x(n-n0)=y(n-n0) with y (n)= t x(n). note: another name of time invariance is shift invariance（移不變性）. example: y(n)=3x(n)+4 tx(n-n0)=3x(n-n0)+4 y(n-n0)= 3x(n-n0)+4 y(n-n0)= tx(n-n0) so this system is time invariance.282.2.4

21、 causality causality（因果性）a discrete-time system is causal if and only if, when x1(n) = x2(n) for n n0, then t x1(n) = tx2(n), for n n0 a causal system is one for which the output at instant n does not depend on any input occurring after n. usually, in the case of a discrete-time signal, a noncausal

22、system is not implementable in real time. however, in some cases a discrete signal does not consist of time samples, a noncausal system can be easily implemented. 292.3 linear time-invariant system an input sequence x(n) can be expressed as a sum of scaled, shifted unit impulses. the output can be e

23、xpressed askknkxnx)()()( ) ( )( ) ()ky nx nx knktt30impulse responses and convolution sums if the system is linear, we can obtain since x(k) in the above equation is just a constant（常數(shù)）, the output iswhere we define( )( ) ()( ) ()kky nx knkx knktt( )( ) ()( ) ()kky nx knkx k h nkt31impulse responses

24、 and convolution sums h(n) = t(n) is referred to as the impulse response（沖激響應(yīng)） of the system. the equationis called a convolution sum or a discrete-time convolution （卷積和，又稱離散卷積和、線性卷積和）. this equation indicates that a linear time-invariant system is completely characterized by its unit impulse respon

25、se h(n).(1.37)32impulse responses and convolution sums the convolution sum can also be written as a shorthand notation for the convolution is and where “ * ” represents the convolution sum.)()()()()(nhnxknhkxnyk)()()()()(nxnhkhknxnykkkhknxny)()()(33examplecompute the linear convolution y(n) = x(n)*h

26、(n), andsolution: ( )1,2,3,1,( )1,4,3,2x nh n( )( )2 (1)3 (2)(3)( )( )4 (1)3 (2)2 (3)( )( )* ( )( ) () ( )2 (1)3 (2)(3) ()4 (1)3 (2)2 (3)kkx nnnnnh nnnnny nx nh nx k h nkkkkknknknknk34cont. ( )4 (1)3 (2)2 (3)2 (1)4 (2)3 (3)2 (4)3 (2)4 (3)3 (4)2 (5) (3)4 (4)3 (5)2 (6)( )6 (1) 14 (2)21 (3)17 (4)9 (5)2

27、 (6)nnnnnnnnnnnnnnnnnnnnnnn35cont.solution ii: 2)6(1432)6(9)5(1432)5(17)4(1432)4(21)3(1432)3(14)2(1432)2(6) 1 (1432)1 (1)0(1432)0(1321)(ylhylhylhylhylhylhylhlx( )1,2,3,1,( )1,4,3,2x nh n362.4 properties of linear time-invariant system suppose now that there are two linear time-invariant system which

28、 are in cascade. that is to say, the output of a system with impulse response h1(n) is the excitation for a system with impulse response h2(n).h1(n)h2(n)y(n)x(n)37systems in cascadethe output of the first system h1(n) is and the output of the second system h2(n) is kknhkxnhnx)()()()(11 lkklnhklhkxnh

29、knhkxnhnhnxny)()()()()()()()()()(21212138systems in cascadeby exchange the summing sequence, we get kllklnhklhkxlnhklhkxny)()()()()()()(2121kkmknhknhkxmhkmnhkxny)()()()()()()(2121get we, lettingmlnthen we can express its output as39systems in cascade conclusion: two linear time-invariant systems in

30、cascade form a linear time-invariant system with an impulse response which is the convolution sum of the two impulse responses.h(n) = h1(n) * h2(n)y(t)x(n)h1(n)h2(n)y(n)x(n)40* discrete-time systems stability（穩(wěn)定性） a system is referred to as bounded-input bounded-output (bibo) stable if, for every in

31、put limited in amplitude, the output signal is also limited in amplitude. （有界輸入產(chǎn)生有界輸出） if x(n) is bounded, i.e., |x(n)| xmax for all n, then linear shift invariant systems are stable if and only if nnh)(max( )( ) ()( )kky nx k h nkxh k（充要條件）41example characterize following system as being either lin

32、ear or nonlinear, time invariant or time varying, causal or noncausal, stable or not stable. 2( )(1)( )(2)y nnxnx nsolution: 1. for so it is not a linear system.222( )(1)( )(2) ( )(1)( )(2)( ) ( )h ax nna xnax nah x na nxnax nh ax nah x n42cont.2. for so it is time varying; 3. because i.e. the outpu

33、t for a certain time t = n of this system not only depends on the input at time t = n, but also depends on the time after n (i.e. t = n+2). so the system is noncausal; 20002000000 ()(1)()(2)()(1)()(2) ()()h x nnnxnnx nny nnnnxnnx nnh x nny nn2( )(1)( )(2)y nnxnx n43cont.4. for a special bounded inpu

34、t we have then the output for system i.e. systems output is unbounded. so the system is unstable.therefore the system is nonlinear, time invariant, noncausal and unstable. ( )( )x nu n1)(nx22( )(1)( )(2)0, 2(1)( )(2)1, 20, 0y nnx nx nnnunu nnnn 442.5 linear constant-coefficient difference equations

35、the input x(n) and the output y(n) of a system described by a linear difference equation（線性差分方程）are generally related by 線性線性：指方程中各y(n-i)和x(n-l)項(xiàng)都只有一次冪，且不存在它們的相乘項(xiàng)。 輸出序列y(n)變量序號(hào)的最高值與最低值之差n稱為差分方程的階數(shù)階數(shù)。mllniilnxbinya00)()(system typedescription of the systemthe continuous-time systemthe differential eq

36、uation（微分方程）the discrete-time systemthe difference equation（差分方程）(2.1)45iir and fir filters equation (2.1) can be rewritten, without loss of generality, considering that a0=1, yielding so the output y(n) is dependent both on samples of the input x(n), x(n-1), x(n-m), and on previous samples of the o

37、utput y(n-1), y(n-2), , y(n-n). mllniilnxbinyany01)()()(mllniilnxbinya00)()(2.2)46iir and fir filters since in order to compute the output, we need the past samples of the output itself, we say that the system is recursive（遞歸的）. when a1=a2=an=0, then the output at sample n depends only on values of

38、the input signal. in such case, the system is called nonrecursive（非遞歸的）. it ismlllnxbny0)()(2.3)47iir and fir filters if we compare the above equation with equation we see that the system in equation (2.3) has a finite-duration impulse response. such discrete-time system are often referred to as fin

39、ite-duration impulse-response (fir) （有限長(zhǎng)單位沖激響應(yīng)）filters. in contrast, when y(n) depends on its past values, as shown in equation (2.2), the impulse response of the system might not be zero when n. therefore, recursive digital system are often referred to as infinite-duration impulse-response (iir)（無限

40、長(zhǎng)單位沖激響應(yīng)）filters.kkhknxny)()()(48review a sequence x(n) is defined to be periodic if and only if there is an integer n0 such that x(n) = x(n + n) for all n. in such a case, n is called the period of the sequence. note, not all discrete cosine functions are periodic. if 2/ is an integer（整數(shù)） or a ratio

41、nal number（有理數(shù)), this sequence will be periodic; if 2/ is an irrational number（無理數(shù)）, this cosine function will not be periodic at all. knknnnnn22),(cos()cos(z49reviewthe characteristics of the discrete-time system y(n) = h x(n) : linearity: if y1(n)= h x1(n), y2(n)= h x2(n),then h ax(n)=ah x(n) and h x1(n)+ x2(n)=h x1(n)+h x2(n) for any constants a and b. time invariance: if y (n)= h x(n),thenh x(n-n0)=y(n