離散時間信號處理DSPIntroduction

1、USSTDigital Signal Processing辦公室：儀表一館113Email: Tel: (M) 55271601(O)1IntroductionDigital Signal Processing (DSP) is used in a wide variety of applications.Telephone & telegramradarAudio signal processingMultimediasystemImage processingMobile telephoneCommunication systemdigitalTVIntroduction3Introduc

2、tionSignalsA 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-dimensional (1-D) signal 一維信

3、號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a color video signal (a RGB television signal) is a 3-D signal. -Multidimensional (M-D) signal 多維信號4IntroductionWhat is Digital Signal Processing?Digital Signal

4、 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 twenty-first century. Su

5、ppose we attach an analog-to-digital converter to a computer, and then use it to acquire a chunk of real world data.5Introduction簡單的說，數(shù)字信號處理是利用計算機(jī)或?qū)Ｓ锰幚碓O(shè)備，以數(shù)值計算的方法對信號進(jìn)行采集 、變換、綜合、估值與識別等加工處理，借以達(dá)到提取信息和便于應(yīng)用的目的。6IntroductionDigital signal processing includes two meanings:Processing analog signals in a dig

6、ital way.Processing digital signals.Advantages:High reliability（可靠性）High agility (靈活性好，易于實現(xiàn)系統(tǒng)性能）High precision（高精度）Low cost （成本低）7IntroductionMain tools:Discrete-time signal representations.Discrete transforms and their fast algorithms (z transform, DFT&FFT).Design and implementation of digital filt

7、er (IIR（無限長單位沖激響應(yīng)）&FIR （有限長單位沖激響應(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.8References程佩青，數(shù)字信號處理教程，清華大學(xué)出版社胡廣書，數(shù)字信號處理理論、算法與實現(xiàn)，清華大學(xué)出版社高西全，丁玉美，闊永紅，數(shù)字信號處理原理、實現(xiàn)及應(yīng)用，電子工業(yè)出版社9Chapter 2 Discrete-Time Signals and System2.1 Discre

8、te-Time Signals: SequencesThe 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.tAmplitudetAmplitudeContinue-time signalDiscrete-time signal11

9、2.1 Discrete-time signals notationsA 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 is a sequence of numbers. For example, the seq

10、uence 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).122.1 Discrete-time signals graphDiscrete-time signals are often depicted graphically.x(n) or x(nT)

11、n or nT13unit sample sequence（單位抽樣序列）The definition of the unit impulse(n) n0114delayed unit sample sequence （延時單位抽樣序列）The definition of the delayed unit sample sequence(n m) n01m15unit step sequence （單位階躍序列）The definition of the unit stepu(n) n01u(n)的后向差分16cosine functionThe definition of the cosin

12、e function is , whose angular frequency（角頻率） is rad/sample.n017Exponential Sequence （實指數(shù)序列）The definition of the real exponential function is n0n18unit ramp（單位斜坡序列）The definition of the unit rampr(n) n0192.1 Discrete-time signalsAn arbitrary sequence can be expressed as a sum of scaled, delayed unit

13、 impulses.The unit step u(n) can be expressed asAnd the unit ramp r(n) can be expressed as 20Example: generate the signal with impulse sequence-3-2-1012345x(n)nn00n0n21Periodic 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

14、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 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. 22Example Determine w

15、hether following discrete signal is periodic or not. If it is, calculate the period of the signal. Solution: if the signal is periodic, then we have Then so system is periodic, its period is 40 (when k=17). 232.2 Discrete-time systemsDefinition: A system is defined mathematically as a unique transfo

16、rmation 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)ExcitationResponseT242.2.1 Memoryless SystemsA system is referred to as memoryless if the output y(n) at every va

17、lue of n depends only on the input x(n) at the same value of n.252.2.2 Linearity SystemsLinearity（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 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

18、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.262.2.3 Time-Invariant SystemsTime invariance（時不變性）A discrete-time system is time invariant if and only if, for any input sequence x (n) and integer n0, thenT x(n-n0)=y(n-n0) wi

19、th 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.272.2.4 CausalityCausality（因果性）A discrete-time system is causal if and only if, when x1(n) = x2(n) for

20、n n0, then T x1(n) = Tx2(n), for n n0A 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 system is not implementable in real time. However, in some cases a discrete signal does not consist

21、 of time samples, a noncausal system can be easily implemented. 282.3 Linear Time-Invariant SystemAn input sequence x(n) can be expressed as a sum of scaled, shifted unit impulses.The output can be expressed as29Impulse responses and convolution sumsIf the system is linear, we can obtain Since x(k)

22、in the above equation is just a constant（常數(shù)）, the output iswhere we define30Impulse responses and convolution sumsh(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 indicate

23、s that a linear time-invariant system is completely characterized by its unit impulse response h(n).(1.37)31Impulse responses and convolution sumsThe convolution sum can also be written asA shorthand notation for the convolution is and where “ * ” represents the convolution sum.32ExampleCompute the

24、linear convolution y(n) = x(n)*h(n), andSolution: 33Cont. 34Cont.Solution II: 352.4 Properties of Linear Time-Invariant SystemSuppose now that there are two linear time-invariant system which are in cascade. That is to say, the output of a system with impulse response h1(n) is the excitation for a s

25、ystem with impulse response h2(n).h1(n)h2(n)y(n)x(n)36Systems in cascadeThe output of the first system h1(n) is And the output of the second system h2(n) is 37Systems in cascadeBy exchange the summing sequence, we getThen we can express its output as38Systems in cascadeConclusion: Two linear time-in

26、variant systems in 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)39* Discrete-time systems stability（穩(wěn)定性）A system is referred to as bounded-input bounded-output (BIBO) stabl

27、e if, for every input 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, thenLinear shift invariant systems are stable if and only if （充要條件）40Example Characterize following system as being either linear or nonlinear, tim

28、e invariant or time varying, causal or noncausal, stable or not stable. Solution: For So it is not a linear system.41Cont.For so it is time varying; Because i.e. the output 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.

29、 t = n+2). So the system is noncausal; 42Cont.for a special bounded input 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. 432.5 Linear Constant-Coefficient Difference EquationsTh

30、e 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：指方程中各y(n-i)和x(n-l)項都只有一次冪，且不存在它們的相乘項。輸出序列y(n)變量序號的最高值與最低值之差N稱為差分方程的階數(shù)。System typeDescription of the systemthe continuous-time systemthe differential equation（微分方程）the discrete-tim

31、e systemthe difference equation（差分方程）(2.1)44IIR and FIR filtersEquation (2.1) can be rewritten, without loss of generality, considering that a0=1, yieldingSo 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 output y(n-1), y(n-2), , y(n-N)

32、. (2.2)45IIR and FIR filtersSince 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 the input signal. In such case, the system is called nonrecursive（非遞歸的）. It

33、 is(2.3)46IIR and FIR filtersIf 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 finite-duration impulse-response (FIR) （有限長單位沖激響應(yīng)）filters.In contrast, when y(n) depends on

34、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)（無限長單位沖激響應(yīng)）filters.47Review A sequence x(n) is defined to be periodic if and only if there is

35、 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 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. 48ReviewThe 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),th