数字信号处理与DSP器件：Discrete-Time Signals and Systems

1、Chapter 3-4. Discrete-Time Signals and SystemsMain ContentsImportant types of signals and their operationsLinear and shift-invariant systemEasier to analyze and implementThe convolution and difference equation representationsRepresentations and implementation of signal and systems using MATLABDiscre

2、te-time signalsAnalog and discrete signalsanalog signal t represents any physical quantity, time in sec.Discrete signal: discrete-time signalN is integer valued, represents discrete instances in timesDiscrete-time signalIn Matlab, a finite-duration sequence representation requires two vectors, and e

3、ach for x and n.Example:Question: whether or not an arbitrary infinite-duration sequence can be represented in MATLAB?Types of sequencesElementary sequence for analysis purposes1. Unit sample sequenceRepresentation in MATLABFunction x,n=impseq(n0,n1,n2)A: n=n1:n2; x = zeros(1,n2-n1+1); x(n0-n1+1)=1;

4、B: n=n1:n2; x = (n-n0)=0; stem(n,x,ro);Program P1_12. Unit step sequenceA: n=n1:n2; x=zeros(1,n2-n2+1); x(n0-n1+1:end)=1;B: n=n1:n2; x=(n-n0)=0;3. Real-valued exponential sequenceFor Example:n=0:10; x=(0.9).n; stem(n,x,ro);Program P1_3MATLAB中数组、矩阵基本运算符 运算 符号 举例加法，a+b + 1 2+3 4 1,2+3减法，a-b - 1 2-3 4

5、1,2-3乘法，a*b * 1,2*3 1,2*3,4 a.*b .* 1,2.*3,4=3,8逆乘，左乘 ax=b x=ab=inv(a)*b 右乘 / xa=b x=b/a=b*inv(a)乘幂，方阵的幂 a2=a*a 元素的幂 . a.2 x.34. Complex-valued exponential sequenceAttenuation: 衰减因子frequency in radians:For Example: n=0:10; x=exp(2+3j)*n);5. Sinusoidal sequencePhase in radiansFor Example:n=0:10; x=3*

6、cos(0.1*pi*n+pi/3)+2*sin(0.5*pi*n)6. Random sequenceRand(1,N)Generate a length N random sequence whose elements are uniformly distributed between 0,1Randn(1,N)Generate a length N Gaussian random sequence with mean 0 and variance 1. en 0,17. Periodic sequenceA sequence x(n) is periodic if x(n)=x(n+N)

7、The smallest integer N is called the fundamental periodFor exampleA: xtilde=x,x,x,xB: xtilde=x*ones(1,P); xtilde=xtilde(:); xtilde=xtilde; transpositionOperations on sequence1. Signal additionSample-by-sample additionx1(n)+x2(n)=x1(n)+x2(n)3. Scalingax(n)=ax(n)5. foldingy(n)=x(n-k)m=n-k; y=x;4. Shif

8、tingy(n)=x(-n)y=fliplr(x); n=-fliplr(n);6. Sample summation ss = sum(x(n1:n2);7. Sample production sp = prod(x(n1:n2);8. Signal energy se = sum(x .* conj(x); or se = sum(abs(x) . 2);9. Signal powerSome useful resultsUnit sample synthesisAny arbitrary sequence can be synthesized as a weighted sum of

9、delayed and scaled unit sample sequence. Even and odd synthesis Even (symmetric): xe(-n)=xe(n)Odd (antisymmetric): xo(-n)=-xo(n)Any arbitrary real-valued sequence can be decomposed into its even and odd component: x (n)=xe(n)+ xo(n) The geometric seriesA one-side exponential sequence of the form an,

10、 n=0, where a is an arbitrary constant, is called a geometric series.Expression for the sum of any finite number of terms of the seriesCorrelations of sequencesIt is a measure of the degree to which two sequences are similar. Given two real-valued sequences x(n) and y(n) of finite energy,Crosscorrel

11、ationAutocorrelationThe index l is called the shift or lag parameter.The special case: y(n)=x(n)Voice recognitionCrosscorrelation outputDiscrete SystemsMathematically, an operation T.y(n) = T x(n)x(n): excitation, input signaly(n): response, output signalClassificationLinear systemsNonlinear systems

12、Linear operation L.Iff L. satisfies the principle of superpositionThe output y(n) of a linear system to an arbitrary input x(n) is called impulse response, and is denoted by h(n,k)h(n,k): the time-varying impulse responseLinear time-invariant (LTI) systemA linear system in which an input-output pair

13、 is invariant to a shift n in time is called a linear times-invariant systemy(n) = Lx(n) - y(n-k) = Lx(n-k)The output of a LTI system is call a linear convolution sumAn LTI system is completely characterized in the time domain by the impulse response h(n).Properties of the LTI systemStabilityA syste

14、m is said to be bounded-input bounded-output (BIBO) stable if every bounded input produces a bounded output.Condition: absolutely summableTo avoid building harmful systems or to avoid burnout or saturation in system operationProperties of the LTI systemCausalityA system is said to be causal if the o

15、utput at index n0 depends only on the input up to and including the index n0The output does not depend on the future values of the inputCondition: h(n) = 0, n 0Such a sequence is termed a causal sequence.To make sure that systems can be built. ConvolutionConvolution can be evaluated in many differen

16、t waysIf the sequences are mathematical functions, then we can analytically evaluate x(n)*h(n) for all n to obtain a functional form of y(n)Graphical interpretation, folded-and-shifted versionFunction form of convolutionThree different conditions under which u(n-k) can be evaluated:Case 1: n0 % the

17、nonzero values of x(n)and y(n) do not overlap.Case 2: 0=n=9 % completely overlapsFolded-and-shiftedSignals x=x(1),x(2),x(3),x(4),x(5)System Impulse Response: h=h(1),h(2)h(3),h(4)y=conv(x,h)y(1)=x(1)*h(1); y(2)=x(1)*h(2)+x(2)*h(1)y(3)=x(1)*h(3)+x(2)*h(2)+x(3)*h(1); x(1),x(2),x(3),x(4),x(5) h(4),h(3),

18、h(2),h(1)Note that the resulting sequence y(n) has a longer length than both the x(n) and h(n) sequence.Sequence correlations revisitedThe correlation can be computed using the conv function if sequences are of finite duration.Example 2.8The meaning of the crosscorrelationThis approach can be used i

19、n applications like radar signal processing in identifying and localizing targets.Difference EquationAn LTI discrete system can also be described by a linear constant coefficient difference equation of the formIf aN = 0, then the difference equation is of order NIt describes a recursive approach for

20、 computing the current output,given the input values and previously computed output values.Solution of difference equationy(n) = yH(n) + yP(n)Homogeneous part: yH(n)Particular part: yP(n)Analytical approach using Z-transform will be discussed in the funtureNumerical solution with Matlaby = filter(b,

21、a,x)Zero-input and Zero-state responseIn DSP the difference equation is generally solved forward in time from n=0. Therefore initial conditions on x(n) and y(n) are necessary to determine the output for n=0.Subject to the initial conditions:Solution:Zero-input and Zero-state responseyZI(n): zero-input solutionA solution due to the ini