Chapt2

Chapter 2. Discrete-Time Signals and Systems,Gao Xinbo School of E.E., Xidian Univ. X ,Main Contents,Important types of signals and their operations Linear and shift-invariant system Easier to analyze and implement The convolution and difference equation representations Representations and implementation of signal and systems using MATLAB,Discrete-time signals,Analog and discrete signals analog signal t represents any physical quantity, time in sec. Discrete signal: discrete-time signal N is integer valued, represents discrete instances in times,Discrete-time signal,In Matlab, a finite-duration sequence representation requires two vectors, and each for x and n. Example: Question: whether or not an arbitrary infinite-duration sequence can be represented in MATLAB?,Types of sequences,Elementary sequence for analysis purposes 1. Unit sample sequence Representation in MATLAB,Function x,n=impseq(n0,n1,n2),A: n=n1:n2; x = zeros(1,n2-n1+1); x(n0-n1+1)=1; B: n=n1:n2; x = (n-n0)=0; stem(n,x,ro);,2. Unit step sequence,A: 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 sequence,For Example:,n=0:10; x=(0.9).n; stem(n,x,ro),4. Complex-valued exponential sequence,Attenuation: 衰减因子 frequency in radians: For Example: n=0:10; x=exp(2+3j)*n);,5. Sinusoidal sequence,Phase in radians For Example: n=0:10; x=3*cos(0.1*pi*n+pi/3)+2*sin(0.5*pi*n),6. Random sequence,Rand(1,N) Generate a length N random sequence whose elements are uniformly distributed between 0,1 Randn(1,N) Generate a length N Gaussian random sequence with mean 0 and variance 1. en 0,1,7. Periodic sequence,A sequence x(n) is periodic if x(n)=x(n+N) The smallest integer N is called the fundamental period For example A: xtilde=x,x,x,x B: xtilde=x*ones(1,P); xtilde=xtilde(:); xtilde=xtilde; transposition,Operations on sequence,1. Signal addition Sample-by-sample addition x1(n)+x2(n)=x1(n)+x2(n),Function y,n=sigadd(x1,n1,x2,n2) n=min(min(n1),min(n2): max(max(n1),max(n2); y1=zeros(1,length(n); y2=y1; y1(find(n=min(n1) ,2. Signal multiplication,Sample-by-sample multiplication Dot multiplication x1(n).x2(n)=x1(n) x2(n),Function y,n=sigmult(x1,n1,x2,n2) n=min(min(n1),min(n2) : max(max(n1),max(n2); y1=zeros(1,length(n); y2=y1; y1(find(n=min(n1) ,3. Scaling,ax(n)=ax(n),5. folding,y(n)=x(n-k) m=n-k; y=x;,4. Shifting,y(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 power,Examples,Ex020100 composite basic sequences Ex020200 operation on sequences Ex020300 complex sequence generation Ex020400 even-odd decomposition,Some useful results,Unit sample synthesis Any arbitrary sequence can be synthesized as a weighted sum of 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),Function xe, x0, m = evenodd(x,n) If any(imag(x) = 0) error(x is not a real sequence); End m = -fliplr(n); m1 = min(m,n); m2 = max(m,n); m=m1:m2; nm = n(1)-m(1); n1 = 1:length(n); x1 = zeros(1, length(m); x1(n1+nm) = x; x = x1; xe = 0.5 * (x + flipflr(x); xo = 0.5*(x - fliplr(x);,The geometric series,A one-side exponential sequence of the form an, 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 series,Correlations of sequences,It 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, Crosscorrelation Autocorrelation,The index l is called the shift or lag parameter.,The special case: y(n)=x(n),Discrete Systems,Mathematically, an operation T. y(n) = T x(n) x(n): excitation, input signal y(n): response, output signal Classification Linear systems Nonlinear systems,Linear operation L.,Iff L. satisfies the principle of superposition The 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 response,Linear time-invariant (LTI) system,A linear system in which an input-output pair is invariant to a shift n in time is called a linear times-invariant system y(n) = Lx(n) - y(n-k) = Lx(n-k) The output of a LTI system is call a linear convolution sum An LTI system is completely characterized in the time domain by the impulse response h(n).,Properties of the LTI system,Stability A system is said to be bounded-input bounded-output (BIBO) stable if every bounded input produces a bounded output. Condition: absolutely summable To avoid building harmful systems or to avoid burnout or saturation in system operation,Properties of the LTI system,Causality A system is said to be causal if the output at index n0 depends only on the input up to and including the index n0 The output does not depend on the future values of the input Condition: h(n) = 0, n 0 Such a sequence is termed a causal sequence. To make sure that systems can be built.,Convolution,Convolution can be evaluated in many different ways If 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 version Matlab implementation Function y,ny=conv_m(x,nx,h,nh) nyb = nx(1)+nh(1); nye = nx(length(x)+nh(length(h); ny = nyb:nye; n = conv(x,h),Function form of convolution,Three different conditions under which u(n-k) can be evaluated: Case 1: n=9 % completely overlaps,Folded-and-shifted,Signals 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),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 revisited,The correlation can be computed using the conv function if sequences are of finite duration. Example 2.8 The meaning of the crosscorrelation This approach can be used in applications like radar signal processing in identifying and localizing targets.,Difference Equation,An LTI discrete system can also be described by a linear constant coefficient difference equation of the form If aN = 0, then the difference equation is of order N It describes a recursive approach for computing the current output,given the input values and previously computed output values.,Solution of difference equation,y(n) = yH(n) + yP(n) Homogeneous part: yH(n) Particular part: yP(n) Analytical approach using Z-transform will be discussed in Chapter 4 Numerical solution with Matlab y = filter(b,a,x) Example 2.9,Zero-input and Zero-state response,In 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 response,yZI(n): zero-in