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1 Introduction to Digital Signal Processing Test 1 Fall 2006 時間: 2006 年 11 月 7 日,140 分鐘 (2:30 – 4:50 PM) 答案卷上請標示清楚題號與頁碼,勿忘記寫上姓名學號。 Close Book 1. (5%) Let x[n] = [n+1]+2[n]+ 3[ n1]+ 2[ n2]+ [ n3], and y[n] = 2[n+1]+[n]5[ n1] [ n2]+ 3[ n3]. Find their convolution x[n]y[n]. 2. To prove the property that time-domain convolution implies z-domain multiplication, let us consider x[n] and y[n] ( n ) that are two discrete-time signals. Their z-transforms are as follows: X(z) = … + z3x[3] + z2x[2] + zx[1] + x[0] + z1x[1] + z2x[2] + z3x[3] + … Y(z) = … + z3y[3] + z2y[2] + zy[1] + y[0] + z1y[1] + z2y[2] + z3y[3] + … The multiplication of X(z) and Y(z) results in a new polynomial W(z) = X(z)Y(z). Let w[n] be the coefficients of this polynomial. W(z) =… + z3w[3] + z2w[2] + zw[1] + w[0] + z1w[1] + z2w[2] + z3w[3] +… (a) (5%) Write down w[1], w[0], w[1], and w[2] by multiplying X(z) and Y(z). (b) (5%) Show that, in general, w[n] is the convolution of x[n] and y[n]. 3. Suppose that three LTI systems are connected in cascade; i.e., the output of S1 is the input of S2, and the output of S2 is the input of S3. The three systems are specified as follows: S1: y1[n] = x1[n] x1[n1], S2: y2[n] = x2[n] 2x2[n1] + x2[n2], S3: y3[n] = x3[n1] + x3[n2], where the output of Si is yi[n] and its input is xi[n]. (a) (10%) Consider the equivalent system that is a single operation from the input x[n] (into S1) to the output y[n] (the output of S3). Thus, x[n] is x1[n] and y[n] is y3[n]. Write down the system function (in z-transform) of the equivalent system. (b) (5%) What is the region of convergence (ROC) of the system function (except (c) (d) (e) (f) (g) possible z = 0 and z = )? (5%) Write down the impulse response of this system. (5%) Is this system FIR or IIR? Explain your answer. (5%) Write down the frequency response of this system. (5%) Is this a causal system? Explain your answer. (5%) Is this system stable? Explain your answer. 2 4. An LTI system is described by the difference equation y[n] = 0.8y[n1] 0.8x[n] + x[n1] (a) (5%) Determine the system function H(z) for this system. (b) (5%) Plot the poles, zeros, and ROC of H(z) on the z-plane. (c) (5%) Is the system stable? (d) (5%) If the input is x[n] = cos[(/4)n], calculate the DTFT Y(ejw) of the output. (hint: decompose cos[(/4)n] into eigenfunctions of an LTI system). (e) (5%) If the input is x[n] = 0.5nu[n], calculate the output y[n] by using the inverse z-transform. 5. Consider the DFT and IDFT that are defined as the following transform pair: 1 N 1 forward DFT: Y [k ] x[n]WNkn , k 0,1..., N 1 N n 0 inverse DFT (IDFT): x[n] 1 N 1 Y [k ]W N k 0 kn N , n 0,1..., N 1 where WN = ej2/N. (a) (5%) Let x be the N-dimensional vector consisting of the time domain sequence, x = [x[0], x[1], …, x[N1]]T and Y be the vector consisting of the frequency components, Y = [Y[0], Y[1], …, Y[N1]]T (where T is the matrix transpose). We can represent the forward transform as Y = Wx, where W is an NN matrix. Similarly, we can also represent the inverse transform as x = AY, where A is an NN matrix. Question: Please write down the NN matrices W and A. (b) (3%) Show that A = W*T, where W* is the matrix obtained by replacing every element in W by its complex conjugate. (c) (7%) The matrix WH W*T is usually called the conjugate-transpose matrix of W. A matrix W satisfying WHW = I is called a unitary matrix, where I is the identity matrix, 1 0 0 0 1 0 I 0 0 1 Question: Prove that W is a unitary matrix by showing that the multiplication of A and W is identity, i.e., I = AW = WHW (or equivalently, A=W1= WH). (hint: You may need to use the sum of a geometric sequence, 1 + r + r2 + … + rN1 = (1 rN)/(1 r).) (continue) 3 (d) (5%) Show that it also holds that WWH = I, and use this property to prove the Parseval theorem ||Y||2 = ||x||2, i.e., the lengths of the vectors Y and x are the same. (hint: You may need to use some properties about conjugate-transpose matrix: Let v be a vector with complex values, then ||v||2 = vHv. Given two squared matrices B and C, (BC)H = CHBH and (BH) H = B).