Problem 1. Suppose we calculate the response of an LTI system to an input signal x(n), using the convolution sum:

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1 EE 438 Homework 4. Corrections in Problems 2(a)(iii) and (iv) and Problem 3(c): Sunday, 9/9, 10pm. EW DUE DATE: Monday, Sept 17 at 5pm (you see, that suggestion box does work!) Problem 1. Suppose we calculate the response of an LTI system to an input signal x(n), using the convolution sum: y(n) = + k= x(k)h(n k), where h(n) is the impulse response of the system. Suppose that both x(n) and h(n) are finiteduration signals: the duration of x(n) is samples, and the duration of h(n) ism samples. How many multiplications and how many additions will it take to calculate y(n) for <n<+? (Your answer should be an expression involving and M.) Problem 2. (a) Consider a DT LTI system with a known frequency response H(e jω ). It was shown in class that the response of such a system to an everlasting complex exponential input signal g(n) =e jωn, <n<+, is: y(n) =g(n)h(e jω ), for <n<+. (1) (In other words, e jωn is an eigenfunction of the system, with eigenvalue H(e jω ).) (i) Does Eq. (1) hold for sinusoidal inputs? In other words, suppose that the input to a DT LTI system is cos(ωn + φ), for < n < +. Is it always the case that the output is cos(ωn + φ)h(e jω ), for <n<+? (ii) Use the above property (1) to derive a simple expression for the response of the system to the following input signal: 2 cos(8n + π/4). (iii) Please disregard what was here previously. Instead, do the following. Answer question (i) for DT LTI systems with even frequency responses, i.e., for systems with H(e jω )=H(e jω ). (iv) Please disregard what was here previously. Instead, do the following. Consider a DT LTI system whose frequency response is an even function. Its frequency response for 0 ω π is: H(e jω )= 1+ae j2ω +2e j4ω 1+be j2ω. Determine the constants a and b, given the two input/output pairs below: when the input is cos(πn/2), the output is cos(πn/2); when the input is sin(πn), the output is 2 sin(πn). 1

2 (b) For each of the two discrete-time systems S 1,S 2 below, the indicated input/output pair represents the results of one experiment with the corresponding system. Decide whether the output y(n) and input x(n) of this system definitely cannot, possibly could, or must satisfy a convolution relationship of the form y(n) = x h(n) for some appropriate impulse response h(n). Choose the statement that applies, and explain your reasoning. If you answer is possibly could or must, determine an impulse response h(n) and frequency response H(e jω ) that would account for the given input/output pair. (i) The response of system S 1 to input e j(π/2)n is 4e j(3π/4)n. (ii) The response of system S 2 to input e j(π/4)n is 4e j(9π/4)(n 1). Problem 3. Consider the following DT complex exponential functions: ( ) j2π(k 1)(n 1) s k (n) = exp, n =1,...,; k =1,...,. In other words, there are functions, s 1 (n),s 2 (n),...,s (n), and each of them is defined for n = 1, 2,...,. As usual, we are identifying each of these functions with a vector: ( ( ) ( ) ( )) j2π(k 1) 0 j2π(k 1) 1 j2π(k 1)( 1) T s k = exp, exp,...,exp. This problem will help you get a feel for what these functions are. (Do not use Matlab in this problem.) (a) Let =3. (i) Draw the three numbers s 1 (1), s 1 (2), and s 1 (3), as points in the complex plane. (ii) Repeat (i) for s 2. (iii) Repeat (i) for s 3. (iv) Calculate s 1, s 3 = s 1 (1)s 3 (1) + s 1(2)s 3 (2) + s 1(3)s 3 (3). (v) Calculate s 2, s 3. (b) Let =4. (i) Draw the four numbers s 1 (1), s 1 (2), s 1 (3), and s 1 (4), as points in the complex plane. (ii) Repeat (i) for s 2. (iii) Repeat (i) for s 3. 2

3 Figure 1: Illustration to Problem 3(g): a regular pentagon inscribed in a unit circle. (iv) Repeat (i) for s 4. (v) Calculate s 2, s 4. (c) Show that your results from Part (a),(iv) and (v), and Part (b),(v), can be generalized as follows: (Hint. s k, s m = = if k m, then s k, s m =0, for any. s k (n)s m(n) ( j2π(k 1)(n 1) exp ) ( exp ow simplify and use the geometric series formula with q = exp (d) Find s k (n). (e) Find s k 2. (The answer will be an expression involving.) ) j2π(m 1)(n 1) ( ) j2π(k m).) (f) Let = 24, and let s(n) = 438 exp (j(πn/ )), for n =1, 2,...,24. Compute the coefficients a 1,a 2,...,a 24 in the expansion: 24 s = a k s k, where s =(s(1),...,s(24)) T, and s k are the complex exponentials defined above. (This expansion is known as a DT Fourier series.) (g) Consider a regular -gon (polygon with equal sides), inscribed in a unit circle. (This is shown in Fig. 1, for = 5.) Find the sum of squared lengths of all diagonals and sides of the -gon. 3

4 Hints. 1. A side of a polygon is a segment connecting two adjacent vertices. 2. A diagonal of a polygon is a segment connecting two non-adjacent vertices. 3. Your answer should be an expression involving. 4. Consider all sides and diagonals which share one particular vertex. Use your observations from Parts (a) and (b) to relate the sum of their squared lengths to the energy of s 1 s 2. x(n) delay by 1 sample delay by 1 sample multiply by C multiply by B multiply by A + y(n) Problem 4. (a) Find a difference equation that describes this system i.e. relates the output of the overall system to its input. (A, B, and C are fixed constants.) (b) Find the frequency response of this system using three methods: (i) from the difference equation; (ii) by calculating the impulse response and finding its DTFT; (iii) by calculating the response to a complex exponential. Verify that all three methods result in the same answer. (c) Suppose that A = B = C = 1, and x(n) = 5, for <n<+. Calculate y(n) using: (i) the difference equation; (ii) the frequency response. Problem 5. Prove the following properties of the inner product and norm in C : (a) g, s = s, g. (b) a 1 s 1 + a 2 s 2, g = a 1 s 1, g + a 2 s 2, g. (c) s,a 1 g 1 + a 2 g 2 = a 1 s, g 1 + a 2 s, g 2. (d) as 2 = a s 2. 4

5 (Boldface letters denote vectors; a, a 1, and a 2 are complex numbers.) Problem 6. A geometric view of linear regression. (a) Pythagoras s theorem in C. Prove Pythagoras s theorem: the sum of energies of two orthogonal vectors is equal to the energy of their sum, i.e., if s, g =0, then s 2 + g 2 = s + g 2. ote that this result generalizes the following 2-D picture: Area = energy of s+g Area = energy of s s s+g g Area = energy of g Figure 2: The sum of squares of two sides of a right triangle is equal to the square of the hypotenuse. (Hint. Use the fact that s + g 2 = s + g, s + g. Write this out using properties of inner products derived in class; then use orthogonality of s and g to cancel some terms.) (b) Projections in C. Suppose that G is a vector subspace of C ; let y C be a vector which does not necessarily belong to G; and let y G be its orthogonal projection onto G. Show that the closest vector in G to y is y G. In other words, show that y y G = min y f f G Hint. Draw a picture of y, its projection y G, and some other vector f G. ow use Pythagoras s theorem from Part (a) to show that y y G 2 y f 2. (c) System identification using linear regression. Consider a discrete-time multi-input single-output memoryless system which takes m input signals x 1,x 2,...,x m and calculates their weighted sum to generate an output y, i.e. m y(n) = w k x k (n) (2) 5

6 In (2), the weights w 1,w 2,...,w m are unknown real numbers. The inputs as well as measurements of the output for n =1, 2,..., are available (here, is some fixed positive integer which we assume to be greater than m). We assume the input to be pairwise orthogonal, i.e. x k (n)x p(n) =0fork p. The goal of this exercise is to estimate the weights w 1,w 2,...,w m from these inputs and the observations of the output. In practice, the measurement noise often makes it impossible to find the weights that would exactly satisfy Eq. (2) for the given inputs and the observed output. Instead, your task is to find the weights which minimize the energy of the difference between the two sides of Eq. (2): find w 1,w 2,...,w m to minimize m y(n) w k x k (n) 2. (3) Hints. Write the input and output signals as -dimensional vectors, to express the above energy as m y 2 w k x k. ow use Part (b) and the results on orthogonal projections derived in class. 6