The Cooper Union Department of Electrical Engineering ECE111 Signal Processing & Systems Analysis Final May 4, 2012 Time: 3

The Cooper Union Department of Electrical Engineering ECE111 Signal Processing & Systems Analysis Final May 4, 2012

Time: 3 hours. Close book, closed notes. No calculators. Part I: ANSWER ALL PARTS. WRITE YOUR ANSWERS IN THE SPACE PROVIDED HERE, NOT IN YOUR EXAM BOOKS. TOTAL: 30 points. Any errors will cause disproportionate penalty. IA: FORMULAS

1. Laplace transform.

2. z-transform.

3. Continuous-time convolution.

4. Discrete-time convolution.

5. CTFT.

1 6. Inverse CTFT.

7. DTFT.

8. Inverse DTFT.

9. DFT.

10. Inverse DFT.

2 IB: Basic Concepts

1. An analog transistor ampliﬁer has simple poles at 2, 3 4j and a double pole at 1; also zeros at 4 j and at +3. True or False− − (no± justiﬁcation): The system− is stable. ______− ±

2. A digital ﬁlter realized as a cascade of several direct form II transposed sections has 1 jπ/3 2 1 j2π/3 2 jπ/4 poles at 2 e± , 3 j, and double poles at 3 e± . It also has zeros at 3 e± , 2, ± 1 jπ/2 − 3, and double zeros at 2 e± . True or False (no justiﬁcation): The system is −stable. ______

3. BIBO stands for (do not deﬁne it; write what the acronym stands for):

3t 4. The Laplace transform of 3δ (t) + 4e− u (t) is (no need to simplify):

5. The discrete-time unit step u [n] is deﬁned as:

6. True or False (no justiﬁcation needed): A transversal ﬁlter is always FIR (can not be IIR). ______

7. An LTI discrete-time system is FIR if its ______response has ﬁnite ______, and is IIR if its ______response has in- ﬁnite ______.

3 IC: Digital Filter Structures Draw the ﬁlters on THIS PAGE in the space provided below.

1. Draw a transversal ﬁlter realization for:

1 2 3 H (z) = 4 + 3z− + 2z− 6z− − 2. Draw a direct-form II realization for: 4z2 + 2z + 1 H (z) = 2z2 4z 3 − −

4 Part II: Answer all parts. Points per problem indicated. Partial credit will be granted “proportionately” when warranted.

1. [8 pts.] Consider the following analog transfer function: s + 2 H (s) = K (s + 0.5) (s + 10)2

First select K so that the DC gain is 20dB. Using that value for K: graph the Bode magnitude plot. Speciﬁcally, graph the straight-line asymptotes, determine corrections and sketch the actual curve. Be sure to label your diagram carefully.

2. [3 pts.] Given the following analog transfer function:

s (s + 30) H (s) = (s + 10)2 (s + 100) (s + 200)

Specify the phase (in degrees) at DC and at . ∞ 3. [6 pts.] Given the following transfer function of an analog system:

s + K s2 + (4 + K) s + 9

where K is a variable gain parameter. It is real, but may be positive, negative, or zero. Assume K is restricted, as necessary, so that the system is stable.

(a) Compute the natural frequency ωn and damping factor ζ, in terms of K. (b) Specify the condition on K for the system to be stable. (Remember, do not reject negative values unless you have to!) (c) For an underdamped system, specify the constraint on ζ and ALSO the corre- sponding constraint on K. Repeat for critically damped, and overdamped cases.

4. [3 pts.] Draw a sketch of the “typical”unit step responses for an underdamped system, and for an overdamped system. In particular, the rise time must be shown, correctly, to be less in one case than the other. Nevertheless, an engineer may select the system with the slower rise time if the preference is to have a system that “behaves” faster- BRIEFLY indicate what this “means”by pointing out features on the sketches.

5 5. [5 pts.] An oscillator circuit exhibits a spectral peak at 10MHz. The spectrum is 3dB below peak at 8MHz and 12MHz. In the following, you can give explicit formulas for the answers, with numbers ready to be plugged into a calculator.

(a) Specify the center frequency, bandwidth and Q. (b) Did you compute Q using an exact or approximate definition? DON’Twrite out the other definition (the one you didn’tuse). (c) Do you have enough information to know if this is a second order circuit or not? What do you base your answer on? (d) Based on your answer to (c), for this case, do the exact and approximate defini- tions of Q give the exact same answer?

6. [10 pts.] Let H (s) be a 2 3 transfer function matrix of an analog system, with state-space realization A, B,× C, D . The eigenvalues of A are 2 with multiplicity 3, and 1 j4 (each simple).{ } − − ± (a) How many inputs, outputs and state variables are there? (b) Is the system internally stable? Is it externally stable? For each case, the answer is yes, no, or not enough information to be certain. No justiﬁcation needed. (c) The matrix that relates the state at time t to the state at time 0, under condition of ______, is called the ______matrix (two words). In terms of A, this matrix is ______. The Laplace transform of this matrix is ______. (d) Referring to the same matrix as in part (c) above: in this case, based on the known information about A, write an explicit formula (in the time domain) for a typical entry, in terms of arbitrary constants. I intend for you to use modal analysis here; I don’tmean set up the computation via matrix functional calculus. (e) Can H (s) have a pole at 4? Can H (s) NOT have a pole at 2? Which of these would be an example− of a hidden pole? −

(f) Suppose we know that this is a minimal realization. Let A0,B0,C0,D0 be any other realization (may or may not be minimal). What is known{ of the dimensions} of A0,B0,C0,D0?

(g) The H23 (s) element of H (s) is the transfer function from the _____ input to the ______output.

6 7. [6 pts.] Let H (z) be the transfer function matrix of a digital system, with state-space realization A, B, C, D . { } (a) Write the state space equations, with ~u [n] input vector, ~y [n] output vector and ~x [n] state vector. (b) Write the formula for H (z). (c) We can write x [n] =?x [0] (ﬁll in the matrix) under the condition that ______. (d) Regarding part (c), x [n] 0 for all initial conditions x [0] if the ______of A are ______(specify→ a region of the complex plane).

8. [6 pts.] Let A be a 4 4 matrix. We want to determine a formula for An, n 0, using matrix functional× calculus. ≥

n M (a) We compute A by first finding a polynomial r (λ) = rM λ + + r1λ + r0, and then making a particular substitution for λ. What should be M···? (b) True or false (no justification): knowing the eigenvalues of A (with multiplicities), not the particular entries of A, is suffi cient to find the rk’s.

9. [8 pts.] The complex-valued WSS random process x [n] satisﬁes the condition that: v [n] = x [n] 0.2x [n 1] + 0.3x [n 2] + 0.4v [n 1] + 0.1v [n 2] (1) − − − − − 2 results in white noise v [n] with σv = 3. Assume x and therefore v are 0-mean for all time.

(a) Write an explicit formula for the PSD Sx (ω) of x. (b) This filter, with x input and v output, is called the ______filter of x. The inverse is called the ______filter of x. (c) Actually, in parts (b), to be technically correct you are assuming the filter in (1) has a particular property, named ______(two words). This requires (only the poles / only the zeros / both poles and zeros) of (1) to be inside the unit circle. Don’t actually check this! Just tell me what WOULD need to be checked. (d) v is called the ______signal of x. (e) Extra credit (10 pts for answering all parts):

1. You should be able to determine the values of E (x [n 1] v∗ [n]) and E (x [n 2] v∗ [n]) easily. What are they, and (brieﬂy) why? (Clear answer− in words is ﬁne) −

2. Use (1) and (i) to compute E (x [n] v∗ [n]). 3. Verify that x [n] v [n] v [n]. − ⊥

10. [3 pts.] Let x (t) be WSS with correlation r (τ). Assuming y (t) = x0 (t) exists, 00 show that ryy (τ) = rxx (τ). Hint: Use the Wiener-Khinchin theorem and invoke a particular property− of the Fourier transform you should know (no, don’t write down the Fourier transform integral).

7 11. [4 pts.] In each case, specify all conditions that apply. If none apply, say so. No justiﬁcation.

(a) x (t) is Gaussian WSS and y (t) = x2 (t). The output is (WSS/SSS/Gaussian). t (b) x (t) is Gaussian WSS and y (t) = e− x (t). The output is (WSS/SSS/Gaussian). (c) x [n] is WSS and y [n] = Q [x [n]] where Q [ ] is a quantizer that rounds x to the nearest integer. Then y [n] is (WSS/SSS). · (d) x [n] is SSS and y [n] = Q [x [n]] where Q [ ] is a quantizer that applies saturation overﬂow with a ﬁxed level A: Q [α] = A· if a A, and Q [α] = α if α < A. Then y [n] is (WSS/SSS). | | ≥ | |

12. [3 pts.] Write what the following abbreviations stand for (deﬁnitions not needed):

(a) PSD (b) WSS (c) SSS

X 2 13. [5 pts.] Consider the real random vector . The mean vector is µ = and Y 1 the correlation matrix is (partially) determined as:

8 ? R = ? 5 (a) Specify the unknown entries that would GUARANTEE that X,Y are orthogonal. If it is not possible to force this just by specifying these values, say so. But no justiﬁcation needed. (b) Specify the unknown entries that would GUARANTEE that X,Y are uncorre- lated. As above, if you can’tforce this condition, say so. (c) Specify the unknown entries that would GUARANTEE that X,Y are indepen- dent. Same as above if you can’t. (d) Suppose you know for certain that X,Y are jointly Gaussian. NOW repeat (a),(b),(c): DO ANY OF YOUR ANSWERS CHANGE? HOW? (e) Now suppose the unknown entries are GIVEN to you. Not necessarily any of the cases above, just “generic”values. Suppose you know X and Y are EACH Gaussian (that’s all you know). Do you have enough information now to write down the pdf for 3X + 4Y ? No justiﬁcation needed.