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6.003 Homework #5 Due at the beginning of recitation on October 12, 2011. Problems 1. DT convolution Let y represent the DT signal that results when f is convolved with g, i.e., y[n] = (f ∗ g)[n] which is sometimes written as y[n] = f [n] ∗ g[n]. Determine closed-form expressions for each of the following: y[n] f [n] ya [n] u[n] yb [n] yc [n] yd [n] u[n] 1 n u[n] 2 n 1 u[n] 2 g[n] u[n] 1 n u[n] 2 n 1 u[n] 3 n 1 u[n] 2 Enter expressions for yi [n] for n ≥ 0 and for n < 0 in the boxes below. ya [n] for n ≥ 0 : ya [n] for n < 0 : yb [n] for n ≥ 0 : yb [n] for n < 0 : yc [n] for n ≥ 0 : yc [n] for n < 0 : yd [n] for n ≥ 0 : yd [n] for n < 0 : 6.003 Homework #5 / Fall 2011 2 2. CT convolution Let y represent the CT signal that results when f is convolved with g, i.e., y(t) = (f ∗ g)(t) which is sometimes written as y(t) = f (t) ∗ g(t). Determine closed-form expressions for each of the following: y(t) f (t) g(t) ya (t) u(t) u(t) yb (t) u(t) e−t u(t) yc (t) e−t u(t) e−2t u(t) yd (t) e−t u(t) e−t u(t) Enter expressions for the non-zero parts of yi (t) and the range of t for which the expression apply in the boxes below. ya (t) for t ≥ 0 : ya (t) for t < 0 : yb (t) for t ≥ 0 : yb (t) for t < 0 : yc (t) for t ≥ 0 : yc (t) for t < 0 : yd (t) for t ≥ 0 : yd (t) for t < 0 : 6.003 Homework #5 / Fall 2011 3 3. Convolutions Consider the convolution of two of the following signals. a(t) 1 b(t) 1 c(t) 1 1 2 1 2 1 2 t t 1 t 1 1 Determine if each of the following signals can be constructed by convolving (a or b or c) with (a or b or c). If it can, indicate which signals should be convolved. If it cannot, put an X in both boxes. Notice that there are ten possible answers: (a ∗ a), (a ∗ b), (a ∗ c), (b ∗ a), (b ∗ b), (b ∗ c), (c ∗ a), (c ∗ b), (c ∗ c), or (X,X). Notice also that the answer may not be unique. ∗ a or b or c or X = t a or b or c or X ∗ a or b or c or X = a or b or c or X = a or b or c or X = a or b or c or X = = a or b or c or X 2 1 2 1 2 1 2 1 2 t 1 4 t 1 4 t a or b or c or X ∗ 1 1 4 a or b or c or X ∗ 2 t a or b or c or X ∗ 1 1 4 a or b or c or X ∗ a or b or c or X 1 4 1 4 t 6.003 Homework #5 / Fall 2011 4 Engineering Design Problems 4. The L operator The R (right-shift) operator can be used to express functional forms for any block diagram that is composed of adders, gains, and delays. Since the delay operation is causal (i.e., its output does not depend on future values of its input) functionals that build on R can only represent causal systems. Deﬁne the L (left-shift) operator to represent “anticipators," which are the anti-causal counterparts to delay elements. If X is an input sequence, then Y = LX means that y[n] = x[n + 1] for all n. Let H1 represent the system described by the following functional: H1 = Y 1 = . X 1 − 3L a. Determine a diﬀerence equation for this system. b. Find the unit-sample response of this system. c. Determine the pole(s) of this system. d. If a system is represented as the ratio of polynomials in R, then the poles of the system are the roots of the denominator polynomial after R is replaced by z1 . Determine an analogous rule for systems represented by a ratio of polynomials in L. Consider a system that is described by the following unit-sample response: |n| 1 for all n . h2 [n] = 2 e. Determine a closed-form functional representation for the system described by h2 [n]. f. Determine the poles of the system in part e. g. Do a series expansion of the system function in part e (it may be helpful to use partial fractions). Explain the relation between the series and the unit-sample response h2 [n]. 6.003 Homework #5 / Fall 2011 5 5. Upsampling One way to enlarge an image is to upsample by linear interpolation, as follows. a. Consider a one-dimensional signal x[n] whose non-zero samples are indexed from 0 to N (shown here for N = 3). x[n] 0 1 2 3 4 5 n First, make a new signal w[n] deﬁned by w[n] = x[ n3 ] n = 0, 3, 6, . . . , 3N 0 otherwise . w[n] 0 1 2 3 4 5 6 7 8 9 n Next, convolve w[n] with f [n] to produce y[n] = (w ∗ f )[n]. y[n] = (w ∗ f )[n] 0 1 2 3 4 5 6 7 8 9 n Determine f [n]. b. Extend the linear interpolation method to two dimensions. Write a program that uses this method to enlarge the following image from its current dimensions (212 × 216) to three times those dimensions. Compare the result of linear interpolation with the result of replicating each pixel value 3 × 3 times (see appendix). The digital form of this image is available on the 6.003 website. The appendix contains sample Python code. 6.003 Homework #5 / Fall 2011 6 6. Smiley Consider the sequence of 1’s and −1’s shown below as x[n]. x[n] 1 n 0 −1 In this x[n], there is a single occurrence of the pattern −1, −1, 1. It occurs starting at n = 1 and ending at n = 3. One method to automatically locate particular patterns of this type is called “matched ﬁltering.” Let p[n] represent the pattern of interest ﬂipped about n = 0. Then instances of the pattern can be found by ﬁnding the times when y[n] = (p ∗ x)[n] is maximized. a. Determine a matched ﬁlter p[n] that will ﬁnd occurrences of the sequence: −1, −1, 1. Design p[n] so that (p ∗ x)[n] has maxima at points that are centered on the desired pattern, i.e., at n = 2 for the sequence above. The same approach can be used to ﬁnd patterns in pictures by generalizing the convolu­ tion operator to two dimensions: y[n, m] = (x ∗ p)[n, m] = ∞ 0 ∞ 0 x[k, l]p[n − k, m − l] . k=−∞ l=−∞ A ﬁle called findsmiley.jpg (which can be downloaded from the 6.003 website) contains a random pattern of white pixels (coded as 255) and black pixels (0) as well as a single instance of the following smiley face: b. Find the row and column of findsmiley that corresponds to smiley’s nose. Note: matched ﬁltering will work best if matching white pixels AND matching black pixels contribute positively to the answer. For that reason, 255 and 0 may not be the optimum choices for the values associated with white and black pixels. c. One advantage of the matched ﬁlter method is that it works even when the signal contains some noise. One can make a noisy version of findsmiley using Python’s gauss function: import random ... pix[i,j] += random.gauss(0,dev) where dev represents the standard deviation of the added noise. Determine the largest value of dev for which your code still ﬁnds smiley. 6.003 Homework #5 / Fall 2011 7 Appendix Python program to increase the size of the zebra image by replicating each pixel 3 × 3 times. import Image as im xx = im.open(’zebra.jpg’) yy = im.new(’L’,[3*xx.size[0],3*xx.size[1]],0) xpix = xx.load() ypix = yy.load() for j in range(yy.size[1]): for i in range(yy.size[0]): ypix[i,j] = xpix[int(i/3),int(j/3)] yy.save(’zebraZOH.jpg’) # # # # Python Imaging Library open input picture create new picture 3x array-type access # loop over rows # loop over columns # copy pixels # create output file The functionality of this code was veriﬁed on the Athena* linux workstations. * Athena is MIT's UNIX-based computing environment. OCW does not provide access to it. MIT OpenCourseWare http://ocw.mit.edu 6.003 Signals and Systems Fall 2011 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.