ECE210 Homework 3- indexing and functions Solved

This homework reviews indexing and functions as covered in class.

1.    Let’s look at images! This question uses imshow (see the help page!) and logical indexing to produce some simple shapes and show how we can use them as masks for image manipulation.

(a)    Create the logical matrix A ∈ M256×256(F2) where aij is true iff

 p(i − 128)2− (j − 128)2 < 64. (Use meshgrid or broadcasting.)

(F2, or Galois-field 2, is the field consisting of only the values {0,1} and whose operations are akin to logical operators. For our purposes, these are the two logical values corresponding to false and true.)

(b)    Create the logical matrix B ∈ M256×256(F2) where bij is true iff

 p(i − 96)2− (j − 96)2 < 64.

(c)    Create the following logical matrices. For each one, use figure and imshow to visualize it. Briefly describe each one in a comment.

i. A ii. B

iii. C = A ∩ BC iv. D = AC ∪ B

(d) Given the following matrices, and visualize them using figure and imshow. What do the .* and + operators do when dealing with logical arrays (“layer masks”)?

E     = rand(256);

F     = linspace(0, 0.25, 256) + linspace(0, 0.25, 256).’;

G    = E .* C + F .* D;

(e)   (Optional) Rather than putting each plot in a new figure, use subplots using subplot or tiledlayout. Label each plot!

(f)    (Optional) Implement a function, generate_circle(x,y,rad) that takes in the coordinates of the circle’s center and its radius, and generates a 256 × 256 logical matrix, and use it for parts (a) and (b). Alternatively, use it to generate arbitrary circles and visually demonstrate the inclusion-exclusion principle by xor-ing all the circles.

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2.    Calculus time!

(a)    Write a function deriv(y, x) and antideriv(y, x) which take vectors y and x, and which perform numerical differentiation and integration on the function y = y(x). These should each output vectors of the same length as the input; you may need to pad your result with an arbitrary value. You did this already; now make it a function.

(b)    Write a function switchsign(x) which takes a vector x and returns a logical array with the same length as x that is true when x switches sign. E.g., for x=[2 3 -1 0 -1 5] it would return [0 0 1 0 0 1]. Do not use a for loop.

(Hint: One way to vectorize this is to use the sign and diff functions. Another way is to write conditions on the vector and use a shifted version of itself. You will probably need to pad the resulting vector to make it the same length as the original vector; one way to do this is to repeat the first element of the resulting vector twice.)

(c)    Write a function extrema(y, x) that uses the first derivative test to find local extrema (minima and maxima). Recall that local extrema on a differentiable function occur when y0(x) changes sign. Use your deriv and switchsign functions here. The output of this function should also be two vectors: one representing the y values of the extrema, and one representing the corresponding x values.

(d)   Write a function inflections(y, x) that uses the second derivative test to find inflection points (i.e., when y00(x) changes sign).

(e)    Let x be a 1001-point vector linearly spaced from −2π to 2π. Use sinc to generate y = sinc(x) over this interval. Generate the following variables using the functions you just wrote:

i.         y_antideriv, the antiderivative of y

ii.       y_deriv, the derivative of y

iii.     y_extr and x_extr, the coordinates of the local extrema of y iv. y_infl and x_infl, the coordinates of the inflection points of y Plot these using:

plot(x,y,x,y_antideriv,x,y_deriv, ... x_extr,y_extr,’r*’,x_infl,y_infl,’bo’);

Title your plot and label the axes. It should look something like:

 

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